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Lesson 1: Basic Concepts and Activity 1 Objects Representing the Undefined
1. These are some of the objects around us that could represent a point or line or plane. Place each object to its corresponding column in the table below. Blackboard Sun Rays
Corner of a table intersection of the side wall and ceiling Electric wire Intersection of the front wall, side wall and ceiling
tip of a needle surface of a table
paper Tip of a ballpen
Wall
Objects that could represent a point
Edge of a table
Objects that could represent a line
Objects that could represent a plane
II. Questions to Ponder: 1. Consider the stars above. Do they represent points? 2. Consider the moon in its fullest form. Would you consider a full moon as a representation of a point? 3. A dot represents a point. How big are a dot that represents a point and a dot that represents a circular region? 4. A point has no dimension. A line has a dimension. How come that a line composed of dimensionless points has a dimension? 5. Does a pencil extend infinitely in both directions? Is a pencil a line? III. Exercises 1. List down 5 other objects that could represent a. a point. b. a line. c. a plane.
Geometry Activity Sheets by Villena-Diaz
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2. Use the figure below, identify what is being asked.
M A
C
B
D
E F
k
G
H p
I
J a. Name the point(s) in the interior region of the circle. b. Name the point(s) in the interior region of the triangle. c. Name the line(s) inside the interior region of the triangle. d. Give other name(s) for line p. e. In the interior of the circle, name the plane that can be formed by the three points not on the same line. f. Name the plane formed by line p and point I. g. Name the points outside the circular region. h. Name the points outside the region bounded by the triangle. i. Name the points of plane M. j. Give other names for plane M.
3. Recall: (a) Two points determine a line. (b) Three points not on the same line determine a plane. (c) Two intersecting lines determine a plane. (d) Two parallel lines determine a plane. Given: The points A, B, C, D, E, F, G, H are positioned this way, such that some points are not on the same plane. (The points are vertices of a cubical box.) A
B
D
C E
H
F G
Answer the following: 1. How many lines are possible which can be formed by these points? (Hint: There are more than 20. Refer to statement (a) above. __________ Geometry Activity Sheets by Villena-Diaz
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2. What are the lines that contain the point A? (Hint: There are more than 3 lines.) ___________________________ 3. Identify the different planes which can be formed by these points. (Hint: There are more than six. Refer to statement (d) above. _______________ 4. What are the planes that contain line DC? __________ 5. What are the planes that intersect at line BF? ________
Activity 2 Other Geometric on Lines Refer to the figure below: Given: The points A, B, C, D, E, F, G, H are positioned this way, such that some points are not on the same plane. (The points are vertices of a cubical box.) A
D
B
C
E H
F G
Intersecting Lines Lines DH and DC are intersecting at point D. Lines CG and GF are intersecting at point G. 1. What other lines intersect with line DH? ___________ 2. What other lines intersect with line CG? ___________ 3. What lines intersect with EF? ________________ Parallel Lines Lines AB and DC are parallel. Lines DH and CG are parallel. 4. What are other lines parallel to line AB? __________ 5. What are other lines parallel to line CG? __________ 6. What lines are parallel to line AD? _____________ Concurrent Lines Lines AD, AB, and AE are concurrent at point A. Lines GH, GF, and GC are concurrent at point G. 7. Name if possible, other lines that are concurrent at point A. ___________ 8. Name if possible, other lines that are concurrent at point G. ___________ 9. What lines are concurrent at point F? __________ Skew Lines Lines DH and EF are two lines which are neither intersecting nor parallel. These two lines do not determine a plane and are being called skew lines. Also, lines AE and GF are skew lines. The lines DH, CG, HE and GF are skew to AB.
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10. What other lines are skew to DH? _____________ 11. What other lines are skew to EF? ______________ 12. What lines are skew to BF? __________________ Activity 3 Definition of a Line Segment
A
B
C
D
AD is a line segment. The points A, B, C, D are on line segment AD. In notation, we write AD . We can also name it as DA .
E
F
G
H
I
J
FH is a segment. The points F, G, H are on line segment FH. The points E, I, J are not on line segment FH. In notation, we write FH . We can also name it as HF .
A B C D E
F G H
I
J
K L M N O P Q R S
T U V
The points E,F,G,J are on line segment EQ or segment QE. The points C,D,T,U are not on line segment EQ.
Answer the following: 1. Name other points which are on line segment EQ. ________________ 2. Name other points which are not on line segment EQ. ________________ Complete the following statements: 3. A line segment is part of a line that has __________. 4. The points on the line segment EQ are ____________________.
Activity 4 Congruent Segments Given the points on the number line: A 1
B 2
3
C D 4
5
6
E F G 7
8
9 10 11
1. Determine the following lengths: a) AB = _______ b) DE = _______ c) BD= _______ d) DF = _______ Geometry Activity Sheets by Villena-Diaz
e) AC= _________ f) DG = _________ g) BE = _________ h) CG = _________ Page 4
2. The following segments are congruent: AB and DE; BD and DF; AC and DG, BE and CG. 3. The following pairs of segments are not congruent: AB and CF; BD and AE; AC and BF; BG and AD. 4. Using the figure below, which segments are congruent? J
K L
M
N
-7 -6 -5 -4 -3 -2 -1
0
O
1
2
3
P Q 4
5
6
7
R 8
9 10
Define congruent segment. Congruent segments are segments __________________.
Activity 5 Definition of a Ray A
B
C
This is ray AB. We can also name it as ray AC. In symbol, we write AC . The points A, B, C are on ray AC. X
Y
Z
This is ray ZY. We can also name it as ray ZX. uuur suuu In symbol, we write ZX . We do NOT write it as XZ . The points X, Y, Z are on ray ZY.
D
E
F
G
This is ray DE. We can also name it as ray DF or ray DG. The points D, E, F, G are on ray DE. Q
R
S
T
This is ray TS. We can also name it as ray TR or ray TQ. The points Q, R, S, T are on ray TS.
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H
I
J
K
L
M
This is ray ML. 1. How else can you name this ray? _________ 2. Name some points on ray ML? ________________ N
O
P
Q
R
S
T
U
The points Q, R, S, T, U are on ray QR. The points N, O, P are not on ray QR. 3. How else can you name ray QR? _______________.
A
B
C
D
E
F
G
H
I
J
4. Name 7 points on ray DE? _______________ 5. Name two points which are not on ray DE? ____________ 6. How else can you name ray DE? _________________
M
N
O
P
Q
R
S
T
U
V
W
X
Y
7. Name 7 points on ray QT? 8. Name 7 points on ray PQ? 9. Name 7 points on ray XU? 10. Name 5 points on ray SP?
In general, how do you describe the points on any ray AC? _____________________________________
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Activity 6 The Union/Intersection of Segments and Rays
Use the figure below to determine the part of the line being described by the union or intersection of two segments, rays or segment and ray: A
B
C
D
E
F
Example: DE ∪ CF is the set of all points on the ray DE and segment CF. Thus, all these points determine ray CD . BC ∩ ED is the set of all points common to ray BC and ray ED. The common points are the points that determine segment BE.
Answer the following: 1) AB ∪ BE = _________ 2) DF ∪ BD = _________ 3) CB ∪ CE = _________ 4) DE ∪ BD = _________ 5) CA ∪ CD = _________ 6) BF ∩ AD = _________ 7) FD ∩ AB = _________ 8) FE ∩ CD = _________ 9) CA ∩ CE =_________ 10) BC ∩ CE =_________
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Lesson 2: Angles Activity 7 Definition of an Angle
The following are angles:
The following are not angles:
Which of these are angles?
An angle is ___________________________________.
I.
Questions to ponder: 1. Is this an angle?
2. Why is this figure, taken as a whole, not an angle?
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Activity 8 Measuring an Angle a) Construct angles with the following measures: 90o, 60o , 30o , 120o b) From the figure, determine the measure of each angle.
1) m∠EHC = __________
6) m∠CHB = __________ 11) m∠BHE = __________
2) m∠CHF = __________
7) m∠DHG = __________ 12) m∠CHI = __________
3) m∠IHA = __________
8) m∠FHI = __________
4) m∠BHD = __________
9) m∠EHF = __________ 14) m∠CHD = __________
5) m∠AHG = __________
10) m∠DHI = __________
13) m∠BHG = __________
15) m∠BHI = __________
Exercise 9. Estimating Angle Measures A. In the drawings below, some of the indicated measures of angles are correct and some are obviously wrong. Estimate the measure of each angle. State which are correct and which are wrong. You are not expected to measure the angles.
Question to ponder:
o
(3n -60)
If angle M is an acute angle, what are the possible values of n ?
M
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Exercise 10: Parts of an Angle
Y
X
Z V
W
Use the given figure to identify the following: 1) The sides of ∠YVW 2) The sides of ∠XVY 3) The angle(s) adjacent to ∠ZVW 4) The angle(s) adjacent to ∠ XVZ 5) The angle(s) adjacent to ∠YVZ 6) The side common to ∠ XVY and ∠YVZ 7) The side common to ∠XVZ and ∠ZVW 8) The side common to ∠XVZ and ∠ZVY 9) The side common to ∠XVY and ∠YVW 10) The common vertex. I.
____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________
Question to Ponder: Why are the angles ∠ XVY and ∠XVZ not considered to be adjacent angles?
Exercise 11: A. Determine the measures of the angles marked with letters. (Note: Figures are not drawn to scales.)
B. Determine whether the statement is true or false. If false, explain why. 9) 20o, 30o, 40o are complementary angles. 10) 100o, 50o, 30o are supplementary angles.
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Activity 12 Angles Formed when Two Parallel Lines are Cut by a Transversal Draw parallel lines g and h. Draw a transversal j so that it forms an 80o angle line with g as shown. Also, draw a transversal k so that it forms a 50o angle with line h as shown. Use your protractor to find the measures of the angles marked with letters.
j g
h
k
80O A D E
H K
A = _____ D = _____ I = _____ K = _____
B C F G
I
50O
J L
M
E = _____ H = _____ L = _____
N
B = _____ G = _____ J = _____ N = _____
C = _____ F = _____ M = _____
In both cases, compare the measures of the: a) corresponding angles b) alternate interior angles c) alternate exterior angles. What did you observe? ________________________ Complete the statements below: When two parallel lines are cut by a transversal, then a) The corresponding angles are __________________. b) The alternate interior angles are _______________. c) The alternate exterior angles are _____________.
I.
Questions to ponder:
Use the figure below to answer the following questions: 1. If lines x and y are parallel and z is a transversal, what can you say about the two angles a) which are both encircled? b) both of which are non-encircled? c) one encircled and one non-encircled?
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2.
If A = (20-2m)o and D = (m-130)o, what is the value of m? z A C E G
B D
x
F
y
H
Exercise 13. Find the angles marked with letters. Lines with arrowheads are parallel. (Note: Figures are not drawn to scales.) 1)
2) j
q
3)
n 75o
105o
p
5) 4) 125
112o
83
70o
6)
o
o
r
7)
t
s
65o
u
8) 109o w w
47o
92o
v
x w
10) 9)
130o b 33o x z
Geometry Activity Sheets by Villena-Diaz
a
c
y
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Lesson 3: Basic Construction
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II. Exercise 14. 1. 2. 3. 4. 5. 6.
Construct ∆ABC such that AB = 6 cm, BC = 8 cm and AC = 7 cm long. Bisect the side BC. Bisect the interior ∠B. Construct the altitude from vertex C. Construct a line through B which is parallel to side AC. Construct an equilateral triangle PQR so that PR and the altitude from vertex C have equal lengths.
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Lesson 4: Polygons . Activity 15 Definition of a Polygon The following are polygons:
The following are not polygons:
Which of these are polygons?
A polygon is ____________________.
Questions to ponder: 1. Can two segments form a polygon? If yes, draw the figure. If no, explain why. (State the condition it failed to satisfy.) 2. What is the minimum number of non-collinear segments needed to satisfy the definition of polygon above? 3. Why are the following figures not considered as polygons?
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Activity 16 Number of Vertices and Interior Angles of a Polygon
For classroom activity: Materials needed: match sticks, paste or glue, paper Consider each piece of matchstick as the side of a polygon. For our purpose, we may draw the following: (Recall: A polygon is ___________________________.) Procedure: 1) Using three pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ 2) Using four pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ 3) Using five pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ 4) Using six pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ Were you able to observe a pattern? ____________ Complete the sentence below: A polygon with n sides has ___ number of vertices and ______ number of interior angles.
Activity 17 Types of Polygon Recall: A polygon is ________________________________________. A polygon is equilateral is _____________________________. A polygon is equiangular if ____________________________. A polygon is regular if ________________________________.
1. Determine if a figure can be constructed using the given condition. If yes, sketch a figure. If no, explain why it cannot be constructed. a) b) c) d) e) f)
A triangle which is equilateral but not equiangular. A triangle which is equiangular but not equilateral A triangle which is regular A quadrilateral which is equilateral but not equiangular. A quadrilateral which is equiangular but not equilateral A quadrilateral which is regular.
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2. In general, a) b) c) d) e) f)
Do all equilateral polygons equiangular? If no, give a counter example. Do all equiangular polygons equilateral? If no, give a counter example. Do all regular polygons equilateral? If no, give a counter example. Do all regular polygons equiangular? If no, give a counter example. Do all equilateral triangles equiangular? Do all equiangular triangles equilateral?
Activity 18 Sum of the Interior Angles of a Polygon Materials needed: pencil, paper, protractor Procedures: 1) Draw a triangle. Using a protractor, determine the measure of its interior angles and determine the sum of the interior angles. 2) Draw a quadrilateral. Then fix a vertex and draw diagonals from this vertex. Then answer the following: a) How many diagonals are drawn from the fixed vertex? b) How many triangles are formed by this/these diagonal(s)? c) Without actually measuring, can you determine the sum of the interior angles of a quadrilateral? 3) Draw a pentagon. Then fix a vertex and draw diagonals from this vertex. Then answer the following: a) How many diagonals are drawn from the fixed vertex? b) How many triangles are formed by this/these diagonal(s)? c) Without actually measuring, can you determine the sum of the interior angles of a pentagon? 4) Continue this with a hexagon and heptagon. 5) Search for a pattern and complete the table below:
No. of sides
No. of diagonals from a fixed vertex
No. of triangles formed by the diagonals drawn from a fixed vertex
Sum of the interior angles
3 4 5 6 7 8 9 10 n
6. Complete this: The sum of the interior angles of a polygon with n sides is ______.
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Activity 19 The Sum of the Exterior Angles of Polygon 1) Given ∆ABC with the exterior angle on each vertex as shown: D d
A a
e E
c
b
C
B
f F
Let the interior angles at A, B, C measure a, b, c respectively while the exterior angles measure d, e, f. Determine the following sum of angles: a + d = _________ (Recall the sum of angles that form a linear pair.) b + e = _________ c + f = _________ (a + d) + (b + e) + (c + f) = _________ (a + b+ c) + ( d + e + f) = _________ a + b + c = _________ (Recall the sum of the interior angles of a triangle.) d + e + f = _________
2. Given the
ABCD and the exterior angle at each vertex as shown: e a h
d
b
f c g
Determine the following sum: a + e = _________ b + f = _________ c + g = _________ d+h (a + e) + (b + f) + (c + g) + (d + h) = _________ (a + b+ c + d) + ( e + f + g + h) = _________ a + b + c + d = _________ (Recall the sum of the interior angles of a quadrilateral.) e + f + g + h = _________ The sum of the exterior angles of a quadrilateral is ______________.
3. Do the same thing with pentagon, hexagon and heptagon. Then complete the following: The sum of the exterior angles of a pentagon is ___________. The sum of the exterior angles of a hexagon is ___________. The sum of the exterior angles of a heptagon is ___________. 4. What conclusion can you formulate about the sum of the exterior angles of a polygon? ________________________________. Geometry Activity Sheets by Villena-Diaz
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I.
Exercise 20
1. For each regular polygon, determine the measure of an exterior angle. a. quadrilateral b. hexagon c. nonagon 2. Determine the sum of the interior angles of the following polygons: a. Pentagon b. heptagon c. octagon 3. Each exterior angle of a regular polygon measures 20o. Determine the sum of its interior angles.
Lesson 5: Triangles Exercise 21. Parts of a Triangle Given ∆ABE with AC ⊥BE and BD = DE. H A
B B
C
D
E
I
G Identify the following, if possible: 1) 2) 3) 4) 5) 6) 7) 8)
vertices sides interior angles exterior angles the remote interior angles of ∠ AEF the remote interior angles of ∠EBG altitude median
______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________
Activity 22. Lengths of Sides of a Triangle For classroom activity: Materials Needed: plastic straws, scissors, ruler For our purpose, we may use our com and draw the segments to form a triangle. Procedure: 1. Cut pieces of straws with the indicated measures. There are three pieces in each set.
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2. With each set of straws, try to form a triangle. Complete the table below:
Lengths of straws (in cm) 3, 3, 7 3, 3, 5 4, 6, 10 4, 6, 9 5, 5, 10 5, 5, 8 6, 7, 11 6, 7, 9 4, 7, 12 4, 7, 10
Do they form a triangle or not?
3. For each set of lengths, add the two shortest lengths. Then, compare the sum with the longest length. What pattern did you observe? ________________________________________
Exercise 24 A. True or False 1. A triangle can have exactly one acute angle. 2. A triangle can have two right interior angles. 3. A triangle can have two obtuse interior angles. 4. A right triangle can be an isosceles triangle. 5. An isosceles triangle can have an obtuse interior angle. 6. An acute triangle can be an isosceles triangle. 7. An obtuse triangle can be a scalene triangle. 8. An acute triangle can be an isosceles triangle. 9. A right triangle can be an equilateral triangle. 10. An obtuse triangle can be an isosceles triangle. B. Determine the measure of the angles marked with letters. Lines with arrowheads are parallel. (Note: Figures are not drawn to scales.) a
1)
70o
d
2)
80o
85o
3) b
60o
b
c
65o
e 4)
5)
109o
f g
Geometry Activity Sheets by Villena-Diaz
x w
g w
i w
j w
20o
h w
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C. Construct the following: 7. Construct a triangle whose sides are 5 cm, 8 cm and 10 cm long. 8. Construct ∆PQR such that PQ = 5 cm, QR = 8 cm, m∠Q = 60o. 9. Construct ∆WXY such that WX = 8 cm, m∠Q = 40o, m∠Q = 80o. D. 1. 2. 3. 4.
Construct 4 different scalene triangles. In the first triangle, construct all the perpendicular bisectors of the sides. In the second triangle, construct all the angle bisectors. In the third triangle, construct all the altitudes. In the fourth triangle, construct a line ing through a vertex and parallel to the opposite side of the chosen vertex.
III. Question to ponder: Try to construct a triangle whose sides are 4 cm, 6 cm and 11 cm. What did you observe? Could explain why?
Lesson 6: Quadrilaterals Exercise 25. Angles in Quadrilateral Find the angles marked with letters. (Note: Figures are not drawn to scales.) 138° 82 °
j
78°
1)
k
6) l
A
m
84°
110° 53° n
2) 7)
54° o
130°
93° q b
p
68°
r
3)
8)
c
120°
32° 21°
s t 84°
.
4) Geometry Activity Sheets by Villena-Diaz
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9) f u
v
88°
68°
62°
53°
e
5) 68° A. On d Parallelograms
10)
g w i
119° 20°
h 62°
35°
x
y
Activity 27 Properties of a Parallelogram
Materials: Pair of scissors, ruler, cardboards or papers Procedures: A. Prepare five models of parallelograms. (Or use the attached sketch of parallelograms.) Name the parallelogram as ABCD.
1.
For the first parallelogram: cut the parallelogram into two so that you can compare ∠A and ∠C; ∠B and ∠D. What do you observe?
Opposite angles of a parallelogram are _______________________. 2.
For the second parallelogram: cut the angles and arrange any two consecutive angles about a point. What do you observe about the sum of any two consecutive angles of a parallelogram?
Consecutive angles of a parallelogram are ____________________. 3.
For the third parallelogram: cut the figure along the diagonal AC. Compare the two triangles formed. Can they be coincided with each other? For the fourth parallelogram: cut the figure along the diagonal BD. Compare the two triangles formed. Can they be coincided with each other?
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In both parallelograms, what do you observe about the triangles formed by the diagonals? Diagonals of a parallelogram divide the parallelogram into ___________. 4.
For the fifth parallelogram: cut the figure along the two diagonals. Then compare the partitioned diagonals. (You may use the com to compare the lengths.)How did one diagonal divide the other diagonal? Diagonals of a parallelogram _________________________.
Activity 28 Special Properties of Rectangle, Square, Rhombus
Materials: two sets of models of rectangles, squares, rhombus. Name each as ABCD. (see attached sheet for the models) Procedure: 1.) Cut the rectangle along the diagonal AC. What type/s of triangle(s) is/are formed? 2.) Cut the rhombus along the diagonal AC. What type/s of triangle(s) is/are formed? 3.) Cut the square along the diagonal AC. What type/s of triangle(s) is/are formed? In which parallelogram does the diagonal divide the parallelogram into two congruent right triangles? ____________________________ 4.) In each figure, draw diagonals AC and BD and let the intersection be point O. In each figure, measure the lengths of the diagonals. (You may use a com to compare the lengths.) In which parallelogram are the diagonals congruent? ______________ 5.) In each figure, draw diagonals AC and BD and let the intersection be point O. Then measure ∠AOD, ∠DOC, ∠COB, ∠BOA. What do you observe? In which parallelogram are the diagonals perpendicular? _________________ 6.) From the results of # 4-5, complete the statements below: Diagonals of a rhombus are _____________________. Diagonals of a rectangle are _____________________. Diagonals of a square are _______________________.
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Models for Activity 27
B
A
C
D
B
A
A
C
D
B
A
Geometry Activity Sheets by Villena-Diaz
B
C
B
C
D
B
A
C
D
C
D
A
D
A
B
C
D
B
A
C
D
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Models for Activity 28 A
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
D
B
C
A
D
A
B
D
B
C
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C
A
D
B
C
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1. These are some of the objects around us that could represent a point or line or plane. Place each object to its corresponding column in the table below. Blackboard Sun Rays
Corner of a table intersection of the side wall and ceiling Electric wire Intersection of the front wall, side wall and ceiling
tip of a needle surface of a table
paper Tip of a ballpen
Wall
Objects that could represent a point
Edge of a table
Objects that could represent a line
Objects that could represent a plane
II. Questions to Ponder: 1. Consider the stars above. Do they represent points? 2. Consider the moon in its fullest form. Would you consider a full moon as a representation of a point? 3. A dot represents a point. How big are a dot that represents a point and a dot that represents a circular region? 4. A point has no dimension. A line has a dimension. How come that a line composed of dimensionless points has a dimension? 5. Does a pencil extend infinitely in both directions? Is a pencil a line? III. Exercises 1. List down 5 other objects that could represent a. a point. b. a line. c. a plane.
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2. Use the figure below, identify what is being asked.
M A
C
B
D
E F
k
G
H p
I
J a. Name the point(s) in the interior region of the circle. b. Name the point(s) in the interior region of the triangle. c. Name the line(s) inside the interior region of the triangle. d. Give other name(s) for line p. e. In the interior of the circle, name the plane that can be formed by the three points not on the same line. f. Name the plane formed by line p and point I. g. Name the points outside the circular region. h. Name the points outside the region bounded by the triangle. i. Name the points of plane M. j. Give other names for plane M.
3. Recall: (a) Two points determine a line. (b) Three points not on the same line determine a plane. (c) Two intersecting lines determine a plane. (d) Two parallel lines determine a plane. Given: The points A, B, C, D, E, F, G, H are positioned this way, such that some points are not on the same plane. (The points are vertices of a cubical box.) A
B
D
C E
H
F G
Answer the following: 1. How many lines are possible which can be formed by these points? (Hint: There are more than 20. Refer to statement (a) above. __________ Geometry Activity Sheets by Villena-Diaz
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2. What are the lines that contain the point A? (Hint: There are more than 3 lines.) ___________________________ 3. Identify the different planes which can be formed by these points. (Hint: There are more than six. Refer to statement (d) above. _______________ 4. What are the planes that contain line DC? __________ 5. What are the planes that intersect at line BF? ________
Activity 2 Other Geometric on Lines Refer to the figure below: Given: The points A, B, C, D, E, F, G, H are positioned this way, such that some points are not on the same plane. (The points are vertices of a cubical box.) A
D
B
C
E H
F G
Intersecting Lines Lines DH and DC are intersecting at point D. Lines CG and GF are intersecting at point G. 1. What other lines intersect with line DH? ___________ 2. What other lines intersect with line CG? ___________ 3. What lines intersect with EF? ________________ Parallel Lines Lines AB and DC are parallel. Lines DH and CG are parallel. 4. What are other lines parallel to line AB? __________ 5. What are other lines parallel to line CG? __________ 6. What lines are parallel to line AD? _____________ Concurrent Lines Lines AD, AB, and AE are concurrent at point A. Lines GH, GF, and GC are concurrent at point G. 7. Name if possible, other lines that are concurrent at point A. ___________ 8. Name if possible, other lines that are concurrent at point G. ___________ 9. What lines are concurrent at point F? __________ Skew Lines Lines DH and EF are two lines which are neither intersecting nor parallel. These two lines do not determine a plane and are being called skew lines. Also, lines AE and GF are skew lines. The lines DH, CG, HE and GF are skew to AB.
Geometry Activity Sheets by Villena-Diaz
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10. What other lines are skew to DH? _____________ 11. What other lines are skew to EF? ______________ 12. What lines are skew to BF? __________________ Activity 3 Definition of a Line Segment
A
B
C
D
AD is a line segment. The points A, B, C, D are on line segment AD. In notation, we write AD . We can also name it as DA .
E
F
G
H
I
J
FH is a segment. The points F, G, H are on line segment FH. The points E, I, J are not on line segment FH. In notation, we write FH . We can also name it as HF .
A B C D E
F G H
I
J
K L M N O P Q R S
T U V
The points E,F,G,J are on line segment EQ or segment QE. The points C,D,T,U are not on line segment EQ.
Answer the following: 1. Name other points which are on line segment EQ. ________________ 2. Name other points which are not on line segment EQ. ________________ Complete the following statements: 3. A line segment is part of a line that has __________. 4. The points on the line segment EQ are ____________________.
Activity 4 Congruent Segments Given the points on the number line: A 1
B 2
3
C D 4
5
6
E F G 7
8
9 10 11
1. Determine the following lengths: a) AB = _______ b) DE = _______ c) BD= _______ d) DF = _______ Geometry Activity Sheets by Villena-Diaz
e) AC= _________ f) DG = _________ g) BE = _________ h) CG = _________ Page 4
2. The following segments are congruent: AB and DE; BD and DF; AC and DG, BE and CG. 3. The following pairs of segments are not congruent: AB and CF; BD and AE; AC and BF; BG and AD. 4. Using the figure below, which segments are congruent? J
K L
M
N
-7 -6 -5 -4 -3 -2 -1
0
O
1
2
3
P Q 4
5
6
7
R 8
9 10
Define congruent segment. Congruent segments are segments __________________.
Activity 5 Definition of a Ray A
B
C
This is ray AB. We can also name it as ray AC. In symbol, we write AC . The points A, B, C are on ray AC. X
Y
Z
This is ray ZY. We can also name it as ray ZX. uuur suuu In symbol, we write ZX . We do NOT write it as XZ . The points X, Y, Z are on ray ZY.
D
E
F
G
This is ray DE. We can also name it as ray DF or ray DG. The points D, E, F, G are on ray DE. Q
R
S
T
This is ray TS. We can also name it as ray TR or ray TQ. The points Q, R, S, T are on ray TS.
Geometry Activity Sheets by Villena-Diaz
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H
I
J
K
L
M
This is ray ML. 1. How else can you name this ray? _________ 2. Name some points on ray ML? ________________ N
O
P
Q
R
S
T
U
The points Q, R, S, T, U are on ray QR. The points N, O, P are not on ray QR. 3. How else can you name ray QR? _______________.
A
B
C
D
E
F
G
H
I
J
4. Name 7 points on ray DE? _______________ 5. Name two points which are not on ray DE? ____________ 6. How else can you name ray DE? _________________
M
N
O
P
Q
R
S
T
U
V
W
X
Y
7. Name 7 points on ray QT? 8. Name 7 points on ray PQ? 9. Name 7 points on ray XU? 10. Name 5 points on ray SP?
In general, how do you describe the points on any ray AC? _____________________________________
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Activity 6 The Union/Intersection of Segments and Rays
Use the figure below to determine the part of the line being described by the union or intersection of two segments, rays or segment and ray: A
B
C
D
E
F
Example: DE ∪ CF is the set of all points on the ray DE and segment CF. Thus, all these points determine ray CD . BC ∩ ED is the set of all points common to ray BC and ray ED. The common points are the points that determine segment BE.
Answer the following: 1) AB ∪ BE = _________ 2) DF ∪ BD = _________ 3) CB ∪ CE = _________ 4) DE ∪ BD = _________ 5) CA ∪ CD = _________ 6) BF ∩ AD = _________ 7) FD ∩ AB = _________ 8) FE ∩ CD = _________ 9) CA ∩ CE =_________ 10) BC ∩ CE =_________
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Lesson 2: Angles Activity 7 Definition of an Angle
The following are angles:
The following are not angles:
Which of these are angles?
An angle is ___________________________________.
I.
Questions to ponder: 1. Is this an angle?
2. Why is this figure, taken as a whole, not an angle?
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Activity 8 Measuring an Angle a) Construct angles with the following measures: 90o, 60o , 30o , 120o b) From the figure, determine the measure of each angle.
1) m∠EHC = __________
6) m∠CHB = __________ 11) m∠BHE = __________
2) m∠CHF = __________
7) m∠DHG = __________ 12) m∠CHI = __________
3) m∠IHA = __________
8) m∠FHI = __________
4) m∠BHD = __________
9) m∠EHF = __________ 14) m∠CHD = __________
5) m∠AHG = __________
10) m∠DHI = __________
13) m∠BHG = __________
15) m∠BHI = __________
Exercise 9. Estimating Angle Measures A. In the drawings below, some of the indicated measures of angles are correct and some are obviously wrong. Estimate the measure of each angle. State which are correct and which are wrong. You are not expected to measure the angles.
Question to ponder:
o
(3n -60)
If angle M is an acute angle, what are the possible values of n ?
M
Geometry Activity Sheets by Villena-Diaz
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Exercise 10: Parts of an Angle
Y
X
Z V
W
Use the given figure to identify the following: 1) The sides of ∠YVW 2) The sides of ∠XVY 3) The angle(s) adjacent to ∠ZVW 4) The angle(s) adjacent to ∠ XVZ 5) The angle(s) adjacent to ∠YVZ 6) The side common to ∠ XVY and ∠YVZ 7) The side common to ∠XVZ and ∠ZVW 8) The side common to ∠XVZ and ∠ZVY 9) The side common to ∠XVY and ∠YVW 10) The common vertex. I.
____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________
Question to Ponder: Why are the angles ∠ XVY and ∠XVZ not considered to be adjacent angles?
Exercise 11: A. Determine the measures of the angles marked with letters. (Note: Figures are not drawn to scales.)
B. Determine whether the statement is true or false. If false, explain why. 9) 20o, 30o, 40o are complementary angles. 10) 100o, 50o, 30o are supplementary angles.
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Activity 12 Angles Formed when Two Parallel Lines are Cut by a Transversal Draw parallel lines g and h. Draw a transversal j so that it forms an 80o angle line with g as shown. Also, draw a transversal k so that it forms a 50o angle with line h as shown. Use your protractor to find the measures of the angles marked with letters.
j g
h
k
80O A D E
H K
A = _____ D = _____ I = _____ K = _____
B C F G
I
50O
J L
M
E = _____ H = _____ L = _____
N
B = _____ G = _____ J = _____ N = _____
C = _____ F = _____ M = _____
In both cases, compare the measures of the: a) corresponding angles b) alternate interior angles c) alternate exterior angles. What did you observe? ________________________ Complete the statements below: When two parallel lines are cut by a transversal, then a) The corresponding angles are __________________. b) The alternate interior angles are _______________. c) The alternate exterior angles are _____________.
I.
Questions to ponder:
Use the figure below to answer the following questions: 1. If lines x and y are parallel and z is a transversal, what can you say about the two angles a) which are both encircled? b) both of which are non-encircled? c) one encircled and one non-encircled?
Geometry Activity Sheets by Villena-Diaz
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2.
If A = (20-2m)o and D = (m-130)o, what is the value of m? z A C E G
B D
x
F
y
H
Exercise 13. Find the angles marked with letters. Lines with arrowheads are parallel. (Note: Figures are not drawn to scales.) 1)
2) j
q
3)
n 75o
105o
p
5) 4) 125
112o
83
70o
6)
o
o
r
7)
t
s
65o
u
8) 109o w w
47o
92o
v
x w
10) 9)
130o b 33o x z
Geometry Activity Sheets by Villena-Diaz
a
c
y
Page 12
Lesson 3: Basic Construction
Geometry Activity Sheets by Villena-Diaz
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Geometry Activity Sheets by Villena-Diaz
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Geometry Activity Sheets by Villena-Diaz
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II. Exercise 14. 1. 2. 3. 4. 5. 6.
Construct ∆ABC such that AB = 6 cm, BC = 8 cm and AC = 7 cm long. Bisect the side BC. Bisect the interior ∠B. Construct the altitude from vertex C. Construct a line through B which is parallel to side AC. Construct an equilateral triangle PQR so that PR and the altitude from vertex C have equal lengths.
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Lesson 4: Polygons . Activity 15 Definition of a Polygon The following are polygons:
The following are not polygons:
Which of these are polygons?
A polygon is ____________________.
Questions to ponder: 1. Can two segments form a polygon? If yes, draw the figure. If no, explain why. (State the condition it failed to satisfy.) 2. What is the minimum number of non-collinear segments needed to satisfy the definition of polygon above? 3. Why are the following figures not considered as polygons?
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Activity 16 Number of Vertices and Interior Angles of a Polygon
For classroom activity: Materials needed: match sticks, paste or glue, paper Consider each piece of matchstick as the side of a polygon. For our purpose, we may draw the following: (Recall: A polygon is ___________________________.) Procedure: 1) Using three pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ 2) Using four pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ 3) Using five pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ 4) Using six pieces of match sticks form a polygon. Paste it on a piece of paper. a) How many sides does it have? _________ b) How many vertices does it have? _______ c) How many interior angles does it have? _______ Were you able to observe a pattern? ____________ Complete the sentence below: A polygon with n sides has ___ number of vertices and ______ number of interior angles.
Activity 17 Types of Polygon Recall: A polygon is ________________________________________. A polygon is equilateral is _____________________________. A polygon is equiangular if ____________________________. A polygon is regular if ________________________________.
1. Determine if a figure can be constructed using the given condition. If yes, sketch a figure. If no, explain why it cannot be constructed. a) b) c) d) e) f)
A triangle which is equilateral but not equiangular. A triangle which is equiangular but not equilateral A triangle which is regular A quadrilateral which is equilateral but not equiangular. A quadrilateral which is equiangular but not equilateral A quadrilateral which is regular.
Geometry Activity Sheets by Villena-Diaz
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2. In general, a) b) c) d) e) f)
Do all equilateral polygons equiangular? If no, give a counter example. Do all equiangular polygons equilateral? If no, give a counter example. Do all regular polygons equilateral? If no, give a counter example. Do all regular polygons equiangular? If no, give a counter example. Do all equilateral triangles equiangular? Do all equiangular triangles equilateral?
Activity 18 Sum of the Interior Angles of a Polygon Materials needed: pencil, paper, protractor Procedures: 1) Draw a triangle. Using a protractor, determine the measure of its interior angles and determine the sum of the interior angles. 2) Draw a quadrilateral. Then fix a vertex and draw diagonals from this vertex. Then answer the following: a) How many diagonals are drawn from the fixed vertex? b) How many triangles are formed by this/these diagonal(s)? c) Without actually measuring, can you determine the sum of the interior angles of a quadrilateral? 3) Draw a pentagon. Then fix a vertex and draw diagonals from this vertex. Then answer the following: a) How many diagonals are drawn from the fixed vertex? b) How many triangles are formed by this/these diagonal(s)? c) Without actually measuring, can you determine the sum of the interior angles of a pentagon? 4) Continue this with a hexagon and heptagon. 5) Search for a pattern and complete the table below:
No. of sides
No. of diagonals from a fixed vertex
No. of triangles formed by the diagonals drawn from a fixed vertex
Sum of the interior angles
3 4 5 6 7 8 9 10 n
6. Complete this: The sum of the interior angles of a polygon with n sides is ______.
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Activity 19 The Sum of the Exterior Angles of Polygon 1) Given ∆ABC with the exterior angle on each vertex as shown: D d
A a
e E
c
b
C
B
f F
Let the interior angles at A, B, C measure a, b, c respectively while the exterior angles measure d, e, f. Determine the following sum of angles: a + d = _________ (Recall the sum of angles that form a linear pair.) b + e = _________ c + f = _________ (a + d) + (b + e) + (c + f) = _________ (a + b+ c) + ( d + e + f) = _________ a + b + c = _________ (Recall the sum of the interior angles of a triangle.) d + e + f = _________
2. Given the
ABCD and the exterior angle at each vertex as shown: e a h
d
b
f c g
Determine the following sum: a + e = _________ b + f = _________ c + g = _________ d+h (a + e) + (b + f) + (c + g) + (d + h) = _________ (a + b+ c + d) + ( e + f + g + h) = _________ a + b + c + d = _________ (Recall the sum of the interior angles of a quadrilateral.) e + f + g + h = _________ The sum of the exterior angles of a quadrilateral is ______________.
3. Do the same thing with pentagon, hexagon and heptagon. Then complete the following: The sum of the exterior angles of a pentagon is ___________. The sum of the exterior angles of a hexagon is ___________. The sum of the exterior angles of a heptagon is ___________. 4. What conclusion can you formulate about the sum of the exterior angles of a polygon? ________________________________. Geometry Activity Sheets by Villena-Diaz
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I.
Exercise 20
1. For each regular polygon, determine the measure of an exterior angle. a. quadrilateral b. hexagon c. nonagon 2. Determine the sum of the interior angles of the following polygons: a. Pentagon b. heptagon c. octagon 3. Each exterior angle of a regular polygon measures 20o. Determine the sum of its interior angles.
Lesson 5: Triangles Exercise 21. Parts of a Triangle Given ∆ABE with AC ⊥BE and BD = DE. H A
B B
C
D
E
I
G Identify the following, if possible: 1) 2) 3) 4) 5) 6) 7) 8)
vertices sides interior angles exterior angles the remote interior angles of ∠ AEF the remote interior angles of ∠EBG altitude median
______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________
Activity 22. Lengths of Sides of a Triangle For classroom activity: Materials Needed: plastic straws, scissors, ruler For our purpose, we may use our com and draw the segments to form a triangle. Procedure: 1. Cut pieces of straws with the indicated measures. There are three pieces in each set.
Geometry Activity Sheets by Villena-Diaz
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2. With each set of straws, try to form a triangle. Complete the table below:
Lengths of straws (in cm) 3, 3, 7 3, 3, 5 4, 6, 10 4, 6, 9 5, 5, 10 5, 5, 8 6, 7, 11 6, 7, 9 4, 7, 12 4, 7, 10
Do they form a triangle or not?
3. For each set of lengths, add the two shortest lengths. Then, compare the sum with the longest length. What pattern did you observe? ________________________________________
Exercise 24 A. True or False 1. A triangle can have exactly one acute angle. 2. A triangle can have two right interior angles. 3. A triangle can have two obtuse interior angles. 4. A right triangle can be an isosceles triangle. 5. An isosceles triangle can have an obtuse interior angle. 6. An acute triangle can be an isosceles triangle. 7. An obtuse triangle can be a scalene triangle. 8. An acute triangle can be an isosceles triangle. 9. A right triangle can be an equilateral triangle. 10. An obtuse triangle can be an isosceles triangle. B. Determine the measure of the angles marked with letters. Lines with arrowheads are parallel. (Note: Figures are not drawn to scales.) a
1)
70o
d
2)
80o
85o
3) b
60o
b
c
65o
e 4)
5)
109o
f g
Geometry Activity Sheets by Villena-Diaz
x w
g w
i w
j w
20o
h w
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C. Construct the following: 7. Construct a triangle whose sides are 5 cm, 8 cm and 10 cm long. 8. Construct ∆PQR such that PQ = 5 cm, QR = 8 cm, m∠Q = 60o. 9. Construct ∆WXY such that WX = 8 cm, m∠Q = 40o, m∠Q = 80o. D. 1. 2. 3. 4.
Construct 4 different scalene triangles. In the first triangle, construct all the perpendicular bisectors of the sides. In the second triangle, construct all the angle bisectors. In the third triangle, construct all the altitudes. In the fourth triangle, construct a line ing through a vertex and parallel to the opposite side of the chosen vertex.
III. Question to ponder: Try to construct a triangle whose sides are 4 cm, 6 cm and 11 cm. What did you observe? Could explain why?
Lesson 6: Quadrilaterals Exercise 25. Angles in Quadrilateral Find the angles marked with letters. (Note: Figures are not drawn to scales.) 138° 82 °
j
78°
1)
k
6) l
A
m
84°
110° 53° n
2) 7)
54° o
130°
93° q b
p
68°
r
3)
8)
c
120°
32° 21°
s t 84°
.
4) Geometry Activity Sheets by Villena-Diaz
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9) f u
v
88°
68°
62°
53°
e
5) 68° A. On d Parallelograms
10)
g w i
119° 20°
h 62°
35°
x
y
Activity 27 Properties of a Parallelogram
Materials: Pair of scissors, ruler, cardboards or papers Procedures: A. Prepare five models of parallelograms. (Or use the attached sketch of parallelograms.) Name the parallelogram as ABCD.
1.
For the first parallelogram: cut the parallelogram into two so that you can compare ∠A and ∠C; ∠B and ∠D. What do you observe?
Opposite angles of a parallelogram are _______________________. 2.
For the second parallelogram: cut the angles and arrange any two consecutive angles about a point. What do you observe about the sum of any two consecutive angles of a parallelogram?
Consecutive angles of a parallelogram are ____________________. 3.
For the third parallelogram: cut the figure along the diagonal AC. Compare the two triangles formed. Can they be coincided with each other? For the fourth parallelogram: cut the figure along the diagonal BD. Compare the two triangles formed. Can they be coincided with each other?
Geometry Activity Sheets by Villena-Diaz
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In both parallelograms, what do you observe about the triangles formed by the diagonals? Diagonals of a parallelogram divide the parallelogram into ___________. 4.
For the fifth parallelogram: cut the figure along the two diagonals. Then compare the partitioned diagonals. (You may use the com to compare the lengths.)How did one diagonal divide the other diagonal? Diagonals of a parallelogram _________________________.
Activity 28 Special Properties of Rectangle, Square, Rhombus
Materials: two sets of models of rectangles, squares, rhombus. Name each as ABCD. (see attached sheet for the models) Procedure: 1.) Cut the rectangle along the diagonal AC. What type/s of triangle(s) is/are formed? 2.) Cut the rhombus along the diagonal AC. What type/s of triangle(s) is/are formed? 3.) Cut the square along the diagonal AC. What type/s of triangle(s) is/are formed? In which parallelogram does the diagonal divide the parallelogram into two congruent right triangles? ____________________________ 4.) In each figure, draw diagonals AC and BD and let the intersection be point O. In each figure, measure the lengths of the diagonals. (You may use a com to compare the lengths.) In which parallelogram are the diagonals congruent? ______________ 5.) In each figure, draw diagonals AC and BD and let the intersection be point O. Then measure ∠AOD, ∠DOC, ∠COB, ∠BOA. What do you observe? In which parallelogram are the diagonals perpendicular? _________________ 6.) From the results of # 4-5, complete the statements below: Diagonals of a rhombus are _____________________. Diagonals of a rectangle are _____________________. Diagonals of a square are _______________________.
Geometry Activity Sheets by Villena-Diaz
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Models for Activity 27
B
A
C
D
B
A
A
C
D
B
A
Geometry Activity Sheets by Villena-Diaz
B
C
B
C
D
B
A
C
D
C
D
A
D
A
B
C
D
B
A
C
D
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Models for Activity 28 A
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
D
B
C
A
D
A
B
D
B
C
Geometry Activity Sheets by Villena-Diaz
C
A
D
B
C
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