Division of Pasig City and San Juan

SAGAD HIGH SCHOOL

E. Angeles St., Sagad Pasig City

STRATEGIC INTERVENTION MATERIALS

IN

MATHEMATICS III

NENET M. PENARANDA

Head Teacher III - Math

LAZARO P. TORRES

Principal

LIGAYA G. INSIGNE

Education Supervisor I – Secondary Math

Prepared and Submitted by:

FEMILA R. PALICAN

EDWIN P. PECO

HELEN U. RAYMUNDO

Journey to the World of…

What have you observed on the figure

on the board ?

What kind of angles

are formed ?

The

figure

formed

a pair

of

angles

Vertical

Angles are

formed by

intersecting

lines

EXPLORE:

Prepare a whole sheet of paper. You can

make a model of different pairs of angle by

folding a rectangular or square paper, by

folding it lengthwise, cross-wise and

diagonally. The folded corner provides a

convenient model of pair of angles.

From the model figure using a piece

of paper, the broken lines are the folded

crease on the paper. The pairs of angles are

a. supplementary angles

b. complementary angles

c. adjacent angles

d. linear pair

e. vertical angles

In geometry, pairs of angles are related to each other in several ways. Some

examples are complementary angles, supplementary angles, vertical angles, adjacent

angles and linear pair.

Two angles are called complementary angles if the sum of their degree measurements equals 90

degrees. One of the complementary angles is said to be the complement of the other.

∠ABC is the complement of ∠CBD

Two angles are called supplementary angles if the sum of their degree measurements equals 180

degrees. One of the supplementary angles is said to be the supplement of the other.

∠ABC is the supplement of ∠CBD

Two pairs of angles are formed by two intersecting lines. Vertical angles are opposite angles in such an

intersection. Vertical angles are equal to each other.

Very often, math questions will require you to work out the values of angles given in diagrams by

applying the relationships between the pairs of angles.

Example 1: Given the diagram below, determine the values of the angles x, y and z.

A B

O

C D

Solution:

Step 1: x is a supplement of 65°.

Therefore, x + 65° =180° ⇒ x = 180° – 65° = 115°

Step 2: z and 115° are vertical angles.

Therefore, z = 115°

Step 3: y and 65° are vertical angles.

Therefore, y = 65°

Answer: x = 115°, y = 65° and z = 115°

Two angles which are adjacent and whose non common side are opposite rays and formed a

straight line are called linear pair.

D

∠DEF and ∠DEG are linear pair

25º 155º

F E G

Two coplanar angles with a common side and a common vertex but, no common interior points.

∠HIJ and ∠JIK are adjacent angles. H

j

I K

1. Theorem 1 - 1 VERTICAL ANGLES – vertical angles are congruent.

∠1 and ∠ 2 are vertical angles, show that ∠1 ∠ 2.

* Angle Addition Postulate *

1 m∠1 + m∠3 = 180, m∠2 + m∠3 = 180

3 4 by substitution, m∠1 + m∠3 = m∠2 + m∠3

2 Subtract m∠3 from each side you will get

m∠1 = m∠2, m∠1 m∠2 or ∠ 1 ∠ 2.

2. Theorem 1 - 2 If two angles are supplements of congruent angles, then the angles

are congruent.

1 2 3

∠1 and ∠2 are supplementary

∠3 and ∠2 are supplementary

show ∠2 ∠3

∠1 and ∠2 are supplementary, m∠1 + m∠2 = 180

∠3 and ∠2 are supplementary, m∠3 + m∠2 = 180

So m∠1 + m∠2 = m∠3 + m∠3

m∠1 = m∠3, m∠1 m∠3, ∠1 ∠3

3. Theorem 1 – 3 If two angles are complements of congruent angles, then the two

angles are congruent.

2

1 3

∠1 and ∠2 are complementary

∠3 and ∠2 are complementary

show ∠2 ∠3

∠1 and ∠2 are complementary, m∠1 + m∠2 = 90

∠3 and ∠2 are complenmentary, m∠3 + m∠2 = 90

So m∠1 + m∠2 = m∠3 + m∠3

m∠1 = m∠3, m∠1 m∠3, ∠1 ∠3

1. 2.

1 2

1 2

3. 30º 4.

150º 50º

40º

Activity 1

Identify the following pair of

angles

b. Activity # 2

Find the measure of the following angles:

1. m1 = _______

m2 = _______

2

1 30º

2. xº = ___________

4xº = __________

5xº = __________

5x 4x

x

3.

x = ___________

1 m 1 = _________

(x + 10) 3 4 (4x – 35) m 2 = _________

2

m 3 = _________

m 4 = _________

4.

x = ____________

m 1 = __________

(12x – 7) 1 2 (3x - 8) m 2 = __________

5.

(4x + 5)

1

2 (3x + 8)

m 1 = ____________

m 2 = ____________

c. Activity # 3

SOLVE:

1. The measure of a supplement of 1 is six times the

measure of a complement of 1. Find the measure of 1, its

supplement, and its complement.

2. A and B are supplementary angles and

mA = 3x + 12 and mB = 2x – 22. Find the measure of both

angles.

3. One angle is twice as large as its complement. Find

the measure of both angles.

T

a. b.

3 1 2 4 P O Q

c. V

W Z

R S T

X =

E C m1 = 2x + 6 B m2 = 3x + 10

m3 = x + 20

mADE = 24x

3 1

2

D

A

a.

x = _____, y = _____

1

(3x + 8 )º 2 3 (5x – 20)º m1 = ________

4 m2 = ________

(5x + 4y) º m3 = ________

m4 = ________

b.

m1 = _____________

2 m2 = _____________

1

25º

1.

2.

Activity Card

Activity Card 1 Activity Card 3

1. Vertical Angle 1. m 1 = 72º

2. Supplementary Angle, Linear Pair complement = 18º

3. Supplementary Angle supplement = 108º

4. Complementary Angle 2.. m A = 126º

5. Adjacent Angle m B = 54º

3. x, 2(90-x) º

Activity Card 2 x = 60º , complement = 30º

1. m1=90º Assessment Card

m2=60º 4. x = 2

2. x =18º m 1 = 10º

4x =72º m 2 = 16º

5x = 90º m 3 = 22º

3. x=15º m ADE = 48º

m 1=155º Enrichment Card

m 3=25º a. x = 14

m 2=155º m 1 = 130º

m 4=25º m 2 = 50º

4. x = 13º m 3 = 50º

m 1 = 149º m 4 = 130º

m 2 = 31º b. m 1 = 90º

5. m 1 = 49º m 2 = 65º

m 2= 41º Activity 4

a. 1 and 3, 2 and 4 are supplementary, linear pair and adjacent

b. TOQ and QOV, QOV and VOP , VOP and POT, POT and TOQ are all right angles and supplementary angles

c. WSZ and ZST are complementary angles while RSZ and ZST are supplementary

.

a. Geometry Activity Book for Enhancement

Skills by Ligaya G. Insigne

b. Geometry by Prentice Hall

c. Geometry by Edwin Moise

d. Geometry by Houghliton Mitten

e. Google and Mozila Firefox