Grade 7 Mathematics Quarter 3 Module 2: Angles

Mathematics Quarter 3 – Module 2: Angles

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CO_Q3_Mathematics 7_Module 2

Mathematics– Grade 7 Alternative Delivery Mode Quarter 3 – Module 2: Angles First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer: Alfredo T. Ondap Jr. Editor: Jenypete Diana Reviewers: Reynaldo C. Tagala Layout Artist: Maylene F. Grigana, Erwin J. Etoc Management Team: Allan G. Farnazo, Gilbert B. Barrera, Arturo D. Tingson Jr., Peter Van C. Ang-ug, Donna S. Panes, Elizabeth G. Torres, Judith B. Alba

Printed in the Philippines by: _________________________

Department of Education – SOCCSKSARGEN – Region XII

Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal Telefax: (083) 2288825/ (083) 2281893 E-mail Address: region12@deped.gov.ph

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Mathematics

Quarter 3 – Module 2: Angles

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

This module was designed and written with you in mind. It is here to help you learn about Angles. The scope of this module suited to different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you study them can be rearranged to correspond with the textbook you are now using.

This module is comprised of only one lesson:

• Lesson – Angles

After going through this module, you are expected to:

1. classify the different kinds of angles; and
2. derive relationships of geometric figures using measurements and by inductive reasoning; supplementary angles, complementary angles, congruent angles, vertical angles, adjacent angles, linear pairs, perpendicular lines, and parallel lines.

What I Know

[transcriber’s note: a stray mark reading “250” appears above the heading]

Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

1. Which of the following statements is TRUE?

   • A. An angle is formed by two noncollinear rays.
   • B. Two points determine an angle.
   • C. An angle divides a plane into two distinct parts.
   • D. Two distinct angles cannot share an interior point.

2. What is the sum of the measures of supplementary angles?

   • A. 45⁰
   • B. 50⁰
   • C. 90⁰
   • D. 180⁰

3. On the figures at the right, ∠RLA and ∠ALF are complementary angles. If m∠ALF = 25, then what is the measure of m∠RLA?

   
   
   
   
   • A. 25
   
   
   • B. 55
   
   
   • C. 65
   
   
   • D. 155
   
   
   
   

   [illustration: rays from point L labeled R (up), A (middle), and F (right), with the angle between A and F marked 25⁰]

   
   

4. Which of the following pairs of angles has sum equal to 90⁰?

   • A. complementary angles
   • B. linear pairs
   • C. obtuse angles
   • D. supplementary angles

5. Which of the following angles has measure less than 90⁰ but greater than 0⁰?

   • A. acute angle
   • B. obtuse angle
   • C. right angle
   • D. straight angle

6. In the figure at the right, m∠XYZ = 75⁰, what kind of angle does it form?

   
   
   
   
   • A. acute angle
   
   
   • B. obtuse angle
   
   
   • C. right angle
   
   
   • D. straight angle
   
   
   
   

   [illustration: rays from point Y labeled X (up) and Z (right), with the angle marked 75⁰]

   
   

7. All of the following are measures of an obtuse angle EXCEPT:

   • A. 95⁰
   • B. 110⁰
   • C. 120⁰
   • D. 190⁰

8. Given the figure at the right, if ∠ABC = 90⁰, which of the following statements is TRUE?

   
   
   
   
   • A. ∠ABC is an acute angle.
   
   
   • B. ∠ABC is a right angle.
   
   
   • C. ∠ABC is an obtuse angle.
   
   
   • D. ∠ABC has no known classification.
   
   
   
   

   [illustration: right angle at vertex B, with rays to A (up) and C (right), marked with a square]

   
   

9. Which of the following is a measure of an acute angle?

   • A. 170⁰
   • B. 150⁰
   • C. 90⁰
   • D. 80⁰

10. If m∠MAT = 135⁰ and m∠HAE = 45⁰, then which of the following could describe the two angles?

    • A. complementary angles
    • B. vertical angles
    • C. supplementary angles
    • D. forms a right angle

11. Given the figures at the right, what can you conclude about the two angles?

    
    
    
    
    • A. linear pairs
    
    
    • B. congruent angles
    
    
    • C. vertical angles
    
    
    • D. perpendicular to each other
    
    
    
    

    [illustration: two angles, one formed by rays from B with vertex at C and labeled A, B, C, marked 35⁰; the other formed by rays from E with vertex at F and labeled D, E, F, marked 35⁰]

    
    

12. Which of the following is true about parallel lines?

    • A. coplanar lines that do not intersect no matter how far they are extended
    • B. two lines with a common point
    • C. three or more lines that have a common point
    • D. straight lines that do not intersect and are not on the same plane

For items No. 13 – 14, consider the figure below:

[chart: diagram; two intersecting lines with the four angles labeled 1 (top), 2 (right), 3 (bottom), and 4 (left)]

13. Which of the following are vertical angles?

    • A. ∠1 and ∠2
    • B. ∠1 and ∠4
    • C. ∠2 and ∠4
    • D. ∠2 and ∠3

14. Which of the following form a linear pair?

    • A. ∠1 and ∠2
    • B. ∠1 and ∠3
    • C. ∠2 and ∠4
    • D. None of these.

15. Which of the following describes the perpendicular lines?

    • A. lines that intersect to form a right angle
    • B. lines that meet at a common point
    • C. lines that do not meet
    • D. lines that meet to form complementary angles

Very Good! You did a good job. You’re now ready for the next set of activities.

Lesson 1 Angles

You are now on the next module! Let’s now study about angles. Angles can be found everywhere. Buildings, establishments, satellite dishes, billboards, and many other objects forming angles. Of course, you also have clocks in your homes, right? Notice the figure formed by the two hands of the clock. They are clearly examples of angles. These sounds interesting. Hello, angles!

What’s In

The concept of rays is useful for us to effectively grasp the lesson in this module. Let us recall some lessons we had on rays.

A ray is a part of a line with an endpoint and extends endlessly in one direction. It is named by its endpoint and any other point on it.

[illustration: ray with endpoint A and a point B on it, arrow pointing beyond B]

Figure 1

Figure 1 shows an example of a ray. Ray AB has endpoint A and goes indefinitely toward the direction of point B. A ray with an endpoint A and another point B on one side of endpoint A is named $\overrightarrow{AB}$ (read as “ray AB”).

Note that ray $\overrightarrow{AB}$ cannot be called $\overrightarrow{BA}$. Ray BA has an endpoint B, as illustrated in Figure 3 below.

[illustration: ray with endpoint B and a point A on it, arrow pointing beyond A]

Figure 2 The given ray is called $\overrightarrow{BA}$.

[transcriber’s note: the text refers to “Figure 3” but the ray diagram is labeled Figure 2, as printed]

Let’s see if you can still remember the concepts about rays. Try the following activity!

Activity 1: Name Me!

Given the figure at the left, name four(4) different rays you can create from it.

[illustration: a line with three points labeled X, Y, and Z, with arrowheads at both ends]

1. 
   
2. 
   
3. 
   
4. 
   

Great! So easy, right? Let’s go on!

What’s New

The measure of an angle indicates how wide the opening is between its two sides. A protractor is used to find the measure of an angle, just like a ruler is used to find the length of a segment.

To find the measure of an angle using a protractor, place the center of the protractor over the vertex of an angle. Then align the mark labeled 0 with one side of the angle and read the scale where the other side of the angle falls. The unit of measurement used for angles is called degree, denoted by the symbol ⁰.

[photo: protractor placed over angle GEO; labels G, E, O]

Fig.3

A protractor usually has outer and inner degree scales. You may use any of these scales depending on the positions of the angle. Using the inner scale of the protractor shown in the figure above, the measure of ∠GEO is equal to 50 degrees, written as

m∠GEO = 50⁰

Now, let us try to draw ∠LAB so that m∠LAB = 80⁰. Use a protractor.

Solution:

• Draw AB
• Place the center of protractor at A with AB coinciding with the line where the 0⁰ mark is.
• Mark L at 80⁰
• Draw AL

[photo: two drawings showing protractor placed at A on line AB and angle LAB measuring 80⁰; labels L, A, B]

Fig. 4

What is It

Definition: An angle is formed by two noncollinear rays with a common endpoint.

BASIC CONCEPTS ON ANGLES

The two rays are called the sides of the angle. The common endpoint is called the vertex.

The symbol for an angle is ∠. An angle can be named in three ways, that is, by using (1) the number assigned to the angle, (2) its vertex, or (3) its vertex and two other points, one from each side of the angle. The vertex of the angle below is B and its sides are BA and BC.

[illustration: angle with vertex B and points A and C; angle labeled 1]

Fig. 5

The angle above can be name as angle 1 (∠1), angle B (∠B), angle ABC (∠ABC), or angle CBA (∠CBA).

Other examples of angles are shown below.

[illustration: (a) angle with points M, A, P; (b) angle with vertex X; (c) right angle labeled 1]

(a) (b) (c)

Fig. 6

The angles are (a) ∠MAP or ∠PAM, (b) ∠X, and (c) ∠1.

KINDS OF ANGLES

Angles can be classified according to their measures. They can be a right angle, an acute angle, or an obtuse angle.

Definition: A right angle is an angle that measures exactly 90⁰. An acute angle is an angle that measures less than 90⁰. An obtuse angle is an angle that measures more than 90⁰ but less than 180⁰

[illustration: (a) right angle marked 90⁰; (b) acute angle marked 35⁰; (c) obtuse angle marked 120⁰]

Fig. 7

Based on the figures above, some angles can be classified as (a) a right angle, (b) an acute angle, or (c) an obtuse angle.

Angle A is a right angle since m∠A = 90⁰. Angle C is an acute angle since m∠C < 90, while angle D is an obtuse angle since 90⁰ < m∠D < 180⁰.

You can approximate the measures of angles by estimating their openings if equal to 90⁰, less than 90⁰, or greater than 90⁰. These approximations of angle measures without using protractors will help you in classifying angles whether they are right, acute, or obtuse.

Example 1

Estimate the measure of each angle. Classify the angle.

a. b. c.

[illustration: (a) angle with points G, O, T; (b) right angle with point M; (c) angle with points S, E, T]

Solution

You can use approximations to determine if the measure of each angle is less than 90⁰, equal to 90⁰, or greater than 90⁰.

a. m∠GOT < 90⁰; acute angle b. m∠M = 90⁰; right angle c. m∠SET > 90⁰; obtuse angle

Example 2

Refer to the given figure. Measure and classify each indicated angle.

a. ∠PQR b. ∠PQT c. ∠RQS

[illustration: lines and rays meeting at Q, with points R, T, P, S, U]

Solution

a. m∠PQR = 90⁰; right angle b. m∠PQT = 30⁰; acute angle c. m∠RQS = 100⁰; obtuse angle

Refer again to the figure in example 2. Use your protractor to measure ∠UQS. Angle UQS measures 30⁰. Thus, m∠PQT = m∠UQS = 30. Angles PQT and UQS are congruent angles, denoted by ∠PQT ≅ ∠UQS, which is read as “angle PQT is congruent to angle UQS.”

Definition: Congruent angles are angles that have the same measure.

Use identical markings to show congruent angles in a figure. Refer to figure 9. If BM is the angle bisector of ∠ABC then ∠ABM ≅ ∠MBC. Also, if ∠ABM ≅ ∠MBC then BM divides ∠ABC into two congruent angles. Thus, BM is the angle bisector of ∠ABC.

[illustration: angle ABC with points A, M, B, C; congruent angle marks shown]

Fig. 8

On the figure above, the identical marks means that ∠ABM ≅ ∠MBC, hence, BM bisects ∠ABC.

Definition: An angle bisector divides an angle into two congruent angles.

Note: Every angle has exactly one angle bisector.

Example 3

Name each angle bisector and all congruent angles in the figure.

1. [illustration: angle CIN with points C, A, N; each half marked 45⁰]
2. [illustration: angle KAE with points K, Y, E; angles marked 30⁰ and 25⁰]
3. [illustration: angle AGL with points A, N, E, L, G; congruent angle marks shown]

Solution

1. Since m∠CIA = m∠AIN = 45⁰, therefore, ∠CIA ≅ ∠AIN. Hence, IA is an angle bisector of ∠CIN.
2. Ray AY is not an angle bisector of ∠KAE since m∠KAY ≠ m∠YAE.
3. Angle AGN is congruent to angle LEG. Rays GN and GE are not angle bisectors of ∠AGL since these two different rays do not divide ∠AGL into two angles.

ANGLE PAIRS

Geometric relationships exist between two angles. These relationships can be used as bases for classifying angle pairs. Angle pairs include adjacent angles, complementary angles, supplementary angles, linear pairs, and vertical angles.

Adjacent Angles

Look at ∠ABD and ∠DBC. What do they have in common?

[illustration: angle ABD and angle DBC sharing ray BD; points A, D, B, C]

Fig. 9

Angles ABD and DBC are adjacent angles with B as their common vertex and ray BD as their common side.

Definition: Adjacent angles are angles on the same plane that have a common side and a common vertex but no common interior points.

Complementary Angles

Look at ∠FEG and ∠GED. What is the sum of their measures? Compare this sum to that of ∠TUV and ∠XYZ. What can you say about the sum of two pairs of angles?

[illustration: (a) angles FEG = 60⁰ and GED = 30⁰ with points F, G, E, D; (b) angles TUV = 45⁰ and XYZ = 45⁰ with points T, V, U, Y, X, Z]

Fig. 10

On the figures above, Angles FEG and GED in (a) are adjacent complementary angles. Angles TUV and XYZ in (b) are nonadjacent complementary angles.

Since m∠FEG + m∠GED = 60⁰ + 30⁰ = 90⁰, then ∠FEG and ∠GED are complementary angles.

Angle GED is the complement of ∠FEG. Also, ∠FEG is the complement of ∠GED. Angles GED and FEG have a common side EG, that is, they are also adjacent angles. Therefore, ∠GED and ∠FEG are adjacent complementary angles.

On the other hand, in figure 10(b),

m∠TUV + m∠XYZ = 45⁰ + 45⁰ = 90⁰

Hence, ∠TUV and ∠XYZ are complementary. They have no common side so they are nonadjacent complementary angles.

Definition: Complementary angles are two angles whose measures have a sum of 90⁰.

Example 1

Find the measure of the complement of an angle whose measure is 55⁰.

Solution

Let x = the measure of the complement of an angle. Hence,

x + 55⁰ = 90⁰ x = 90⁰ − 55⁰ By SPE: Subtract 55 from both sides. x = 35⁰.

Check your answer.

35⁰ + 55⁰ = 90⁰ True

Hence, the measure of the complement of a 55⁰ angle is a 35⁰ angle.

Example 2

Find the measures of the complementary angle of a 12⁰ angle less than twice the other.

Solution

Let x = the measure of the first angle, 2x − 12 = the measure of the second angle.

Since the two angles are complementary.

x + (2x − 12) = 90 3x − 12 = 90 Combine similar terms. 3x = 90 + 12 By APE: Add 12 to both sides. 3x = 102 3x/3 = 102/3 By DPE: Divide both sides by 3. x = 34.

It follows that the second angle measures

2x − 12 = 2(34) − 12 = 68 − 12 = 56.

Thus, the two complementary angles measure 34⁰ and 56⁰, respectively.

Supplementary Angles

Look at the following angles pairs: ∠MNO and ∠ONP, and ∠ABC and ∠RST. What is the sum of the measures of each angle pair?

[illustration: (a) angle MNO = 135⁰ and angle ONP = 45⁰ on straight line MNP with point O; (b) angle ABC = 60⁰ and angle RST = 120⁰ on straight line ABST with points C, R]

Fig.11

In Figure 11(a), Angles MNO and ONP are adjacent supplementary angles, while in (b), Angles ABC and RST are nonadjacent supplementary angles.

The sum of the measures of ∠MNO and ∠ONP is

m∠MNO + m∠ONP = 135⁰ + 45⁰ = 180⁰.

Hence, ∠MNO and ∠ONP are supplementary angles. These two angles are supplements of each other. Notice also that these angles are adjacent (since they have a common side NO) and, at the same time, supplementary.

On the other hand, in figure 11(b),

m∠ABC + m∠RST = 60 + 120 = 180.

Hence, ∠ABC and ∠RST are also supplementary. Notice also that these two angles are supplementary but not adjacent.

Definition: Supplementary angles are two angles whose measures have a sum of 180⁰.

Example

The measure of the supplement of an angle is 25⁰ more than 4 times the measure of the angle. Find the measure of each angle.

Solution

Let x = the measure of an angle, 180 − x = the measure of its supplement.

Then solve for x in the equation 180 − x = 4x + 25.

180 − x = 4x + 25 −x − 4x = 25 − 180 By SPE: Subtract 4x and 180 from both sides. −5x = −155 −5x/5 = −155/−5 DPE: Divide both sides by -5. x = 31

Then substitute 31 for x in 180 − x to get the measure of the supplement of the angle.

180 − x = 180 − 31 = 149

Check your answer.

31 + 149 = 180 True

Therefore, the angle measures 31⁰ and its supplement measures 149⁰.

Linear Pairs

Look at ∠VXY and ∠XYZ. How will you classify this angle pair?

[illustration: straight line V–X–Z with ray XY above; points V, X, Y, Z]

Fig.12

In the figures above, Angles VXY and YXZ are adjacent and supplementary angles with a common side XY.

You call ∠VXY and ∠YXZ a linear pair.

Definition: A linear pair consists of two adjacent angles whose noncommon sides are opposite rays.

Example

Find the measure of each angle in the linear pair.

[illustration: straight line P–A–R with ray AM above; angle PAM labeled (3x − 15)⁰ and angle MAR labeled (x + 5)⁰]

Solution

Since ∠PAM and ∠MAR form a linear pair, they are supplementary. Thus,

(3x − 15) + (x + 5) = 180 4x − 10 = 180 Combine similar terms. 4x = 180 + 10 By APE: Add 10 to both sides. 4x = 190 4x/4 = 190/47.5 By DPE: Divide both sides by 4.

Substituting 47.4 for x, the two angles measure

x + 5 = 47.5 + 5 and 3x − 15 = 3(47.5) − 15 x = 52.5 x = 127.5

[transcriber’s note: the printed text reads “47.4” in the substitution line, while the division step above gives 47.5; the value is transcribed as printed]

[transcriber’s note: the printed division line reads “190/47.5” as shown]

Vertical Angles

When two lines intersect, they form a pair of nonadjacent angles called vertical angles.

[illustration: two intersecting lines forming angles numbered 1, 2, 3, 4]

Definition: Vertical angles are two nonadjacent angles formed by two intersecting lines.

Example

In the figure below, name each indicated angle pair.

1. complementary angles
2. supplementary angles
3. vertical angles
4. adjacent angles
5. Linear pairs

[illustration: two perpendicular lines and a third line crossing them, with angles numbered 1, 2, 3, 4, 5; right-angle mark shown]

Solution

1. ∠1 and ∠2
2. ∠4 and ∠5
3. ∠1 and ∠4

Perpendicular Lines

When two lines intersect to form right angles, the two lines are said to be perpendicular to each other.

[illustration: two perpendicular lines through point O, with points A, B, C, and D on the four rays]

In the accompanying figure, $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersect at point $O$ to form right angles. We say that $AB$ is perpendicular to $CD$. In symbols, we write $AB \perp CD$.

Since $\angle AOC$ and $\angle BOC$ form a linear pair, they are supplementary. That is

$m\angle AOC + m\angle BOC = 1800$

Since $m\angle AOC = 90^0$, therefore, $m\angle BOC = 90^0$.

4. ∠1 and ∠2, ∠1 and ∠5, ∠2 and ∠3, ∠3 and ∠4, and ∠4 and ∠5
5. ∠4 and ∠5

Vertical angles are congruent.

Example

Find the measures of all vertical angles in the figure where $m\angle 1 = 2x + 15$ and $m\angle 2 = 85 – 5x$.

[illustration: two intersecting lines with angles numbered 1, 2, 3, and 4; angles 1 and 2 are opposite each other]

Solution

Since $\angle 1$ and $\angle 2$ are vertical angles, $\angle 1 \cong \angle 2$. Hence,

$m\angle 1 = m\angle 2$ Definition of congruent angles
$2x + 15 = 85 – 5x$ By substitution
$2x + 5x + 15 = 85$ By APE: Add 5x to both sides.
$7x + 15 = 85$ Combine like terms.
$7x = 85 – 15$ By SPE: Subtract 15 from both sides.
$7x = 70$
$\frac{7x}{7} = 70$ By DPE: Divide both sides by 7.
$x = 10.$

By substituting 10 for $x$, you get the measures of $\angle 1$ and $\angle 2$.

$m\angle 1 = 2x + 15 = 2(10) + 15 = 20 + 15 = 35$
$m\angle 2 = 85 – 5x = 85 – 5(10) = 85 – 50 = 35$

Since $\angle 1$ and $\angle 3$ form a linear pair, then $\angle 1$ and $\angle 3$ are supplementary angles. Thus,

$m\angle 1 + m\angle 3 = 180$
$35 + m\angle 3 = 180$ By Substitution
$m\angle 3 = 180 – 35$ By SPE: Subtract 35 from both sides.
$m\angle 3 = 145.$

Since $\angle 3$ and $\angle 4$ are vertical angles, $\angle 3 \cong \angle 4$. Thus,

$m\angle 3 = m\angle 4$ Definition of congruent angles
$145 = m\angle 4$ By substitution
$m\angle 4 = 145$ By symmetric property of equality

Hence,

$m\angle 1 = m\angle 2 = 35$ and $m\angle 3 = m\angle 4 = 145.$

Since $m\angle AOC = 90^0$, therefore, $m\angle BOC = 90^0$. In the same manner, $m\angle AOD = 90^0$ and $m\angle BOD = 90^0$. Thus, perpendicular lines form four right angles.

Parallel Lines

[illustration: two parallel lines labeled m and n, each with arrows at both ends]

Parallel lines are lines that do not intersect no matter how far they are extended. Lines $m$ and $n$ do not intersect, and are parallel. You write this as $m \parallel n$. The symbol $\parallel$ is read as “is parallel to”.

Definition: Parallel lines are coplanar lines that do not intersect.

What’s More

Let us now try use the knowledge we have gained from the concepts presented earlier in answering series of activities below. Write your answer on a separate sheet of paper.

A. Determine if each statement is TRUE or FALSE.

1. An angle is formed by two noncollinear rays.
2. All right angles are congruent.
3. All acute angles are congruent.
4. Two distinct angles cannot share an interior point.
5. Point $O$ is the vertex of $\angle GEO$.
6. All congruent angles are right angles.
7. An obtuse angle measures less than $90^0$.
8. An angle divides a plane into two distinct parts.
9. The sum of the measures of two acute angles is always greater than $90^0$.
10. The sum of the measures of two obtuse angles is greater than $180^0$.

B. Estimate the measure of each angle. Classify the angles whether RIGHT, ACUTE or OBTUSE ANGLE.

1. [illustration: angle with vertex X and a right-angle mark]
2. [illustration: angle with vertex Y]
3. [illustration: angle with vertex Z]
4. [illustration: angle with vertex A]

C. Identify whether the following pairs of angles are ADJACENT ANGLES, LINEAR PAIRS, COMPLEMENTARY ANGLES, SUPPLEMENTARY ANGLES or VERTICAL ANGLES. Refer to the given figure. (You may have one or more possible answers in each item.)

[illustration: rays OA, OB, OC, OD, and OE meeting at point O, with a right-angle mark at O; points A, B, C, D, and E are labeled]

1. $\angle DOE$ and $\angle EOA$ = ______________________________
2. $\angle AOB$ and $\angle AOE$ = ______________________________
3. $\angle EOD$ and $\angle DOC$ = ______________________________
4. $\angle EOD$ and $\angle BOD$ = ______________________________
5. $\angle DOE$ and $\angle BOA$ = ______________________________

D. Find the value of $x$ in each figure. Copy on a separate sheet of paper and complete the answer in the table.

Item	Answer
1. [illustration: two intersecting lines with angles $(2x + 11)^0$ and $(3x – 45)^0$ opposite each other]	Your Answer Here: $2x + 11 = 3x – 45$
2. [illustration: a ray meeting a straight line, with angles $(2x – 5)^0$ and $(4x – 7)^0$ forming a linear pair]	Your Answer Here: $(2x – 5) + (4x – 7) = 180$
3. [illustration: a right angle with a ray inside it; angles $(x – 20)^0$ and $(3x + 6)^0$ are labeled]	Your Answer Here: $(x – 20) + (3x + 6) = 90$

Good job! Get ready for another learning battle ahead!

What I Have Learned

Write always, sometimes, or never in each blank to complete each statement. Write your answer in a separate sheet of paper.

1. Angles complementary to the same angle or to congruent angles are _____________________ congruent.
2. Two right angles are _____________________ congruent.
3. Two congruent angles are _____________________ right.
4. Angles supplementary to the same angle or congruent angles are _____________________ congruent.
5. Two complementary angles _____________________ form a right angle.
6. Two angles that form a right angle are _____________________ complementary.
7. Two vertical angles are _____________________ nonadjacent.

Nice work! Now you’re up for the next challenge of this lesson.

What I Can Do

Here is another activity that will let you apply what you have learned about angles.

Solve each word problem.

1. What is the measure of the complement of an angle whose measure is $72^0$?
2. Find the measure of the supplement of an angle whose measure is $72^0$?
3. Two angles form a linear pair. If the measure of the first angle is twice that of the second angle, what is the measure of each angle?

Assessment

For you to determine how much you’ve learned, please answer the questions by choosing the letter of the best answer.

1. Which of the following statements is NOT TRUE?
   a. An angle is formed by two noncollinear rays.
   b. Two points determine an angle.
   c. An angle is a union of two rays..
   d. An angle can be a right angle if its measure is equal to $90^0$.

   
   

2. If two angles form a linear pair, then they are supplementary. What is the sum of the measures of these two angles?
   a. $180^0$
   b. $90^0$
   c. $50^0$
   d. $45^0$

   
   

3. On the figures at the right, $\angle YVA$ and $\angle AVN$ are complementary angles. If $m\angle YVA = 25$, then what is the measure of $m\angle RLA$?
   [illustration: angle at V between rays VY and VN with $75^0$ marked; point A is on the ray between them, with labels Y, A, N, and V]
   a. $285^0$
   b. $105^0$
   c. $15^0$
   d. $5^0$

   
   

4. Which of the following pairs of angles whose sum of the measures is equal to $180^0$?
   a. complementary angles
   b. acute angles
   c. obtuse angles
   d. supplementary angles

   
   

5. Which of the following angles has measure greater than $90^0$ but lesser than $180^0$?
   a. acute angle
   b. obtuse angle
   c. right angle
   d. straight angle

   
   

6. If an angle measures $90^0$, what kind of angle does it form?
   a. acute angle
   b. obtuse angle
   c. right angle
   d. straight angle

   
   

7. All of the following are measures of an acute angle EXCEPT:
   a. $90^0$
   b. $85^0$
   c. $78^0$
   d. $5^0$

   
   

8. Given the figure at the right, if $m\angle ABC = 120^0$, which of the following statements is TRUE?
   [illustration: angle ABC with vertex B, points A and C labeled, and $120^0$ marked]
   a. $\angle ABC$ is an acute angle.
   b. $\angle ABC$ is a right angle.
   c. $\angle ABC$ is an obtuse angle.
   d. $\angle ABC$ has no known classification.

   
   

9. Which of the following is a measure of an obtuse angle?
   a. $35^0$
   b. $65^0$
   c. $90^0$
   d. $160^0$

   
   

10. If $m\angle MAT = 35^0$ and $m\angle HAE = 55^0$, then which of the following could describe the two angles?
    a. complementary angles
    b. vertical angles
    c. supplementary angles
    d. forms a right angle

    
    

11. Given the figures at the right, what can you conclude about the two angles?
    [illustration: a ray meeting a straight line, with $145^0$ and $35^0$ marked as the two angles]
    a. They are congruent.
    b. They form a linear pair.
    c. They form vertical angles.
    d. They are perpendicular to each other.

    
    

12. Which of the following is true about perpendicular lines?
    a. Perpendicular lines are lines are coplanar lines that do not intersect no matter how far they are extended.
    b. Perpendicular lines are two lines with a common point.
    c. Perpendicular lines are lines that intersect each other to form right angles.
    d. Perpendicular lines are straight lines that do not intersect and are not on the same plane.

    
    

For items No. 13 – 14, consider the figure below:

[illustration: two intersecting lines with angles numbered 1, 2, 3, and 4]

13. Which of the following are vertical angles?
    a. $\angle 1$ and $\angle 2$
    b. $\angle 1$ and $\angle 4$
    c. $\angle 2$ and $\angle 4$
    d. $\angle 2$ and $\angle 3$
14. Which of the following forms a linear pair?
    a. $\angle 1$ and $\angle 2$
    b. $\angle 1$ and $\angle 3$
    c. $\angle 3$ and $\angle 4$
    d. None of these.
15. Which of the following describes the parallel lines?
    a. intersect to form a right angle
    b. meet at a common point
    c. lines that do not meet
    d. lines that meet to form complementary angles

Good Job! You did well on this module! Keep going!

Additional Activities

This section includes supplementary activities related to operations involving polynomials.

Do as indicated. Refer to the given figure.

[illustration: point A with rays AB, AC, AD, AE, and AF; the line through D, A, and C is straight; angles 1 and 2 are marked]

1. Give another name for $\angle 1$.
2. Name all angles with $\overleftrightarrow{AF}$ as a side.
3. Determine how many acute angles there are with point $A$ as their vertex. Name two of these acute angles.

Answer Key

What I Know

1. A
2. D
3. C
4. A
5. A
6. A
7. D
8. B
9. D
10. C
11. B
12. A
13. C
14. A
15. A

What’s In

Activity 1:

1. XY
2. XZ
3. ZX
4. ZY

What’s More

A.

1. True
2. True
3. False
4. False
5. False
6. False
7. False
8. False
9. False
10. True

B.

1. Right angle
2. Acute angle
3. Obtuse angle
4. Obtuse angle

C.

1. Linear pair, adjacent angles, supplementary angles
2. Linear pair, adjacent angles, supplementary angles
3. Adjacent angles, complementary angles
4. Linear pair, adjacent angles, supplementary angles
5. Vertical angles

D.

1. $x = 56^0$
2. $x = 32^0$
3. $x = 26^0$

Assessment

1. B
2. A
3. C
4. D
5. B
6. C
7. A
8. C
9. D
10. A
11. B
12. C
13. A
14. B
15. C

Additional Activities

1. ∠CAB or ∠BAC
2. ∠FAD or ∠DAF, ∠FAE or ∠EAF [?]
3. 4; ∠DAF or ∠FAD, ∠CAB or ∠BAC [?]

[transcriber’s note: the answer key on this page is printed upside down. The Additional Activities entries were read with some uncertainty, so they are marked [?] where the reading is not certain.]

What I Have Learned

1. sometimes
2. always
3. sometimes
4. sometimes
5. always
6. always
7. always

What I Can Do

1. $18^0$
2. $108^0$
3. $60^0, 120^0$

References

De Leon, Cecile M., et.al, Integrated Mathematics Textbook for First Year, 2000, ISBN: 971-07-1389-2

Jose-Dilao, Soledad, et.al, Intermediate Algebra Textbook for Second Year Revised Edition,2009, ISBN: 978-971-0315-55-0

Department of Education-Bureau of Learning Resources (DepEd-BLR) (2016) Grade 7 Mathematics Teacher’s Guide. Lexicon Press Inc., Philippines

Oronce, Orlando A., et.al, E-Math Elementary Algebra Revised Edition 2010, ISBN: 978-971-23-5479-3, pp.192-195, pp.208-211

Oronce, Orlando A., et.al, E-Math Intermediate Algebra Revised Edition 2010, ISBN:978-971-23-5480-9, pp.70-72, pp.76-83, pp.86-95

For inquiries or feedback, please write or call:

Department of Education – Bureau of Learning Resources (DepEd-BLR)

Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph