This is the Grade 7 Mathematics Quarter 3, Module 3 self-learning module, Parallel Lines Cut by Transversal. It is for Grade 7 learners studying at home or in class. After it, learners can name the angle pairs formed when a transversal crosses two parallel lines, and use those pairs to find unknown angle measures.
The module is a First Edition (2020) Alternative Delivery Mode resource from DepEd SOCCSKSARGEN (Region XII). It has one lesson, plus a pretest, activities, an assessment, and an answer key.
On this page 8 sections
What learners will learn
The module states one learning objective. After going through it, learners are expected to:
- derive relationships among angles formed by parallel lines cut by a transversal, using measurement and inductive reasoning.
Key terms to review
The lesson starts by reviewing these kinds of lines:
- Intersecting lines: coplanar lines that share a point.
- Perpendicular lines: lines that meet at right angles.
- Parallel lines: coplanar lines that never meet.
- Skew lines: lines that are not coplanar and never meet.
- Transversal: a line that crosses two or more coplanar lines at different points.
Angle pairs formed by a transversal
A transversal crossing two lines makes eight angles, numbered 1 to 8 in the module’s figure. The module sorts them like this, using its figure where transversal x crosses lines y and z:
| Type | Angles in the module’s example |
|---|---|
| Interior angles | 2, 4, 6, 8 |
| Exterior angles | 1, 3, 5, 7 |
| Corresponding angles | 1 and 6, 3 and 8, 2 and 5, 4 and 7 |
| Alternate interior angles | 2 and 8, 4 and 6 |
| Alternate exterior angles | 1 and 7, 3 and 5 |
| Same-side interior angles | 2 and 6, 4 and 8 |
The module defines the main pairs this way:
- Alternate interior angles: two non-adjacent interior angles on opposite sides of the transversal.
- Corresponding angles: a non-adjacent interior angle and exterior angle on the same side of the transversal.
- Same-side interior angles: interior angles on the same side of the transversal.
The numbering changes from figure to figure. Always check the diagram in front of you before naming a pair.
The key rules when the lines are parallel
When two parallel lines are cut by a transversal:
- Corresponding angles are congruent (equal in measure).
- Alternate interior angles are congruent.
- Same-side interior angles are supplementary (they add up to 180°).
The activities also use two facts that are true for any two crossing lines. Vertical angles are congruent. Angles that form a linear pair are supplementary.
Worked examples from the module
Example 1. Lines a and b are parallel and cut by transversal c, with m∠1 = 60°.
- m∠2 = 120°, because ∠1 and ∠2 are supplementary.
- m∠3 = 120° and m∠4 = 60°, by vertical angles.
- m∠5, ∠6, ∠7, and ∠8 then follow from corresponding angles: 60°, 120°, 120°, and 60°.
Example 2. Lines p and q are parallel and cut by transversal r, with m∠3 = 125°. The same steps give:
- 125° for ∠1, ∠5, and ∠8.
- 55° for ∠2, ∠4, ∠6, and ∠7.
The takeaway: once you know one angle, every other angle is either equal to it or 180° minus it.
Later items show the same idea with algebra. For example, the assessment gives m∠3 = (2x + 5)° and m∠8 = 75°, and asks for x.
Parts of the module
- What I Know: a 10-item pretest on transversals, angle pairs, and angle measures.
- What’s In: a review of line types and angle pairs, with Activity 1, where learners identify the transversal and the angle pairs in a figure.
- What Is It: the three rules for parallel lines and the two worked examples.
- What’s More: Activity 2 (Find My Missing Parts), where learners complete angle measures and find all angles given m∠a = 108°.
- What I Have Learned: Activity 3 (Fill Me), a fill-in-the-blank recap.
- What I Can Do: Activity 4 (Solve Me), where learners identify angle pairs and find all angles given m∠3 = 135°.
- Assessment: a 10-item post-test.
- Additional Activities: two algebra problems using angle expressions.
- Answer Key: answers to the pretest, activities, assessment, and additional activities.
How to use this module
- Learners: Take the pretest first to see what you already know. Then work through the parts in order. Write all answers on a separate sheet of paper, not on the module. Check your work with the answer key only after you finish, and be honest with yourself when you use it. Finish with the assessment to check your learning.
- Parents and facilitators: Help your child read each direction carefully. Encourage them to ask the teacher when a task is hard. The module notes that it includes notes for facilitators and parents on supporting home-based learning.
- Teachers: The pretest and assessment cover the same skills, so you can compare results to measure progress. The module says its lessons can be read in an order that matches the textbook you use. Review the answer key before you use it for checking.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 7 Mathematics, Quarter 3, Module 3. It has a single lesson, Parallel Lines Cut by Transversal.
What is a transversal?
A transversal is a line that crosses two or more coplanar lines at different points. Where it crosses, it forms eight angles that can be grouped into pairs.
What happens to the angles when parallel lines are cut by a transversal?
Corresponding angles are congruent, and so are alternate interior angles. Same-side interior angles are supplementary, which means they add up to 180 degrees.
Does the module have an answer key?
Yes. It has answers for the pretest, Activities 1 to 4, the assessment, and the additional activities. The module asks learners to use the key honestly, after answering.
How do I find all eight angles if I know only one?
Use vertical angles and linear pairs to find the angles at one crossing. Then use corresponding angles to copy those measures to the other crossing. For example, if one angle is 60 degrees, each of the others is either 60 or 120 degrees.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 3 – Module 3: Parallel Lines Cut by Transversal
[illustration: cover art of students studying outdoors and at a table, a girl holding a globe, and a boy writing by a lantern]
7
Mathematics
Quarter 3 – Module 3: Parallel Lines Cut by Transversal
GOVERNMENT PROPERTY NOT FOR SALE
CO_Q3_Mathematics 7_Module 3
Copyright and Development Team
Mathematics – Grade 7 Alternative Delivery Mode Quarter 3 – Module 3: Parallel Lines Cut By a Transversal First Edition, 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
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
Development Team of the Module
| Writer: | Ana Cristy P. Mercado |
| Editors: | Hyacinth R. Moneva |
| Reviewers: | Reynaldo C. Tagala |
| Layout Artist: | Maylene F. Grigana, Erwin J. Etoc |
| Management Team: | Allan G. Farnazo Gilbert B. Barrera Arturo D. Tiongson Jr. Peter Van C. Ang-ug Donna S. Panes Elizabeth G. Torres Judith B. Alba |
7
Mathematics
Quarter 3 – Module 3: Parallel Lines Cut by Transversal
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.
This module was designed and written with you in mind. It is here to help you master how to derive relationships among angles formed by parallel lines cut by a transversal using measurement and by inductive reasoning. The scope of this module permits it to be used in many 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 read them can be changed to correspond with the textbook you are now using.
The module is comprised of only one lesson:
- Derives relationships among angles formed by parallel lines cut by a transversal using a measurement and by inductive reasoning.
After going through this module, you are expected to:
- derive relationships among angles formed by parallel lines cut by a transversal using a measurement and by inductive reasoning.
What I Need to Know
[illustration: hand pointing with a finger inside a circle]
What I Know
[illustration: hand holding a pencil inside a circle]
Let us check what you already know about this lesson! Read and understand each item carefully. Write your answer on a separate sheet of paper.
A line that intersects two or more coplanar lines in different points.
a. Transversal
b. Skew Line
c. Parallel Line
d. Perpendicular LineWhen parallel lines are cut by a transversal, corresponding angles are ________.
a. Unequal
b. Congruent
c. Complementary
d. SupplementaryWhen parallel lines are cut by a transversal, same side interior angles are _____.
a. Congruent
b. Unequal
c. Complementary
d. SupplementaryWhen parallel lines are cut by a transversal, alternate interior angles are ________.
a. Supplementary
b. Complementary
c. Congruent
d. Unequal
For items 5-10, suppose r ∥ s, give what is asked.
[chart: diagram; figure for items 5–10 showing two parallel lines r and s cut by a transversal, with angles numbered 1 to 8. Line r carries angles 1, 2, 3, 4 and line s carries angles 5, 6, 7, 8.]
How many pairs of vertical angles are there?
a. 4
b. 6
c. 8
d. 10How many linear pairs are there?
a. 4
b. 6
c. 8
d. 16Which of these angles is congruent to ∠5?
a. ∠1
b. ∠4
c. ∠6
d. ∠7What do you call ∠4 and ∠6?
a. Corresponding Angles
b. Alternate Interior Angles
c. Alternate Exterior Angles
d. Same-Side Interior AnglesWhat do you call ∠1 and ∠2?
a. Linear Pair
b. Vertical Angles
c. Corresponding Angles
d. Same-Side Interior AnglesSuppose m∠3 = 150º, what is the measure of ∠1?
a. 30º
b. 150º
c. 130º
d. 170º
Lesson 1: Parallel Lines Cut By Transversal
Parallel lines have various applications in real life. In fact, you can see these lines all around you. Now you will explore the relationships between pairs of angles formed when parallel lines are cut by a transversal.
What’s In
[illustration: hand holding a puzzle piece inside a circle]
Let’s review!
Recall that:
Intersecting lines are coplanar lines that have a point in common.
Perpendicular lines are lines that intersect at right angles.
Parallel lines are coplanar lines that do not intersect.
Skew lines are non-coplanar lines that do not intersect.
Angles Formed by Transversal
A transversal is a line intersecting two or more coplanar lines at different points.
Alternate Interior Angles – a pair of non-adjacent interior angles on opposite sides of a transversal.
Corresponding Angles – a pair of non-adjacent interior and exterior angles on the same-side of transversal.
Same-Side Interior Angles – interior angles on the same-side of transversal.
Transversal creates different types of angle pairs. To illustrate, let x be the transversal of the lines y and z.
[chart: diagram; transversal x crossing lines y and z, not labeled with values]
[illustration: two lines cut by a transversal, with angles numbered 1 to 8; the upper line is labeled y at its left end and x at its right end, and the word “Exterior” is printed above the upper intersection]
The interior angles are: <2, <4, <6 and <8.
The exterior angles are : <1, <3, <5 and <7.
The pairs of corresponding angles are : <1 and <6, <3 and < 8, <2 and <5, <4 and <7.
The pairs of alternate interior angles are : <2 and <8, <4 and <6.
The pairs of alternate exterior angles are : <1 and <7, <3 and <5.
The pairs of same side interior angles are: <2 and <6, <4 and <8.
Activity 1. Identify the following using the given figure.
[illustration: two lines labeled a and b and a transversal labeled c, with angles numbered 1 to 8]
a. the transversal b. 4 pairs of corresponding angles c. 2 pairs of alternate interior angles d. 2 pairs of alternate exterior angles e. 2 pairs of same side interior angles
What is It
When two parallel lines are cut by transversal;
- The corresponding angles are congruent.
- The alternate interior angles are congruent.
- The same side interior angles are supplementary.
[illustration: two parallel lines r and s cut by transversal t, with angles numbered 1 to 8; the label “r ll s” is printed at the upper right]
<1≅<5 <3≅<6 m<4 + m<6= 180°
Illustrative Examples
- Lines a and b are cut by a transversal c and m<1 is 60°. Assume that line a is parallel to line b. Find the measures of all the other angles.
[illustration: lines a and b cut by transversal c, with angles numbered 1 to 8]
Solution:
There are many angle pair relationships that you can use to obtain the measures of the different angles. Here is one way of doing it.
m<2 = 120° since <1 and <2 are supplementary angles m<3 = 120° since <2 and <3 are vertical angles m<4 =60° since <1 and <4 are vertical angles m<5 =60° since <1 and <5 are corresponding angles m<6 = 120° since <2 and <6 are corresponding angles m<7 = 120° since <3 and <7 are corresponding angles m<8 = 60° since <4 and <8 are corresponding angles
- Lines p and q are cut by transversal r and m< 3 is 125°. Assume that p is parallel to q. Find the measurements of all the other angles.
[illustration: lines p and q cut by transversal r, with angles numbered 1 to 8]
Solution:
m<1 = 125° since <1 and <3 are vertical angles m<2 = 55° since <3 and <2 are supplementary angles m<4 = 55° since <2 and <4 are vertical angles m<5 = 125° since <1 and <5 are corresponding angles m<6 = 55° since <2 and <6 are corresponding angles m<7 = 55° since <4 and <7 are corresponding angles m<8 = 125° since <3 and <8 are corresponding angles
What’s More
Activity 2. Find my missing parts.
Let’s apply what you have just learned!
- In the diagram, a║b. Complete the following statements.
[illustration: parallel lines a and b cut by transversal c, with angles numbered 1 to 8]
a. If m<1 =100°, then m<5 = ______________________. b. If m<2 =85°, then m<6 = ______________________. c. If m<3 =105°, then m<7 = ______________________. d. If m<4 =73°, then m<8= ______________________. e. If m<5 =67°, then m<8= ______________________.
- Suppose y║z and m<a =108°. Find the measurements of the other angles and justify your answer.
[illustration: parallel lines y and z cut by transversal x, with angles labeled a to h]
What I Have Learned
Activity 3. Fill Me
Let’s recap!
Complete the statements below by expressing your understanding on parallel lines cut by transversal.
- If m<4 + m<5 = 180°, then the two angles are ________________________.
- If a pair of parallel lines are cut by transversal then the vertical angles are ______________________.
- When two parallel lines are cut by transversal then the alternate interior angles are______________________.
- If a pair of parallel lines are cut by transversal then the same side interior angles are ______________________.
- Transversal is a _______________________________________________________.
What I Can Do
Activity 4. Solve Me.
- Assume that b║c. Find the following given the figure below.
[illustration: parallel lines b and c cut by transversal d, with angles numbered 1 to 8]
a. the transversal b. interior angles c. exterior angles d. pairs of vertical angles e. pairs of alternate interior angles f. pairs of same side interior angles g. pairs of corresponding angles
- Assume that e║f and g is the transversal. Find the measures of the other angles, given m<3 =135°. Justify your answer.
[illustration: parallel lines e and f cut by transversal g, with angles numbered 1 to 8]
Assessment
Read and understand each item carefully. Write your answer on a separate sheet of paper.
When parallel lines are cut by transversal, vertical angles are ______.
A. Supplementary Angle
B. Complementary Angle
C. Congruent Angle
D. Unequal AngleWhen parallel lines are cut by transversal, same side interior angles are______.
A. Supplementary Angle
B. Complementary Angle
C. Congruent Angle
D. Unequal AngleA line that intersects two or more coplanar lines in different points.
A. Perpendicular Line
B. Parallel Line
C. Skew Line
D. Transversal
For numbers 4-10, lines m and n are cut by transversal q. Assume that m║n. Give what is asked.
[illustration: parallel lines m and n cut by transversal q, with angles numbered 1 to 8]
Which of the following angle pairs is vertical angles?
A. <1 and <2
B. <2 and <3
C. <4 and <5
D. <3 and <8Which of the following angles pairs are supplementary angles?
A. <1 and <2
B. <1 and <4
C. <1 and <7
D. <1 and <8If m<3=70°, what is the m<7?
A. 70°
B. 80°
C. 90°
D. 110°If m<4 =125°, what is the m<7?
A. 55°
B. 65°
C. 115°
D. 125°Which of the following angle pairs is corresponding angles?
A. <1 and <3
B. <2 and<3
C. <3 and <5
D. <3 and <7If m<6=105°, what is the m<4?
A. 65°
B. 75°
C. 100°
D. 105°If m<3 = (2x+5)°, what is the value of x if m<8 =75°?
A. 35°
B. 45°
C. 65°
D. 75°
Additional Activities
Let us make use of your newly acquired skill! Given the figure below, give what is asked.
Lines a and b are cut by transversal d. Assume a║b.
[illustration: parallel lines a and b cut by transversal d, with angles numbered 1 to 8]
What is the value of x if m<5 = (2x+10)°and m<1 = (4x-2)°?
Given m<1 = (2y-10)° and m<6 =(3y+5)°, what is the value of <1?
Answer Key
- m<1 = 45°, <1 and <3 are supplementary angles
m<2 = 45°, <2 and <3 are supplementary angles
m<4 = 135°, <3 and <4 are supplementary angles
m<5 = 45°, <1 and <5 are alternate interior angles
m<6 = 45°, <2 and <6 are corresponding angles
m<7 = 135°, <3 and <7 are corresponding angles
m<8 = 135°, are corresponding angles.
WHAT I KNOW!
- A
- B
- D
- C
- A
- C
- A or D
- B
- A
- B
ACTIVITY 1
A. A
B. <1 and <5, <3 and <7, <4 and <8, <2 and <6.
C. <2 and <5, <3 and <8
D. <1 and <6, <4 and <7
E. <2 and <8, <3 and <5
ACTIVITY 2
A. 100°
B. 85°
C. 105°
D. 73°
E. 113°
ACTIVITY 3
- linear pair
- congruent
- congruent
- supplementary
- line that intersects two or more lines
ACTIVITY 4
- a. d
b. <1, <2, <5, and <6
c. <3, <4, <7 and <8
d. <1 and <3, <2 and <4, <5 and <8, <6 and <7.
e. <1 and <5, <2 and <6
f. <2 and <5, <1 and <6
g. <1 and <8, <2 and <7, <3 and <5, <4 and <6.
ASSESSMENT
- C
- A
- D
- D
- A
- D
- D
- C
- D
- A
ADDITIONAL ACTIVITIES
x=6
m<1 = 64°
<b = 72°, <a and <b are supplementary angle
<c = 108°, <a and <c are vertical angles
<d = 72°, <b and <d are vertical angles
<e = 108°, <a and <e are corresponding angles
<f = 72°, <b and <f are corresponding angles
<g = 108°, <c and <g are corresponding angles
<h = 72°, <d and <h are corresponding angles
References
Department of Education – Instructional Materials Council Secretariat (DepEd-IMCS) (2013) Mathematics – Grade 7. ISBN: 978-971-9990-60-4.
Nivera, Gladys C., et.al, Grade 7 Mathematics Patterns and Practicalities 2013. SalesianaBooks by Don Bosco Press, Inc.
Oronce, Orlando A., et.al, E-Math I Elementary Algebra Revised Edition 2010. Rex Book Store, Inc.
For inquiries or feedback, please write or call:
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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
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