kxdm8 Math 8 Q4 Week 4 1

ERICA RIO G. SIOSON

NAME: ____________________________________ YR & SEC: _____________________

Competency:

The learner proves properties of parallel lines cut by a transversal (M8GE-IVd-1).

To the Learners:

Before you start with this module, you need to set aside other tasks that will cause disturbance while enjoying the lessons. Read the instructions below to understand clearly the objectives of this module. Enjoy learning, Have a good day!

1. Follow the directions carefully before applying definite actions to avoid corrections.

2. Prepare your pen and paper to jot down terms which needed clarifications.

3. Answer all the activities/tests with honesty.

4. Don’t hesitate to ask your parents/guardian/teacher if you have any questions.

5. Be motivated and enthusiastic while studying. Enjoy learning!

Expectations

This module is designed to help you understand how to prove properties of parallel lines cut by a transversal.

After going through this module, you are expected to:

• Identify the different angles formed by parallel lines cut by a transversal.

• Find the measures of angles formed by parallel lines cut by a transversal.

• Prove properties of parallel lines cut by a transversal.

• Demonstrate appreciation about angles formed by parallel lines cut by a transversal by relating it to real- life situations.

Directions: Read and analyse each item/question carefully. Choose the letter of the correct answer. Write your answer on the blank before each number

_____1. Which of the following statement is true about parallel lines cut by a transversal?

A. Corresponding angles are supplementary.

B. Alternate exterior angles are congruent.

C. Same-side interior angles are complementary.

D. Alternate interior angles are supplementary.

_____2. In the figure shown, what is the relationship between ∠4 and ∠5?

A. ∠4 and ∠5 are corresponding angles and they are ≅.

B. ∠4 and ∠5 are alternate interior angles and they are ≅.

C. ∠4 and ∠5 are alternate exterior angles and they are ≅.

D. ∠4 and ∠5 are same-side interior angles and they are supplementary.

_____3. Parallel lines 𝑚 and 𝑛 are cut by transversal 𝑡. If 𝑚∠4 = 135°, what is the measure of ∠5?

A. 135° B. 145° C. 225° D. 45°

_____4. Which of the following reasons should be used to prove statement number 3?

Given: 𝑎 ∥ 𝑏 Prove: ∠3 ≅ ∠5

Pre-test

Mathematics 8 Quarter 4 Week 4

STATEMENT REASON 1. 𝑎 ∥ 𝑏 1. Given

2. ∠3 ≅ ∠1 2. Vertical Angle Theorem 3. ∠1 ≅ ∠5 3.

4. ∠3 ≅ ∠5 4. Transitive Property of Congruence A. Corresponding Angles Postulate

B. Alternate Interior Angles Theorem C. Alternate Exterior Angles Theorem D. Same-side Interior Angles Theorem

_____5. Which of the following reasons should be used to prove statement number 5?

Given: 𝑙₁∥ 𝑙₂

Prove: 𝑚∠𝑑 + 𝑚∠𝑒 = 180°

STATEMENT REASON

1. 𝑙₁∥ 𝑙₂ 1. Given

2. ∠𝑑 ≅ ∠ℎ 2. Corresponding Angles Postulate 3. 𝑚∠ℎ + 𝑚∠𝑒 = 180° 3. Linear Pair Postulate

4. 𝑚∠𝑑 = 𝑚∠ℎ 4. Definition of Congruent Angles 5. 𝑚∠𝑑 + 𝑚∠𝑒 = 180° 5.

A. Transitive Property of Congruence B. Associative Property of Equality C. Identity Property of Equality D. Substitution Property of Equality

Looking Back to your Lesson

Before you proceed to the main lesson, you’ll have to recall the following pairs of angles and their properties that you will use for this lesson.

PAIR OF ANGLES ILLUSTRATION PROPERTIES

Two angles are supplementary if the sum of their measures is 180°

If two angles are

supplementary, then the sum of their measures is 180.

Vertical angles are nonadjacent angles formed by two

intersecting lines.

If two angles are vertical, then they are congruent.

If two angles are adjacent and supplementary, then they are called a linear pair.

If the sum of two adjacent angles is 180, then these angles form a linear pair.

Introduction of the Topic

A transversal is a line that intersects coplanar lines, each at a different point.

Parallel lines are coplanar lines which never intersect and have the same direction. They do not intersect because they are equidistant.

ERICA RIO G. SIOSON

When parallel lines are cut by a transversal, several pairs of angles are formed based upon the locations in relation to the line.

These pairs of angles are…

1.CORRESPONDING ANGLES

- the pair of angles found on the same side of the transversal, one interior and one exterior, but not adjacent.

∠1 𝑎𝑛𝑑 ∠5 are corresponding angles.

∠4 𝑎𝑛𝑑 ∠8 are corresponding angles.

∠2 𝑎𝑛𝑑 ∠6 are corresponding angles.

∠3 𝑎𝑛𝑑 ∠7 are corresponding angles.

When lines 𝑎 and 𝑏 are parallel, the measures of the corresponding angles are equal.

𝑚∠1 ≅ 𝑚∠5 and 𝑚∠4 ≅ 𝑚∠8 𝑚∠2 ≅ 𝑚∠6 and 𝑚∠3 ≅ 𝑚∠7

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

2.ALTERNATE INTERIOR ANGLES

- the pair of angles found in the “interior” (between the parallel lines) and “alternate” sides of the transversal.

∠4 𝑎𝑛𝑑 ∠6 are alternate interior angles.

∠3 𝑎𝑛𝑑 ∠5 are alternate interior angles.

When lines 𝑎 and 𝑏 are parallel, the measures of the alternate interior angles are equal.

𝑚∠4 ≅ 𝑚∠6 and 𝑚∠3 ≅ 𝑚∠5 Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

3.ALTERNATE EXTERIOR ANGLES

- the pair of angles found in the “exterior” (outside the parallel lines) and “alternate” sides of the transversal.

∠1 𝑎𝑛𝑑 ∠7 are alternate exterior angles.

∠2 𝑎𝑛𝑑 ∠8 are alternate exterior angles.

When lines 𝑎 and 𝑏 are parallel, the measures of the alternate exterior angles are equal.

𝑚∠1 ≅ 𝑚∠7 and 𝑚∠2 ≅ 𝑚∠8 Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.

4.SAME-SIDE INTERIOR ANGLES

- the pair of angles found in the “interior” (between the parallel lines) and on the same side of the transversal.

∠4 𝑎𝑛𝑑 ∠5 are same-side interior angles.

∠3 𝑎𝑛𝑑 ∠6 are same-side interior angles.

When lines 𝑎 and 𝑏 are parallel, the measures of the interior angles on the same side of the transversal are supplementary.

𝑚∠4 + 𝑚∠5 = 180° and 𝑚∠3 + 𝑚∠6 = 180°

Same-side Interior Angles Theorem

If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.

5.SAME-SIDE EXTERIOR ANGLES

- the pair of angles found in the “exterior” (outside the parallel lines) and on the same side of the transversal.

∠1 𝑎𝑛𝑑 ∠8 are same-side exterior angles.

∠2 𝑎𝑛𝑑 ∠7 are same-side exterior angles.

When lines 𝑎 and 𝑏 are parallel, the measures of the exterior angles on the same side of the transversal are supplementary.

𝑚∠1 + 𝑚∠8 = 180 and 𝑚∠2 + 𝑚∠7 = 180 Same-side Exterior Angles Theorem

If two parallel lines are cut by a transversal, then same-side exterior angles are supplementary.

PROVING PROPERTIES OF PARALLEL LINES CUT BY A TRANSVERSAL

In this section, you will be able to apply the properties of parallel lines cut by a transversal to prove the following pairs of angles.

CORRESPONDING ANGLES: In the figure shown, 𝑙₁𝑎𝑛𝑑 𝑙₂ are parallel lines cut by a transversal 𝑡. Prove that ∠1 ≅ ∠5.

ALTERNATE INTERIOR ANGLES: The photo below shows an example of a food for breakfast. It shows how two parallel lines, 𝑎 and 𝑏 are cut by a transversal 𝑡. Prove that ∠4 ≅ ∠6.

ALTERNATE EXTERIOR ANGLES: Study the photo below. If line 𝑚 and line 𝑛 are parallel and cut by a transversal 𝑡, prove that ∠2 ≅ ∠8.

STATEMENT REASON

1. 𝑙₁∥ 𝑙₂ 1. Given

2. ∠5 ≅ ∠7 2. Vertical Angles Theorem

3. ∠7 ≅ ∠1 3. Alternate Exterior Angles Theorem 4. ∠1 ≅ ∠5 4. Transitive Property of Congruence

STATEMENT REASON

1. 𝑎 ∥ 𝑏 1. Given

2. ∠4 ≅ ∠8 2. Corresponding Angles Postulate 3. ∠8 ≅ ∠6 3. Vertical Angles Theorem

4. ∠4 ≅ ∠6 4. Transitive Property of Congruence

STATEMENT REASON

1. 𝑚 ∥ 𝑛 1. Given

2. ∠2 ≅ ∠4 2. Vertical Angles Theorem

3. ∠4 ≅ ∠8 3. Corresponding Angles Postulate 4. ∠2 ≅ ∠8 4. Transitive Property of Congruence

If

ERICA RIO G. SIOSON

SAME-SIDE INTERIOR ANGLES: The photo below shows a part of a stair. It also shows that line 𝑙₁ is parallel to 𝑙₂ and cut by a transversal 𝑠. Prove that 𝑚∠𝑒 + 𝑚∠𝑑 = 180°.

SAME-SIDE EXTERIOR ANGLES: The photo below shows a person reading a bible. If lines 𝑎 and 𝑏 are parallel and cut by a transversal 𝑡, prove that 𝑚∠1 + 𝑚∠8 = 180°.

Activities

DIRECTIONS: Identify the locations of the crayons scattered on top of a table. Shade the shapes before the names of pairs of angles that correspond to the color of the crayons.

Corresponding Angles Alternate Exterior Angles Same-side Exterior Angles

Alternate Interior Angles Same-side Interior Angles

STATEMENT REASON

1. 𝑙₁∥ 𝑙₂ 1. Given

2. ∠𝑎 ≅ ∠𝑒 2. Corresponding Angles Postulate 3. 𝑚∠𝑎 + 𝑚∠𝑑 = 180° 3. Linear Pair Postulate

4. 𝑚∠𝑎 = 𝑚∠𝑒 4. Definition of Congruent Angles 5. 𝑚∠𝑒 + 𝑚∠𝑑 = 180° 5. Substitution Property of Equality

STATEMENT REASON

1. 𝑎 ∥ 𝑏 1. Given

2. ∠1 ≅ ∠5 2. Corresponding Angles Postulate 3. 𝑚∠5 + 𝑚∠8 = 180° 3. Linear Pair Postulate

4. 𝑚∠1 = 𝑚∠5 4. Definition of Congruent Angles 5. 𝑚∠1 + 𝑚∠8 = 180° 5. Substitution Property of Equality

If

DIRECTIONS: Cut out the emoticons found at the last page of this module and paste it on the circle that corresponds to the answers in the sentences that follow.

A happy emoticon and a 140° angle are corresponding angles. They are congruent.

A wink emoticon and a 60° angle are alternate exterior angles. They are congruent.

A wow emoticon and a 140° angle are alternate interior angles. They are congruent.

A blessed emoticon and a 40° angle are same-side exterior angles. They are supplementary.

A blissful emoticon and a 120° angle are same-side interior angles. They are supplementary.

The photo below shows the dispersion of each mobile legend heroes on the first part of the game.

Help Layla in proving that the given heroes are in Congruent angular location along the map, given that lines 𝑎 and 𝑏 are parallel cut by transversal 𝑡.

PROVE: ≅

STATEMENT REASON

1. 𝑎 ∥ 𝑏 1.

2.

≅

2.

3.

≅

3.

4.

≅

4.

ERICA RIO G. SIOSON

Remember

Your conjectures in this module can be summarized below.

If two parallel lines are cut by a transversal, then

… corresponding angles are congruent.

… alternate interior angles are congruent.

… alternate exterior angles are congruent.

… same-side interior angles are supplementary.

… same-side exterior angles are supplementary.

Check your Understanding

Find the measures of unknown angles and explain your reasoning.

1. 2. 3.

Post-test

Directions: Read and analyse each item/question carefully. Choose the letter of the correct answer. Write your answer on the blank before each number.

_____1. Which of the following statement is true about parallel lines cut by a transversal?

A. Corresponding angles are supplementary.

B. Alternate exterior angles are congruent.

C. Same-side interior angles are complementary.

D. Alternate interior angles are supplementary.

_____ 2. In the figure below, line 𝑎 and line 𝑏 are parallel cut by a transversal 𝑡. Which of the following are corresponding angles?

A. ∠1 𝑎𝑛𝑑 ∠3, ∠2 𝑎𝑛𝑑 ∠4 B. ∠4 𝑎𝑛𝑑 ∠6, ∠3 𝑎𝑛𝑑 ∠5 C. ∠4 𝑎𝑛𝑑 ∠8, ∠3 𝑎𝑛𝑑 ∠7 D. ∠1 𝑎𝑛𝑑 ∠7, ∠2 𝑎𝑛𝑑 ∠8

_____3. Parallel lines 𝑚 and 𝑛 are cut by transversal 𝑡. If 𝑚∠4 = 135°, what is the measure of ∠5?

A. 135° B. 145° C. 225° D. 45°

_____4. In the figure shown, what is the relationship between ∠4 and ∠5?

A. ∠4 and ∠5 are corresponding angles and they are ≅.

B. ∠4 and ∠5 are alternate interior angles and they are ≅.

C. ∠4 and ∠5 are alternate exterior angles and they are ≅.

D. ∠4 and ∠5 are same-side interior angles and they are supplementary.

_____5. In the given figure, which pair of angles are same-side exterior angles?

A. ∠4 and ∠6 B. ∠1 and ∠7 C. ∠2 and ∠7

D. ∠3 and ∠6

_____6. Which of the following statements is always true?

A. All intersecting lines are parallel.

B. Parallel lines are equidistant.

C. Lines that intersect and form a right angle are parallel lines.

D. Skew lines are parallel lines.

_____7. In the figure below, 𝑎 ∥ 𝑏 and 𝑡 is a transversal. Which angles are congruent to ∠4?

A. ∠1, ∠6, 𝑎𝑛𝑑 ∠8 B. ∠2, ∠5, 𝑎𝑛𝑑 ∠7 C. ∠3, ∠5, 𝑎𝑛𝑑 ∠7 D. ∠2, ∠6, 𝑎𝑛𝑑 ∠8

_____8. If 𝑙₁∥ 𝑙₂ , which of the following statements is/are true?

I. ∠𝑎 ≅ ∠𝑒

II. ∠𝑓 𝑎𝑛𝑑 ∠ℎ 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 III. ∠𝑏 𝑎𝑛𝑑 ∠𝑔 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 IV. ∠𝑑 ≅ ∠𝑓

A. I only B. II and III only C. I, III, and IV only D. I, II, and III only

_____9. Which of the following reasons should be used to prove statement number 3?

Given: 𝑎 ∥ 𝑏 Prove: ∠3 ≅ ∠5

STATEMENT REASON

1. 𝑎 ∥ 𝑏 1. Given

2. ∠3 ≅ ∠1 2. Vertical Angle Theorem

3. ∠1 ≅ ∠5 3.

4. ∠3 ≅ ∠5 4. Transitive Property of Congruence A. Corresponding Angles Postulate

B. Alternate Interior Angles Theorem C. Alternate Exterior Angles Theorem D. Same-side Interior Angles Theorem

_____10. Which of the following reasons should be used to prove statement number 5?

Given: 𝑙₁∥ 𝑙₂

Prove: 𝑚∠𝑑 + 𝑚∠𝑒 = 180°

STATEMENT REASON

1. 𝑙₁∥ 𝑙₂ 1. Given

2. ∠𝑑 ≅ ∠ℎ 2. Corresponding Angles Postulate 3. 𝑚∠ℎ + 𝑚∠𝑒 = 180° 3. Linear Pair Postulate

4. 𝑚∠𝑑 = 𝑚∠ℎ 4. Definition of Congruent Angles 5. 𝑚∠𝑑 + 𝑚∠𝑒 = 180° 5.

A. Transitive Property of Congruence B. Associative Property of Equality C. Identity Property of Equality D. Substitution Property of Equality

ERICA RIO G. SIOSON

Additional Activities

A. FINDING MEASURES OF NUMBERED ANGLES

The photo on the left shows the portion of LRT along Edsa-Munoz. Use the figure to find the measures of numbered angles.

B. This photo shows an aerial view of an intersection in a certain city. It also shows two parallel lines 𝑟 and 𝑠 cut by a transversal 𝑡. Prove that ∠7 ≅ ∠8.

Reflection :

For EMOJINGLES ACTIVITY

STATEMENT REASON

1. 𝑟 ∥ 𝑠 1. Given

2. ∠7 ≅ ∠6 2.

3. ∠6 ≅ ∠8 3.

4. ∠7 ≅ ∠8 4. Transitive Property of Congruence