kxdm8 Math 8 Q4 Week 5

NAME: ____________________________________ YR & SEC: _____________________

Competency:

The learner determines the conditions under which lines and segments are parallel or perpendicular.

(M8GE-IVe-1)

To the Learners:

Before starting the module, I want you to set aside other tasks that will disturb you while enjoying the lessons. Read the simple instructions below to successfully enjoy the objectives of this kit. Have fun!

1. Follow carefully all the contents and instructions indicated in every page of this module.

2. Writing enhances learning. Keep this in mind and take note of the important concepts in your notebook.

3. Perform all the provided activities in the module.

4. Let your facilitator/guardian assess your answers using the answer key card.

5. Analyze the post-test and apply what you have learned.

6. Enjoy studying!

Expectations

This module was designed to help you master on how to determine the conditions under which lines and segments are parallel or perpendicular.

After going through this module, you are expected to:

1. demonstrate understanding of the key concepts of parallel lines;

2. determine and prove the conditions under which lines and segments are parallel;

3. demonstrate understanding of the key concepts of perpendicular lines;

4. determine the conditions under which lines and segments are perpendicular.

Pre-test

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

1. Which statement is not true?

A. Perpendicular lines always form a right angle.

B. Lines that form a right angle are parallel lines.

C. Parallel lines are always the same distance apart.

D. Two lines are parallel if and only if they lie in the same plane and do not intersect.

2. Refer to Figure 1, a ║ d with e as the transversal. What must be true about ∠3 and ∠4 if b ║ c with e, also as the transversal?

A. ∠3 is greater than ∠4.

B. ∠3 is congruent to ∠4.

C. ∠3 is a supplement of ∠4. Figure 1 D. ∠3 is a complement of ∠4.

3. Refer to Figure 2. Lines m and n are parallel cut by transversal t which is also perpendicular to m and n. Which statement is not correct?

A. ∠1 and ∠6 are congruent.

B. ∠2 and ∠3 are supplementary.

C. ∠3 and ∠5 are congruent angles.

D. ∠1 and ∠4 form a linear pair. Figure 2

MATH 8 QUARTER 4 Week 5

4. Refer to Figure 3. What theorem proves, if a ⊥ c and b ⊥ c, then a ║ b?

A. Two lines are parallel if they do not intersect.

B. Two lines are perpendicular if they intersect at right angles.

C. In a plane, if two lines are perpendicular to the same line, the the two lines are parallel.

D. Given a line and a point on the line, there is only one line through Figure 3 the given point that is perpendicular to the given line.

5. Refer to Figure 4. 𝑙₁ and 𝑙₂ are cut by transversals m and n.

What value of x will make m ⊥ 𝑙₂?

A. 6 B. 9 C. 12 D. 15

Figure 4

Looking Back to your Lesson

From your previous lesson, you have learned the definition of parallel and transversal lines. You also studied how to prove the Theorems of parallel lines cut by a transversal. The following are the points to remember When the transversal intersects parallel lines, there are special relationships between these pair of angles.

Postulate/Theorems Examples Figure

Corresponding Angles Postulate - If two parallel lines are cut by a

transversal, then corresponding angles are congruent.

∠1 ≅ ∠5

∠2 ≅ ∠6

∠3 ≅ ∠7

∠4 ≅ ∠8 Alternate Interior Angles Theorem

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

∠3 ≅ ∠6

∠4 ≅ ∠5 Alternate Exterior Angles Theorem

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

∠1 ≅ ∠8

∠2 ≅ ∠7 Same-Side Interior Angles Theorem

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

∠3 & ∠5 are supplementary

∠4 & ∠6 are supplementary Same-Side Exterior Angles Theorem

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

∠1 & ∠7 are supplementary

∠2 & ∠8 are supplementary

Write a two-column proof of Alternate Interior Angles Theorem.

Given: m∥n; line t is a transversal line.

Prove: ∠1 ≅ ∠8

∠2 ≅ ∠7

Introduction of the Topic

Have you ever wondered how carpenters, architects and engineers design their work? What factors are being considered in making their design? The use of parallelism and perpendicularity of lines in real life necessitates the establishment of these concepts.

This module seeks to answer the question: “How can we establish parallelism or perpendicularity of lines?”

Lesson 1: Parallelism

Two lines are parallel if and only if they lie in the same plane and do not intersect. Parallel lines never cross

.

Parallel lines are marked with "feathers" (arrows) such as > or >>.

The notation to indicate parallel lines are two vertical bars ∥.

Line m being parallel to line n is written m ∥ n.

Parallel lines are always the same distance apart, which is referred to as being "equidistant".

For our purposes, parallel lines will always be straight lines that go on indefinitely.

The parallel lines may be horizontal, vertical or slanted.

When lines intersect, a series of angles are formed. Certain angles are given specific "names" based upon their locations in relation to the lines. These specific names may be used whether the lines involved are parallel or not parallel.

To prove that the two lines or segments are parallel, you must show that one of the following is true.

Converse of Corresponding Angles Postulate

- If two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel.

Converse of Alternate Interior Angles Theorem

- If two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.

Given: Lines r and s are cut by a transversal line t;

∠3 ≅ ∠6 Prove: r∥s

Proof:

Statements Reasons

1. Lines r and s are cut by a

transversal line t; ∠3 ≅ ∠6 Given

2. ∠6 & ∠7 are vertical angles. Definition of Vertical Angles

3. ∠6 ≅ ∠7 Vertical Angles Theorem

4. ∠3 ≅ ∠7 Transitive Property of Congruence

5. r∥s Converse of Corresponding Angles Postulate

Converse of Alternate Exterior Angles Theorem

- If two lines are cut by a transversal so that a pair of alternate exterior angles is congruent, then the lines are parallel.

Given: Lines r and s are cut by a transversal line t;

∠1 ≅ ∠8

Prove: r∥s

Proof:

Statements Reasons

1. Lines r and s are cut by a

transversal line t; ∠1 ≅ ∠8 Given

2. ∠5 & ∠8 are vertical angles. Definition of Vertical Angles

3. ∠5 ≅ ∠8 Vertical Angles Theorem

4. ∠1 ≅ ∠5 Transitive Property of Congruence

5. r∥s Converse of Corresponding Angles Postulate

Converse of Same-Side Interior Angles Theorem

- If two lines are cut by a transversal so that a pair of same-side interior angles is supplementary, then the lines are parallel.

Given: Lines r and s are cut by a transversal line t;

∠3 𝑎𝑛𝑑∠5 are supplementary.

Prove: r∥s Proof:

Statements Reasons

1. Lines r and s are cut by a transversal line t; ∠3 𝑎𝑛𝑑∠5 are supplementary.

Given

2. 𝑚∠3 + 𝑚 ∠5 = 180° Definition of Supplementary Angles 3. ∠5 and ∠6 form a linear pair. Definition of Linear Pair

4. ∠5 and ∠6 are supplementary. Linear Pair Postulate

5. 𝑚∠5 + 𝑚 ∠6 = 180° Definition of Supplementary Angles 6. 𝑚∠3 + 𝑚 ∠5 = 𝑚∠5 + 𝑚 ∠6 Transitive Property of Equality 7. 𝑚∠3 = 𝑚 ∠6 Addition Property of Equality

8. ∠3 ≅ ∠6 Definition of Congruent Angles

9. r∥s Converse of Alternate Interior Angles Theorem

Converse of Same-Side Exterior Angles Theorem

- If two lines are cut by a transversal so that a pair of same-side exterior angles is supplementary, then the lines are parallel.

Given: Lines r and s are cut by a transversal line t;

∠1 𝑎𝑛𝑑∠7 are supplementary.

Prove: r∥s Proof:

Statements Reasons

1. Lines r and s are cut by a transversal line t; ∠1 𝑎𝑛𝑑∠7 are supplementary.

Given

2. 𝑚∠1 + 𝑚 ∠7 = 180° Definition of Supplementary Angles 3. ∠7 and ∠8 form a linear pair. Definition of Linear Pair

4. ∠7 and ∠8 are supplementary. Linear Pair Postulate

5. 𝑚∠7 + 𝑚 ∠8 = 180° Definition of Supplementary Angles 6. 𝑚∠1 + 𝑚 ∠7 = 𝑚∠7 + 𝑚 ∠8 Transitive Property of Equality 7. 𝑚∠1 = 𝑚 ∠8 Addition Property of Equality

8. ∠1 ≅ ∠8 Definition of Congruent Angles

9. r∥s Converse of Alternate Exterior Angles Theorem

Example 1. Apply the Converse

Find the value of x so that 𝑐 ∥ 𝑠.

Solution:

(7𝑥 − 24)° and (6𝑥 − 5)° are measures of two alternate exterior angles. If they are equal it follows that the alternate exterior

angles are congruent and 𝑐 ∥ 𝑠.

(7𝑥 − 24)° = (6𝑥 − 5)°

7𝑥 − 24 = 6𝑥 − 5 7𝑥 − 6x = 24 − 5 𝑥 = 19

Answer: If 𝑥 = 19, then 𝑐 ∥ 𝑠.

Example 2. Proving Lines are Parallel Write a two-column proof that shows 𝑚 ∥ 𝑒.

Given: 𝑠 ∥ 𝑏 ∠7 ≅ ∠10 Prove: 𝑚 ∥ 𝑒 Proof:

Statements Reasons

1. 𝑠 ∥ 𝑏

∠7 ≅ ∠10 Given

2. ∠2 and ∠10 are corresponding

angles Definition of Corresponding Angles

3. ∠2 ≅ ∠10 Corresponding Angles Postulate

4. ∠2 ≅ ∠7 Transitive Property of Congruence

5. 𝑚 ∥ 𝑒 Converse of Alternate Interior Angles Theorem

Lesson 1: Perpendicularity

Two lines are perpendicular if and only if they form a right angle.

Perpendicular lines always form a right angle of 90º. While perpendicular lines may appear in a slanted or a horizontal position on paper, the lines must always intersect to form a right angle.

(slanted positions) (horizontal/vertical positions)

Perpendicular lines are marked with a "square" (a box) drawn at the point of intersection.

The notation to indicate perpendicular lines is ⊥. Line m being perpendicular n is written m ⊥ n.

Segments and rays can also be perpendicular. A perpendicular bisector of a segment is a line or a ray or another segment that is perpendicular to the segment and intersects the segment at its midpoint. The distance between two parallel lines is the perpendicular distance between one of the lines and any point on the other line.

To prove that two lines are perpendicular, you must show that one of the following is true:

1. If two lines are perpendicular to each other, then they form four right angles.

If 𝑚 ⊥ 𝑛, then ∠1, ∠2, ∠3 and ∠4 are right angles.

2. If the angles in a linear pair are congruent, then the lines containing their sides are perpendicular.

If ∠1 and ∠2 form a linear pair and ∠1 ≅ ∠2,

then 𝑙₁⊥ 𝑙₂.

3. If two angles are adjacent and complementary, the non-common sides are perpendicular.

If ∠𝐶𝐴𝑅 and ∠𝐸𝐴𝑅 are complementary and adjacent, then 𝐴𝐶̅̅̅̅ ⊥ 𝐴𝐸̅̅̅̅.

Example 1. Find the value of x given that 𝒍 ⊥ 𝒂 Solution:

The two angles are congruent and form a linear pair.

(7𝑥 − 1)° + (5𝑥 + 25)° = 180°

7𝑥 + 5𝑥 = 180 + 1 − 25 12𝑥 = 156

𝑥 = 13 Answer: If 𝑥 = 13, then 𝑙 ⊥ 𝑎.

Example 2. Proving Segments are Perpendicular Write a two-column proof that shows 𝐴𝑁̅̅̅̅ ⊥ 𝐼𝑇̅̅̅.

Given: 𝐴𝐼̅̅̅ ≅ 𝐴𝑇̅̅̅̅

∠𝑁𝐴𝐼 ≅ ∠𝑁𝐴𝑇 Prove: 𝐴𝑁̅̅̅̅ ⊥ 𝐼𝑇̅̅̅

Proof:

Statements Reasons

1. 𝐴𝐼̅̅̅ ≅ 𝐴𝑇̅̅̅̅

∠𝑁𝐴𝐼 ≅ ∠𝑁𝐴𝑇 Given

2. 𝐴𝑁̅̅̅̅ ≅ 𝐴𝑁̅̅̅̅ Reflexive Property of Congruence

3. ∆𝐴𝑁𝐼 ≅ ∆𝐴𝑁𝑇 SAS Congruence Postulate

4. ∠𝐴𝑁𝐼 ≅ ∠𝐴𝑁𝑇 CPCTC

5. 𝐴𝑁̅̅̅̅ ⊥ 𝐼𝑇̅̅̅ If the angles in a linear pair are congruent, then the lines containing their sides are perpendicular.

Activities

Activity 1

Given the following information, determine which lines, if any, are parallel. State the postulates or theorems that support your answer.

1. ∠13 ≅ ∠6 ___________________________________

2. ∠8 ≅ ∠16 ___________________________________

3. ∠5 ≅ ∠7 ____________________________________

4. 𝑚∠7 + 𝑚∠11 = 180° _________________________

5. 𝑚∠3 + 𝑚∠8 = 180° __________________________

Activity 2

A. Find the value of x so that 𝑠 ∥ 𝑐.

1. 2.

B. Find the value of x so that 𝑗 ⊥ 𝑟.

1. 2.

Activity 3

Fill the missing statements and reasons.

Given: 𝑝 ⊥ 𝑗 𝑛 ⊥ 𝑗 Prove: 𝑝 ∥ 𝑛

Proof:

Statements Reasons

1. Given

∠1 is a right angle. 2.

3. Definition of Perpendicular Lines

∠1 ≅ ∠2 4.

𝑝 ∥ 𝑛 5.

Remember

Two lines are parallel if and only if they lie in the same plane and do not intersect. Parallel lines never cross.

To prove that the two lines or segments are parallel, you must show that one of the following is true.

Converse of Corresponding Angles Postulate

- If two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel.

Converse of Alternate Interior Angles Theorem

- If two lines are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.

Converse of Alternate Exterior Angles Theorem

- If two lines are cut by a transversal so that a pair of alternate exterior angles is congruent, then the lines are parallel.

Converse of Same-Side Interior Angles Theorem

- If two lines are cut by a transversal so that a pair of same-side interior angles is supplementary, then the lines are parallel.

Converse of Same-Side Exterior Angles Theorem

- If two lines are cut by a transversal so that a pair of same-side exterior angles is supplementary, then the lines are parallel.

Two lines are perpendicular if and only if they form a right angle.

To prove that two lines are perpendicular, you must show that one of the following is true:

1. If two lines are perpendicular, then they form four right angles.

2. If the angles in a linear pair are congruent, then the lines containing their sides are perpendicular.

3. If two angles are adjacent and complementary, the non-common sides are perpendicular.

Check your Understanding

A. Lines p and q are cut by a transversal, find the value of x that makes p ∥ q.

1. ∠4 and ∠5 are alternate interior angles, m∠4 = 5x − 11 and m ∠5 = 3𝑥 + 7.

2. ∠4 and ∠6 are same side interior angles, m∠4 = 6x + 3 and m ∠6 = 4𝑥 + 7.

3. ∠2 and ∠6 are corresponding angles, m∠2 = 3x − 15 and m ∠6 = 𝑥 + 55.

B. Find the value of x so that m⊥n.

1. 2.

C. Write a two-column proof for each.

1. Given: 𝑙 ∥ 𝑜

∠8 ≅ ∠7 Prove: 𝑎 ∥ 𝑑

Proof:

Statements Reasons

Post Test

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

1. Which statement is not true?

A. Perpendicular lines always form a right angle.

B. Lines that form a right angle are parallel lines.

C. Parallel lines are always the same distance apart.

D. Two lines are parallel if and only if they lie in the same plane and do not intersect.

2. Refer to Figure 1, a ║ d with e as the transversal. What must be true about ∠3 and ∠4 if b ║ c with e, also as the transversal?

A. ∠3 is greater than ∠4.

B. ∠3 is congruent to ∠4.

C. ∠3 is a supplement of ∠4. Figure 1 D. ∠3 is a complement of ∠4.

3. Refer to Figure 2. Lines m and n are parallel cut by transversal t which is also perpendicular to m and n. Which statement is not correct?

A. ∠1 and ∠6 are congruent.

B. ∠2 and ∠3 are supplementary.

C. ∠3 and ∠5 are congruent angles. Figure 2 D. ∠1 and ∠4 form a linear pair.

4. Refer to figure 3. What theorem proves, if a ⊥ c and b ⊥ c, then a ║ b?

A. Two lines are parallel if they do not intersect.

B. Two lines are perpendicular if they intersect at right angles.

C. In a plane, if two lines are perpendicular to the same line, the the two lines are parallel.

D. Given a line and a point on the line, there is only one line through Figure 3 the given point that is perpendicular to the given line.

5. Refer to Figure 4. 𝑙₁ and 𝑙₂ are cut by transversals m and n.

What value of x will make m ⊥ 𝑙₂?

A. 6 B. 9 C. 12 D. 15

Figure 4

Additional Activities

To understand more the lesson, visit the following website:

- https://www.khanacademy.org/math/basic-geo/basic-geo-lines/parallel-perp/v/parallel-and- perpendicular-lines-intro

- http://www.nbisd.org/users/0006/docs/Textbooks/Geometry/geometrych3.pdf - http://www.regentsprep.org/Regents/math/geometry/GP8/PracParallel.htm

Reflection

You have accomplished the task successfully. This shows that you learned the important concepts in this module. To end this lesson meaningfully and to welcome you to the next module, I want you to

accomplish this.

- In this unit I learned about … - These concepts can be used in … - I understand that …

References

Reference Books

Advincula R.C., Gonowon, R.R. (2019). Global Mathematics 8, K to 12 Edition. Quezon City:

The Library Publishing House, Inc.

Oronce O.A., Mendoza, M.O. (2018). Exploring Mathematics 8. Sampaloc, Manila:

Rex Book Store, Inc.

Alferez, G.S., Alferez, M.S. and Lambino, A.E. (2007). MSA Geometry. Quezon City:

MSA Publishing House.

Nivera G.C., Buzon, O.N. and Lapinid, M.C. (2007). Geometry: Patterns and Practicalities. Manila:

Don Bosco Press. Inc.

Website Links

http://www.nbisd.org/users/0006/docs/Textbooks/Geometry/geometrych3.pdf http://www.regentsprep.org/Regents/math/geometry/GP8/PracParallel.htm https://mathbitsnotebook.com/Geometry/ParallelPerp/PPdefinitions.html

http://siprep.org/faculty/pmaychrowitz/documents/SI-Geometry-Honors-Edition-Chp3.pdf

https://www.khanacademy.org/math/basic-geo/basic-geo-lines/parallel-perp/v/parallel-and-perpendicular- lines-intro

Answer Key

Activity 1

1. Lines are not parallel.

2.

ℎ ∥ 𝑚 − Converse of Corresponding Angles Postulate

3.

𝑘 ∥ 𝑝 − Converse of Corresponding Angles Postulate

4.

ℎ ∥ 𝑚 −

Converse of Same-Side Interior Angles Theorem 5. Lines are not parallel.

Pretest/Post Test 1. B

2. B 3. D 4. C 5. D

Activity 2

A. B.

1.

𝑥 = 11 1. 𝑥 = 8

2.

𝑥 = 12²₃

2. 𝑥 = 20

Activity 3 1.

𝑝 ⊥ 𝑗; 𝑛 ⊥ 𝑗

2.

Definition of Perpendicular lines

3.

∠2 is a right angle.

4. All right angles are congruent 5. Converse of Corresponding

Angles Postulate

Check your Understanding A. B.

1.

𝑥 = 9

1.

𝑥 = 14

2.

𝑥 = 17

2.

𝑥 = 15

3.

𝑥 = 35

C.

Statements Reasons

1. 𝑙 ∥ 𝑜 ∠8 ≅ ∠7

Given

2. ∠6 and ∠8 are supplementary angles Same-Side Interior Angles Theorem 3. 𝑚∠6 + 𝑚∠8 = 180° Definition of Supplementary Angles

4. 𝑚∠8 = 𝑚∠7 Definition of Congruent Angles

5. 𝑚∠6 + 𝑚∠7 = 180° Substitution

6. ∠6 and ∠7 are supplementary angles Definition of Supplementary Angles

7. 𝑎 ∥ 𝑑 Converse of Same-Side Interior Angles

Theorem