Mathematics 8 Quarter 4 Module 5: Conditions for Parallel or Perpendicular Lines and Segments

Mathematics Quarter 4 – Module 5: Determining the Conditions under which Lines and Segments are Parallel or Perpendicular

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Mathematics

Quarter 4 – Module 5:

Determining the Conditions under which Lines and Segments are Parallel or Perpendicular

[illustration: learners studying outdoors and at home, a teacher with a learner, and a globe on a table]

CO_Q4_Mathematics 8_Module 5

Mathematics – Grade 8 Alternative Delivery Mode Quarter 4 – Module 5: Determining the Conditions under which Lines and Segments are Parallel or Perpendicular First Edition, 2019

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

Printed in the Philippines __________________________

Department of Education – Caraga Region

Office Address: Learning Resource Management Section (LRMS), J.P. Rosales Avenue, Butuan City, Philippines 8600 Telefax Nos.: (085) 342-8207 / (085) 342-5969 E-mail Address: caraga@deped.gov.ph

Development Team of the Module

Writers: Ruth Ann Maglasang and Nilbeth S. Merano Language Editor: Victorino S. Nimes Content Editors: Merjorie G. Dalagan, Rosalita A. Bastasa, and Fritch A. Paronda Lay-out Editor: Devina P. Malinao Reviewers: Rhea J. Yparraguirre, Villaflor D. Edillor, Shirley C. Compe, Imelda B. Llasos, Caren L. Alamin, and Jayrold V. Navarra Illustrators: Ruth Ann Maglasang and Fritch A. Paronda Layout Artists: Ruth Ann Maglasang and Erwin J. Etoc Management Team: Francis Cesar B. Bringas, Isidro M. Biol, Jr., Maripaz F. Magno, Josephine Chonie M. Obseñares, Josita B. Carmen, Celsa A. Casa, Regina Euann A. Puerto, Bryan L. Arreo, Elnie Anthony P. Barcena, Leopardo P. Cortes, Jr., Claire Ann P. Gonzaga

8

Mathematics

Quarter 4 – Module 5:

Determining the Conditions under which Lines and Segments are Parallel or Perpendicular

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

[illustration: hand pointing with a finger in a circle]

This module was designed and written with you in mind. It is here to help you master the skills in determining the conditions under which lines and segments are parallel or perpendicular. You are provided with varied activities to process the knowledge and skills learned and to deepen and transfer your understanding of the lesson. The scope of this module enables you to use it in many different learning situations. The lesson is 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.

This module contains:

Lesson 1- Determining the Conditions under which Lines and Segments are Parallel or Perpendicular.

After going through this module, you are expected to:

1. Identify the properties of parallel lines, perpendicular lines, and quadrilaterals;
2. prove the conditions under which lines and segments are parallel or perpendicular;
3. use the properties of parallel lines, perpendicular lines, and quadrilaterals to find the measures of angles and sides involving quadrilaterals; and
4. apply properties of parallel and perpendicular lines to real life situations.

What I Know

[illustration: hand holding a pencil in a circle]

PRE-ASESSMENT

Directions: Answer each of the following items accurately. Write the letter of the best answer on a separate sheet of paper. You may skip the module if you get 100% correct answers, otherwise, proceed.

For items 1 – 2, refer to the characteristics below.

I. Lines are coplanar. II. Lines are non-coplanar. III. Lines do not intersect. IV. Lines intersect and form right angles.

1. Which of the following statements above is/are NOT true about two parallel lines?
   A. I only B. I and III only C. III only D. II and IV only

   
   

2. Which is/are true about perpendicular lines?
   A. II only B. II and IV only C. I and IV only D. II and III only

   
   

For items 3 – 6, refer to the figure at the right.

[chart: diagram; figure for items 3 – 6, two horizontal lines labeled x and y cut by two transversals labeled w and z. Labels as printed: w, z, x, y; a, b, c, d (near line x); 1, 2, 3, 4 (near transversal z and line x); e, f, g [?], h (near line y); 5, 6, 7, 8 (near transversal z and line y)]

3. If x ∥ y, which pair of angles are supplementary?
   A. ∠2 and ∠7 C. ∠2 and ∠3
   B. ∠b and ∠4 D. ∠b and ∠c

   
   

4. If x ∥ y, which pair of angles are congruent?
   A. ∠d and ∠h C. ∠c and ∠f
   B. ∠f and ∠6 D. ∠4 and ∠5

   
   

5. What is m∠2 if x ∥ y, m∠3 = 3x − 15° and m∠5 = x + 55°?
   A. 75° B. 80° C. 85° D. 90°

   
   

6. If m∠f = m∠8, which statement is true?
   A. x ∥ y B. x ⊥ y C. w ∥ z D. w ⊥ z

   
   

7. Which of the following statements is true about trapezoid?
   A. A trapezoid has four right angles.
   B. A trapezoid has four congruent sides.
   C. A trapezoid has pairs of opposite sides parallel.
   D. A trapezoid has exactly one pair of opposite sides parallel.

   
   

8. If □MATH is a rhombus and m∠A = 88°, what is the measure of ∠H?
   A. 45° B. 87° C. 88° D. 92°

   
   

CO_Q4_Mathematics 8_Module 5

9. Which of the following are properties of a rhombus?
   I. All sides are congruent. IV. Diagonals are congruent.
   II. All angles are right angles. V. No sides are parallel.
   III. Diagonals are perpendicular to each other.

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

10. Anna was asked by her teacher to find the perimeter of the floor of their classroom with a length of 18 m and a width of 12 m. Is Anna correct when she got a perimeter of 60 m?
    A. No, the perimeter of the floor of the classroom should be 216 m.
    B. Yes, since the floor of the classroom is a quadrilateral.
    C. Yes, since the floor of the classroom is a rectangle.
    D. No, the perimeter should be 30 m.

    
    

11. Given the parallelogram FARM, if m∠M = 42°, what is the measure of ∠F?
    A. 138° B. 128° C. 48° D. 42°

    
    

For items 12 – 15, refer to the figure at the right and complete the proof on the table. Write the letter of the correct answer from the list of statements below.

A. m∠4 + m∠3 = 180° B. m∠1 = m∠4 C. Vertical Angle Theorem D. Interior Angles Same Side-Parallel Theorem E. Substitution Property of Equality F. Corresponding Angles-Parallel Theorem G. Transitive Property of Equality

Given: $\overrightarrow{EM} \parallel \overrightarrow{TO}$ and ∠1 ≅ ∠4 Prove: $\overrightarrow{MN} \parallel \overrightarrow{ET}$

[chart: diagram; figure for items 12 – 15, two horizontal and vertical rays with labels E, M, O, T, R, N and angle labels 1, 2, 3, 4]

Proof:

Statements	Reasons
1. $\overrightarrow{ME} \parallel \overrightarrow{OT}$	1. Given
2. ∠2 and ∠3 are supplementary angles	2. If two parallel lines are cut by a transversal, then same-side interior angles are supplementary (Interior Angles Same Side-Parallel Theorem).
3. m∠2 + m∠3 = 180°	3. Definition of supplementary angles
4. ∠1 ≅ ∠2	4. (12)
5. m∠1 + m∠3 = 180°	5. Substitution Property of Equality
6. ∠1 ≅ ∠4	6. Given
7. m∠1 = m∠4	7. Definition of Congruent Angles
8. (13)	8. (14)
9. ∠4 and ∠3 are supplementary angles	9. Definition of supplementary angles
10. $\overrightarrow{MN} \parallel \overrightarrow{ET}$	10. (15)

CO_Q4_Mathematics 8_Module 5

Lesson 1

Determining the Conditions under which Lines and Segments are Parallel or Perpendicular

Do you like sports? Tennis, basketball, volleyball, badminton, and soccer are some of the sports that we Filipinos like playing or watching. Everybody loves sports but have you ever been curious of the design of each court? If you take a closer look at the courts, you will notice that they consist of lines: parallel lines and perpendicular lines. Parallel lines and perpendicular lines are evident in our surroundings, in our house, the perimeter fence, designs in our clothing, and many others. They play important roles in geometry and in real life.

[photo: wooden perimeter fence] Perimeter Fence

[photo: volleyball court with players and net] Volleyball Court

[photo: tennis court with net] Tennis Court

Ultimately, this lesson provides understanding of concepts in geometry that lead to prove the conditions under which lines and segments are parallel or perpendicular.

CO_Q4_Mathematics 8_Module 5

What’s In

Activity: Supply Me!

You have learned in module 4 the different conditions that guarantee lines are parallel and the relationships between angles formed when two parallel lines are cut by a transversal. This time, you are encouraged to remember the angle pairs formed when parallel lines are cut by a transversal.

Directions: Use the figure below in answering each of the following questions. Supply the reason. The first item is done for you.

[chart: diagram; two parallel lines cut by a transversal, with angles numbered 1 to 8. Top line: 1 (upper left), 2 (upper right), 3 (lower left), 4 (lower right). Bottom line: 5 (upper left), 6 (upper right), 7 (lower left), 8 (lower right).]

1. If the $m\angle 3 = 110°$, what is the $m\angle 6$?
   Answer: 110°
   Reason: Alternate interior angles are congruent.

   
   

2. If the $m\angle 3 = 70°$, what is the $m\angle 5$?
   Answer: _____________________
   Reason: _____________________

   
   

3. If the $m\angle 2 = 105°$, what is the $m\angle 7$?
   Answer: _____________________
   Reason: _____________________

   
   

4. If the $m\angle 7 = 75°$, what is the $m\angle 6$?
   Answer: _____________________
   Reason: _____________________

   
   

5. If the $m\angle 8 = 115°$, what is the $m\angle 4$?
   Answer: _____________________
   Reason: _____________________

   
   

6. If the $m\angle 7 = 95°$, what is the $m\angle 1$?
   Answer: _____________________
   Reason: _____________________

   
   

What’s New

Activity 1: Am I Perpendicular or Not?

Directions: Given the figures below, determine whether the lines in each item are perpendicular or not. Write PL if they are perpendicular and PN if not.

1. [illustration: two lines crossing at a right angle, with a right-angle mark, one line vertical and one horizontal]

   
   

2. [illustration: two lines crossing at an oblique angle, with no right-angle mark]

   
   

3. [illustration: a vertical line and a horizontal line meeting at a right angle, with a right-angle mark]

   
   

4. [illustration: two lines crossing at a right angle, with a right-angle mark, drawn diagonally]

   
   

Questions:

1. What is your common observation in the figures given above?
2. What makes figures 1, 3 and 4 different from the second figure?
3. Which among the four figures show perpendicularity? You can check by using a protractor to measure the four figures.
4. Define perpendicular lines based on your observation in the figure above.
5. Cite at least five (5) real life examples that you can think perpendicular lines are used.

Activity 2: Name Me!

Directions: Below is the tree diagram that shows the relationship of some quadrilaterals. Complete the diagram. The first step is done for you.

[chart: tree diagram (flowchart); top box labeled “QUADRILATERALS”. Below it, two shapes: a parallelogram (left) and a trapezoid (right). Under the parallelogram, two shapes: a blank rectangle (left) and a parallelogram-shaped box (right). Both of these point to a blank square at the bottom. The boxes below the top level are unlabeled.]

Directions: Put a check on the characteristics applicable to the given quadrilaterals below.

Properties	Parallelogram	Rectangle	Rhombus	Square	Trapezoid
All sides are congruent.
All angles are congruent.
Diagonals are perpendicular to each other.
Diagonals are congruent.
Exactly one pair of parallel sides.
Two pairs of opposite sides are congruent.

Questions:

1. Did you find difficulty in answering the tasks above?
2. What can you say about the relationship of the quadrilaterals presented on the tree diagram and on the given table?
3. What do you think are the properties common to rectangle, square, and rhombus?
4. Why are parallelograms considered special quadrilaterals?
5. Compare and contrast the properties of parallelograms and list all properties that you found.

B. Complete Me!

Directions: Given the figures below, complete each table of proof.

1. Given: $\angle 1 \cong \angle 2$
   Prove: $c \parallel d$

[chart: diagram; two lines labeled c (upper) and d (lower) cut by a transversal, with angles numbered 1 (upper left, on the transversal near c), 2 (lower left, near d), and 3 (upper right, above c).]

Proof:

Statements	Reasons
1. $\angle 1 \cong \angle 2$	1. _______________
2. $\angle 1 \cong \angle 3$	2. Vertical Angle Theorem
3. _______________	3. Transitive Property
4. $c \parallel d$	4. _______________

2. Given: $\angle 1 \cong \angle 2$
   Prove: $a \parallel b$

[chart: diagram; two lines labeled a (upper) and b (lower) cut by a transversal, with angles numbered 1 (upper left, near a), 3 (lower right, near b), and 2 (between the lines).]

Proof:

Statements	Reasons
1. $\angle 1 \cong \angle 2$	1. _______________
2. $\angle 2 \cong \angle 3$.	2. Vertical Angles are congruent.
3. _______________	3. Transitive Property
4. $a \parallel b$	4. _______________

What is It

Are you already done answering the activities and questions provided above? Are you ready to check if your answers are correct or not? If so, let us now discover more about parallel lines, perpendicular lines, and quadrilaterals. Along the way, you can finally uncover the theorems and postulates on the conditions under which lines and segments are parallel including some properties of parallelogram.

Theorems on Conditions under which Lines and Segments are Parallel or Perpendicular

Recall that two lines are parallel if and only if they are coplanar and do not intersect. This concept is considered as a unique fundamental principle in geometry. Let us start with the Parallel Postulate.

Parallel Postulate

Given a line and a point not on the given line, there is exactly one line through the given point parallel to the given line.

[chart: diagram; line m (bottom) with point P above it, and lines q, s, and t drawn through or near P. Labels: q (upper left), P (center), s (right), t (lower left), m (lower right).]

Given, line $m$ and point $P$ not in $m$. Only one line $s \parallel m$. Lines $t$ and $q$ are not parallel to $m$ through $P$.

Now, let us discuss further the theorems and postulates that prove that two lines are parallel or perpendicular. These theorems are converses of the theorems discussed in module 4.

Postulate and Theorems

Alternate Interior Angle-Parallel Postulate

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

Alternate Exterior Angles-Parallel Theorem

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

Corresponding Angles-Parallel Theorem

If two lines are cut by a transversal and a pair of corresponding angles are congruent, then the lines are parallel.

Interior Angles Same Side-Parallel Theorem

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

Perpendicular Same Line-Parallel Theorem

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

Example 1

Given: $\angle 1 \cong \angle 4$ Prove: $m \parallel n$

[chart: diagram; lines m (upper) and n (lower) cut by transversal t, with angles numbered 1 (upper left, above m), 2 (upper right, above m), 3 (lower left, below n), and 4 (lower right, below n).]

Proof:

Statements	Reasons
1. $\angle 1 \cong \angle 4$	1. Given
2. $\angle 1 \cong \angle 2$	2. Vertical angles are congruent.
3. $\angle 2 \cong \angle 4$	3. Transitive Property of Congruence
4. $\angle 4 \cong \angle 3$	4. Vertical angles are congruent.
5. $\angle 2 \cong \angle 3$	5. Transitive Property of Congruence
6. $m \parallel n$	6. Alternate Interior Angles-Parallel Postulate

Example 2

Given: $\angle 2 \cong \angle 3$ Prove: $c \parallel d$

[chart: diagram; lines c (upper) and d (lower) cut by transversal t, with angles numbered 1 (upper right, above c), 2 (upper left, above c), and 3 (lower left, below c).]

Proof:

Statements	Reasons
1. $\angle 2 \cong \angle 3$	1. Given
2. $\angle 1 \cong \angle 2$	2. Vertical angles are congruent.
3. $\angle 1 \cong \angle 3$	3. Transitive Property of Congruence
4. $c \parallel d$	4. Corresponding Angles-Parallel Theorem

Example 3

Given: $\angle 1 \cong \angle 2$ Prove: $a \parallel b$

[chart: diagram; lines a (upper) and b (lower) cut by transversal t, with angles numbered 1 (upper right, above a), 3 (upper left, above b), and 2 (lower right, below b).]

Proof:

Statements	Reasons
1. $\angle 1 \cong \angle 2$	1. Given
2. $\angle 2 \cong \angle 3$	2. Vertical angles are congruent.
3. $\angle 1 \cong \angle 3$	3. Transitive Property of Congruence
4. $a \parallel b$	4. Corresponding Angles-Parallel Theorem

Here are two other theorems involving parallel lines.

The Three Parallel Lines Theorem

In a plane, if two lines are both parallel to a third line, then they are parallel.

[transcriber’s note: the last word of this theorem is partly clipped in the image; it reads “parallel.”]

The proofs of these theorems are left for you to prove in activity found in What’s More.

The Two Perpendicular Lines Theorem

If two coplanar lines are perpendicular to a third line, then they are parallel to each other.

[transcriber’s note: on this page, the sentence above overlaps the “What’s More.” label printed in the margin, so the two are partly obscured.]

Theorems on Perpendicular Lines

At this point, let us discuss and discover more about perpendicular lines.

Definition

Perpendicular Lines are defined as two lines that intersect to form right angles. Line segments and rays can also be perpendicular.

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

Theorem 1

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

In the accompanying figure, $\overleftrightarrow{SM}$ and $\overleftrightarrow{EU}$ intersect at point O to form right angles. We can read it as “$\overleftrightarrow{SM}$ is perpendicular to $\overleftrightarrow{EU}$”. In symbols, we write as $\overleftrightarrow{SM} \perp \overleftrightarrow{EU}$. Since, $m\angle MOE = 90°$, therefore $m\angle SOE = 90°$.

In the same manner, $m\angle SOU = 90°$ and $m\angle MOU = 90°$. Thus, perpendicular lines form four right angles.

[chart: diagram; perpendicular lines meeting at point O, with E (top), U (bottom), S (left), M (right), and a right-angle mark at O.]

If $\overleftrightarrow{EU}$ bisects $\overline{SM}$, then $\overleftrightarrow{EU}$ is called the perpendicular bisector of $\overline{SM}$.

Definition

A perpendicular bisector of a line segment is a line or a ray or another line segment that is perpendicular to the line segment and intersects it at its midpoint.

Theorem 2

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

Refer to the figure above. Since [illegible] and [illegible] form a linear pair and are congruent, thus, [illegible] and [illegible] are perpendicular to each other. That is, $\angle SOE \cong \angle MOE$; and they are linear pair, $m\angle SOE + m\angle MOE = 180°$ therefore, $\overleftrightarrow{SM} \perp \overleftrightarrow{EU}$.

[transcriber’s note: the names of the angles and segments in the sentence above are not legible in the image and are marked [illegible].]

Applying the theorems and postulates on parallel lines and perpendicular lines to parallelograms, can you still determine which lines and segments are parallel or perpendicular? Let’s take a look at the following examples.

EXAMPLE 4

Given: $\overline{PE} \cong \overline{EN}$, $\overline{LA} \cong \overline{AN}$

[chart: diagram; a figure with points L (top), P (bottom left), N (bottom right), A (on the right side, between L and N), and E (on the base, below A). Right-angle marks are at L, A, and E.]

From the figure above, we can state the following: $\overline{PL} \perp \overline{LN}$, $\overline{AE} \perp \overline{LN}$, $\overline{PL} \parallel \overline{AE}$ since they are both perpendicular to the same line segment $\overline{LN}$. $\overline{PE} \perp \overline{LE}$, $\overline{EL}$ and $\overline{AE}$ are perpendicular bisectors of $\overline{PN}$ and $\overline{LN}$, respectively. Also, the measure of $\angle EAL$ is $90°$ and $m\angle PEL = 90°$ while, $\angle PEL$ and $\angle NEL$ are supplementary angles and at the same time adjacent, therefore they are linear pair.

Moreover, we can say that PEAL is a quadrilateral since it is a four-sided figure. Can you say that $\square PEAL$ is a parallelogram?

This time let us discover the relationships of parallel and perpendicular lines in parallelograms.

Definition

A quadrilateral is a polygon with four sides. The symbol $\square$ is used to indicate a quadrilateral. For example, $\square PEAL$, this is read as “Quadrilateral PEAL.”

Quadrilaterals are classified according to the number of pairs of parallel sides.

1. Parallelogram – is a quadrilateral with two pairs of parallel sides. It is further classified as:
   a. Rhomboid – is a parallelogram with no right angle and any two consecutive sides are not congruent.
   b. Rectangle – is a parallelogram with four right angles.
   c. Rhombus – is a parallelogram with four congruent sides.
   • Square – is a rectangle with four congruent sides. It is both a rectangle and a rhombus because it satisfies the definition of a rectangle and a rhombus.
2. Trapezoid – is a quadrilateral with exactly one pair of parallel sides. If the non-parallel sides, called legs, are congruent, then the trapezoid is isosceles.
3. Trapezium – is a quadrilateral with no pair of parallel sides.

Given the definitions, we can now deduce the following properties.

Points to Remember

A quadrilateral is a parallelogram if:

• A pair of opposite sides are both parallel and congruent.
• Two pairs of opposite sides are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.
• Opposite angles are congruent.

Let’s take a look at this example.

EXAMPLE 5

1. Given the parallelogram at the right, find the value of $y$.

[illustration: parallelogram with angles labeled (y + 30)° at the upper right and (2y − 45)° at the lower left]

Solution:

In the given quadrilateral, we know that the two angles given are opposite angles and opposite angles are congruent in a parallelogram. Hence, we can formulate an equation out of this.

$(2y – 45°) = (y + 30°)$ Equate the two expressions.

$(2y – y) – 45° = (y – y) + 30°$ Add (-y) to both sides of the equation, (Addition Property of Equality).

$y + (-45° + 45°) = (30° + 45°)$ Add (+45), (Addition Property of Equality)

$y = (30° + 45°)$ Simplify

$y = 75°$ Result

What’s More

Activity 1: Justify Me!

A. Determine whether each pair of lines $l_1$ and $l_2$ are parallel or not and justify your answer.

1. [illustration: lines $l_1$ and $l_2$ cut by transversal $l_3$, with angles 95° and 120° marked]
2. [illustration: lines $l_1$ and $l_2$ cut by transversal $l_3$, with angles 56° and 78° marked]
3. [illustration: lines $l_1$ and $l_2$ cut by transversal $l_3$, with angles 100° and 100° marked]
4. [illustration: lines $l_1$ and $l_2$ cut by transversal $l_3$, with angles 130° and 130° marked]

B. Refer to the figure at the right, explain why $a \parallel b$. Justify your answer with a theorem or postulate. Choose your answers from the choices provided in the box.

[illustration: transversal lines b and a cut by a horizontal transversal; angles numbered 1 to 8, with b labeled at the upper left and a at the upper right]

1. If $m\angle 1 = 50°$, then $m\angle 5 = 50°$
2. If $m\angle 7 = 78°$, then $m\angle 2 = 78°$
3. If $m\angle 4 = 120°$, then $m\angle 7 = 60°$
4. If $m\angle 3 = 99°$, then $m\angle 6 = 99°$
5. If $m\angle 6 = 85°$, then $m\angle 1 = 95°$

[Word box, printed above Activity 2; transcriber’s note: the box has no title and is reproduced as a list in row order]

• Corresponding Angles-Parallel Theorem
• Law of Substitution
• Interior Angles Same Side-Parallel Theorem
• Reflexive Property
• Alternate Interior Angles-Parallel Theorem
• Supplementary Angles
• The Three Parallel Lines Theorem
• Vertical Angle Theorem
• Alternate Exterior Angles-Parallel Theorem
• Linear Pair Postulate
• Transitive Property of Equality
• Exterior Angles Same Side-Parallel Theorem

Activity 2: Think More!

Direction: Given the figure at the right, find the length of each specified segment.

[chart: geometry figure; parallelogram FAME with vertices A (top left), M (top right), E (lower right), F (lower left), diagonals meeting at D; segment labels (2f) cm on AM side of the upper diagonal, (x + 40) cm, (3x + 20) cm, and (3f − 7) cm]

1. ☐ $FAME$ is a parallelogram.
   a. $\overline{AD}$
   b. $\overline{AE}$
   c. $\overline{FM}$
   d. $\overline{FD}$
2. In ☐ $FAME$, if $m\angle AFE = 6a – 45°$ and $m\angle EMA = 4a + 15°$, what is $m\angle FEM$? (Show your solution).

Activity 3: Prove Me!

Given the figure below, complete each proof:

[chart: three parallel lines r, s, and t cut by a transversal; angles numbered 1, 2, and 3 at the intersections]

1. Given: $r \parallel t$, $s \parallel t$

   
   

   Prove: $r \parallel s$

   
   

Proof:

Statements	Reasons
1. $r \parallel t$	1. _________________
2. $\angle 1 \cong \angle 3$	2. If two parallel lines are cut by transversal, then corresponding angles are congruent.
3. __________	3. Given
4. $\angle 3 \cong \angle 2$	4. _________________
5. __________	5. Transitive Property
6. $r \parallel s$	6. ___________________

2. Given: $r \parallel s$, $r \perp t$

   
   

   Prove: $s \perp t$

   
   

[chart: lines r and s, both crossed by a vertical transversal t; the right angles at the intersections are numbered 1 and 2]

Proof:

Statements	Reasons
1. $r \parallel s$, $r \perp t$	1. Given
2. $m\angle 1 = 90°$	2. _________________
3. _________________	3. Corresponding angles are congruent.
4. $m\angle 1 = m\angle 2$	4. _________________
5. $m\angle 2 = 90°$	5. Transitive Property of Equality
6. $s \perp t$	6. _________________

What I Have Learned

Directions: Tell whether each statement is true or false by checking the box that corresponds to your answer. If false, change the underlined word/s to make the statement true.

	True	False

1. Parallel lines are coplanar lines that do not intersect. ☐ True ☐ False ______

   
   

2. Two intersecting are always perpendicular. ☐ True ☐ False ______

   
   

3. If two parallel lines are cut by a transversal and a pair of alternate interior angles are supplementary, then the transversal is perpendicular to the two parallel lines. ☐ True ☐ False ______

   
   

4. Line 1 ($l_1$) is a transversal if it intersects two coplanar lines $l_2$ and $l_3$ at the same point/s. ☐ True ☐ False ______

   
   

5. If two lines are cut by a transversal and a pair of alternate interior angles are congruent, then the lines are parallel. ☐ True ☐ False ______

   
   

6. If a transversal is perpendicular to two parallel lines, then the interior angles on the same side of the transversal are congruent. ☐ True ☐ False ______

   
   

7. If a line is the perpendicular bisector of a line segment, then the line intersects the segment at any point. ☐ True ☐ False ______

   
   

8. If the angles in a linear pair are congruent, then the lines containing their sides are perpendicular. ☐ True ☐ False ______

   
   

9. If two coplanar lines are perpendicular to a third line, then the two coplanar lines are perpendicular to each other. ☐ True ☐ False ______

   
   

10. Given a line and a point not on the line, there is exactly one line through the point parallel to the given line. ☐ True ☐ False ______

    
    

What I Can Do

Directions: Refer to the figure below. Make a two – column proof to prove the following. The first part is done for you.

1. Given: $m\angle 1 \cong m\angle 3$.

   
   

   Prove: $x \parallel y$

   
   

[chart: parallel lines x and y cut by transversal t; angles 1, 2, and 3 numbered at the intersections]

Statements	Reasons
1. $m\angle 1 \cong m\angle 3$
2. $m\angle 1 + m\angle 2 = 180°$
3.
4.

2. Given: $\overline{GV}$ bisects $\overline{EI}$; $\overline{EI}$ bisects $\overline{GV}$

   
   

   Prove: $GIVE$ is a parallelogram

   
   

[chart: parallelogram GIVE with diagonals meeting at D; angles numbered 1, 2, 3, and 4]

Statements	Reasons
1. $\overline{GV}$ bisects $\overline{EI}$; $\overline{EI}$ bisects $\overline{GV}$	1. Given
2. $\overline{ED} \cong \overline{ID}$; $\overline{GD} \cong \overline{VD}$	2. Definition of segment bisector.
3.	3.
4.	4.
5.	5.
6.	6.
7.	7.

Assessment

POST-ASSESSMENT

Directions: Read and answer each question carefully and write the letter of the correct answer on a separate sheet of paper.

1. The symbol used to indicate parallelism is _____.
   • A. ⊥
   • B. ∥
   • C. ≅
   • D. ↔
2. Lines that intersect to form right angles are said to be _____.
   • A. coinciding
   • B. intersecting
   • C. parallel
   • D. perpendicular

For items 3 – 7, refer to the figure at the right.

[illustration: lines w and z crossing a transversal line, with lines x and y crossing; angles labeled a through h and 1 through 8]

3. Which statement is true if $\angle f \cong \angle d$?

   • A. $w \parallel z$
   • B. $w \perp z$
   • C. $x \parallel y$
   • D. $x \perp y$

4. Which pair of angles are supplementary?

   • A. $\angle 2$ and $\angle 3$
   • B. $\angle 2$ and $\angle 7$
   • C. $\angle b$ and $\angle 4$
   • D. $\angle b$ and $\angle c$

5. If $x \parallel y$, which pair of angles are congruent?

   • A. $\angle 1$ and $\angle 6$
   • B. $\angle 4$ and $\angle 5$
   • C. $\angle c$ and $\angle f$
   • D. $\angle d$ and $\angle g$

6. If $x \parallel y$, what is the value of $x$ if $m\angle 3 = 3x + 16°$ and $m\angle 5 = x + 66°$?

   • A. 25°
   • B. 35°
   • C. 50
   • D. 70°

7. If $x \parallel y$, $m\angle c = 13y + 9°$, $m\angle d = 6y$ and $m\angle 2 = 11y + 1°$, what is the measure of $\angle 8$?

   • A. 60°
   • B. 80°
   • C. 100°
   • D. 120°

For items 8 – 9, refer to the figure at the right.

[illustration: perpendicular lines w and z crossing; angles 1, 2, 3, and 4 numbered around the intersection]

8. If $w \perp z$, all the following statements are true EXCEPT:
   • A. $m\angle 1 = 90°$
   • B. $m\angle 2 = 90°$
   • C. $m\angle 1 + m\angle 2 = 180°$
   • D. $m\angle 3 + m\angle 4 = 200°$
9. If $w \perp z$, $m\angle 1 = 20x + 2y$ and $m\angle 2 = 25x – 2y$, find the values of $x$ and $y$.
   • A. $x = 2°$, $y = 4°$
   • B. $x = 3°$, $y = 5°$
   • C. $x = 4°$, $y = 5°$
   • D. $x = 4°$, $y = 7°$
10. In the figure at the right, which of the following guarantees that $a \parallel b$?

[illustration: parallel lines a and b cut by transversal t; angles 1 to 8 numbered at the intersections]

• A. $\angle 1$ and $\angle 2$ are congruent.
• B. $\angle 2$ and $\angle 5$ are supplementary.
• C. $\angle 3$ and $\angle 7$ form a linear pair.
• D. $\angle 4$ and $\angle 8$ are supplementary.

11. Given that $\overleftrightarrow{JE} \perp \overrightarrow{VR}$ at $V$, find the possible algebraic expressions to represent $\angle JVR$ and $\angle EVR$ such that the values for the variables $x$ and $y$ are 7 and 13, respectively.

[illustration: line JE perpendicular to ray VR at V, with a right-angle mark at V]

• A. $m\angle JVR = (8x – 14°)$, $m\angle EVR = (12y + 6°)$
• B. $m\angle JVR = (9x – 25°)$, $m\angle EVR = (13y – 1°)$
• C. $m\angle JVR = (12x + 6°)$, $m\angle EVR = (8y – 14°)$
• D. $m\angle JVR = (13x – 1°)$, $m\angle EVR = (9y – 25°)$

For items 12-15, refer to the figure given below and complete the proof. Choose the letter of your answer from the box provided.

Given: $\overrightarrow{OC} \parallel \overrightarrow{SN}$ and $\angle 1 \cong \angle 4$

Prove: $\overrightarrow{CE} \parallel \overrightarrow{OS}$

[chart: lines OC and SN, parallel, with ray CE, and ray OS; angles 1, 2, 3, and 4 numbered at the intersections]

Proof:

Statements	Reasons
$\overrightarrow{OC} \parallel \overrightarrow{SN}$	Given
1. $\angle 2$ and $\angle 3$ are same-side interior angles	1. Definition of Same-Side Interior Angles
2. $\angle 2$ and $\angle 3$ are supplementary angles	2. If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.
3. $m\angle 2 + m\angle 3 = 180°$	3. Definition of Supplementary Angles
4. $\angle 3$ and $\angle 4$ are vertical angles	4. Definition of Vertical Angles
5. $\angle 3 \cong \angle 4$	5. (12)
6. $m\angle 3 = m\angle 4$	6. Definition of Congruent Angles
7. $m\angle 2 + m\angle 4 = 180°$	7. Substitution Property of Equality
8. $\angle 1 \cong \angle 4$	8. Given
9. $m\angle 1 = m\angle 4$	9. Definition of Congruent Angles
10. (13)	10. (14)
11. $\angle 1$ and $\angle 2$ are supplementary angles	11. Definition of Supplementary Angles
12. $\angle 1$ and $\angle 2$ are same-side interior angles	12. Definition of Same-Side Interior Angles
13. $\overrightarrow{CE} \parallel \overrightarrow{OS}$	13. (15)

• A. $m\angle 1 = m\angle 4$
• B. $m\angle 1 + m\angle 2 = 180°$
• C. Vertical Angles Theorem
• D. Transitive Property of Equality
• E. Substitution Property of Equality
• F. Corresponding Angles-Parallel Postulate
• G. Interior Angles Same Side-Parallel Theorem

Additional Activities

Construct Me!

Refer to the figure below and construct a two-column proof to prove the following given.

1. Given: ∠1 and ∠2 are supplementary angles.
   ∠3 and ∠4 are supplementary angles.
   Prove: P̅L̅ ∥ E̅T̅

[chart: diagram; figure for item 1 showing points P, E, S, L, T, A with angles 1, 2, 3, and 4 labeled]

2. What is the value of x, for which p ∥ q

[chart: diagram; parallel lines p and q cut by a transversal, with angles 1, 2, 3, and 4 labeled]

a. m∠2 = 3x − 15°, m∠4 = 4x − 20°

b. m∠1 = 3x + 46°, m∠3 = 2x + 24°

c. m∠1 = 3x − 29°, m∠4 = 2

Answer Key

POST-ASSESSMENT

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

What I Can Do

Reasons	Statements
Given	m∠1 ≅ m∠3
Definition of Linear pair.	m∠1 + m∠2 = 180°
Definition of Supplementary Angle	m∠2 + m∠3 = 180° [?]
Interior Angles Same Side-Parallel Theorem	x ∥ y

What I Have Learned

1. True
2. False, sometimes
3. True
4. False, different
5. True
6. True
7. False, midpoint
8. True
9. False, parallel
10. True

What’s More

Activity 1

1. Not parallel
2. parallel
3. not parallel
4. parallel

Activity 2

1. a. 14 cm
   b. 28 cm
   c. 100 cm
   d. 50 cm
2. m∠FEM = 45°

Activity 3

Reasons	Statements
1. Given	1. t ∥ r
2. Corresponding Parallel – Angles Theorem	2. ∠1 ≅ ∠3
3. Given	3. s ∥ t
4. Corresponding Parallel – Angles Theorem	4. ∠2 ≅ ∠3
5. Transitive Property	5. ∠2 ≅ ∠1
6. Corresponding Parallel – Angles Theorem	6. s ∥ r

Activity 1 (Parallel Lines Reasons)

A. 1. Corresponding Angles Parallel Postulate 2. Alternate Interior Angles Theorem 3. Interior Angles Same Side Parallel Theorem 4. Alternate Exterior Angles Parallel Theorem 5. Exterior Angles Same Side Parallel Theorem

B. [transcriber’s note: the printed key lists these reasons beside the letters “A.” and “B.” in the rotated layout; the mapping of letters to items is not clear in the scan]

What I Know

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

What’s In

1. [illegible]
2. 110°, interior angles on the same side of transversal are supplementary.
3. 105°, alternate exterior angles are congruent
4. 75°, vertical angles are congruent
5. 115°, corresponding angles are congruent
6. 85°, same-side exterior angles are supplementary

What’s New

Activity A

1. PL
2. PN
3. PL
4. PL

Activity B

• Parallelogram: Rectangle
• Trapezoid – Rhombus
• Square

Activity C

Properties	Trapezoid	Square	Rhombus	Rectangle	Parallelogram
All sides are congruent.		√	√
All angles are congruent.		√		√
Diagonals are perpendicular to each other.		√	√
Diagonals are congruent.		√		√
Exactly one pair of parallel sides.	√
Two pairs of opposite sides are congruent.		√	√	√	√

[transcriber’s note: the check marks in Activity C are read from a rotated, low-resolution scan; their column placement should be verified against the printed page]

What’s More: Activity 3

Reasons	Statements
1. Given	1. t ⟂ r, s ⟂ r [?]
2. Definition of perpendicular lines	2. m∠1 = 90°
3. Corresponding angles are congruent.	3. ∠2 ≅ ∠1
4. Definition of Congruent angles	4. m∠2 = m∠1
5. Transitive Property of Equality	5. m∠2 = 90°
6. Definition of perpendicular lines	6. t ∥ s [?]

What I Can Do

Reasons	Statements
1. Given	1. E̅I̅ bisects G̅V̅; G̅V̅ bisects E̅I̅
2. Definition of segment bisector.	2. E̅D̅ ≅ I̅D̅, G̅D̅ ≅ V̅D̅
3. Vertical Angles are congruent.	3. ∠1 ≅ ∠2
4. SAS	4. ΔVDI ≅ ΔGDE
5. CPCTC	5. I̅V̅ ≅ E̅G̅, ∠4 ≅ ∠3
6. Alternate Interior Angles Parallel Postulate	6. I̅V̅ ∥ E̅G̅
7. If a pair of opposite sides of a quadrilateral are congruent and parallel, it is a parallelogram	7. GIVE is a parallelogram.

Additional Activities

Proof 1.

Reasons	Statements
1. Given	1. ∠1 and ∠2 are supplementary angles.
2. Interior Angles Parallel – Same Side Theorem	2. P̅L̅ ∥ S̅A̅
3. Given	3. ∠3 and ∠4 are supplementary angles.
4. Interior Angles Same Side Parallel Theorem	4. E̅T̅ ∥ S̅A̅
5. The Three Parallel Lines Theorem	5. E̅T̅ ∥ P̅L̅

2. a. x = 5 b. x = 22 c. x = 30

What’s New: Activity C

Properties	Trapezoid	Square	Rhombus	Rectangle	Parallelogram
All sides are congruent.		√	√
All angles are congruent.		√		√
Diagonals are perpendicular to each other.		√	√
Diagonals are congruent.		√		√
Exactly one pair of parallel sides.	√
Two pairs of opposite sides are congruent.		√	√	√	√

[transcriber’s note: the check marks in this Activity C table are read from a rotated scan and should be verified against the printed page]

References

Abuzo, Emmanuel P., Bryant, Merden L., Cabrella, Jem Boy B. Caldez, Belen P., Callanta, Melvin M., Castro, AnastaciaPreserfina I., Halabaso, Alicia R., Javier, Sonia P., Nocom, Roger T., and Ternida, Conception S. (2013). Grade 8 Mathematics Learners Module. pp. 441-480. Philippines. Book Media Press, Inc. and Printwell, Inc.

Bernabe, J. & Jose-Dilao, S (2009). Geometry Textbook for Third Year. pp. 70-71 and 115-141. Araneta Avenue, Quezon City, Philippines. SD Publications, Inc.

Boyd, C. (1998). Geometry Integrated, Applications and Connections. pp. 288-320. Columbus, United States of America. Glencoe/McGraw – Hill.

Diaz, Zenaida B., Maharlika Mojica P. Manalo, Catalina B., Suzara, Josephine L. Mercado, Jesus P., Esparrago, Marlo S., Reyes, Jr., Nestor V. Next Century Mathematics 8. pp. 547-599. Philippines. Phoenix Publishing House.

Gabriel, Judy L., and Mallari, Ma. Theresa G. (2017). Mathematics for Grade 8: A Spiral Approach. pp. 182 – 188. Cubao, Quezon City, Philippines. Educational Resources Corporation.

Nivera, Gladys C. (2014). Grade 8 Mathematics Pattern and Practicalities. pp. 398-417. Salesiana Books by Don Bosco Press. Makati City, Philippines.

Romero, Karl Freidrich Jose D. (2003). Geometry in the Real World. pp. 87-93. Makati City, Philippines. Salesiana Publishers, Inc.

Website Links

https://www.slideshare.net>mobile. Angles formed by parallel lines cut by transversal – slideshare. December 17, 2019

https://www.mathisfun.com>parallel lines, pairs of angles – Math is fun. December 17, 2019

https://www.mathopenref.com. Constructions. Retrieved on December 21, 2019

https://www.slideshare.net>mobile. Proving Lines are Perpendicular. Retrieved on December 21, 2019

https://www.nj01001706.schoolwrites.net.pdf. Proofs with Perpendicular Lines 3.4. Retrieved on December 20, 2019

For inquiries or feedback, please write or call:

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

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

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