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Bisectors LP

BS Education (Isabela State University)

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Detailed Lesson Plan

MATHEMATICS
Grade 8
I. CONTENT STANDARD
a. demonstrates understanding of key concepts of axiomatic structure of geometry
and triangle congruence.

II. PERFORMANCE STANDARDS

a. is able to communicate mathematical thinking with coherence and clarity in
formulating, investigating, analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate representations

III. LEARNING OBJECTIVES

At the end of the lesson, the students should be able to:
a. recall conditions on triangle congruence postulates and theorems;
b. illustrate perpendicular lines and angle bisectors by construction; and
c. apply triangle congruence involving perpendicular lines and angle
bisectors.

IV. LEARNING CONTENT

a. Topic: Applies triangle congruence to construct perpendicular lines
and angle bisectors.
b. Learning Resources: Visual Aid and Power point presentation
c. References: Mathematics Learner’s Module 8
https://depedtambayan.net/wp-content/uploads/2022/01/MATH8-
Q3MODULE8.

V. LEARNING PROCEDURE
Teacher’s Activity Student’s Activity
A. Preliminary Activities

Good morning Grade 8.

Good morning ma’am.

Before we start, everybody stands and let us feel the

presence of the Lord. Would you please lead the
prayer, Mona? Lord, thank you for today. Thank
you for the ways in which you
provided to all of us. Bless our
teacher and help us to focus our
hearts and minds now on what we
are about to learn. Guide us by your
eternal light as we discover more
about the world around us. We ask

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all of these in Jesus name. Amen.

Okay, please be seated.

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May I ask the secretary to check the attendance

Okay ma’am.

B. Review

Let’s have a short review about what you have done

last meeting.
What is acronym of CPCTC?
(Students raising their hands)

Yes, Lyka?
Corresponding parts of congruent
triangle are congruent.

Okay, very good.

C M
N

What is reflexive property?

(Students raise their hands)
Yes? Reflexive property of equality
states that a number is always equal
to itself.

Very good.
In the given figure above, 𝐴𝑁̅̅̅̅≅ ̅𝐴𝑁̅̅̅nis an example
of? What property, again? (Students raise their hands.)
Reflexive property of equality
Yes, Mark? ma`am

Very good.

C. Motivation
Before we start the discussion for our lesson

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today, we are going to play a game that called

Fix Me! This is to provide you a glimpse on
what we are discuss today.

This game consists of 2 groups. And here are

the mechanics: You will be given cutted
pieces of paper then you are going to arrange
it to form a figure. This figure is usually use
during summer. I will only give 20 seconds
to answer.

To all who will participate in this game will

be given additional points to our quiz. Ready
class?
Yes ma`am

What Figure that you formed?

(Students raise their hands)
Yes, Group 1?

A KITE ma`am

Okay, Very good

Thank you Class for participating on the

game. You are all given plus point on the
quiz.

D. Presentation of the Lesson

What did you notice to our activity? Anyone?
Ma’am

Yes, Kaye? I observed ma’am that there are

things to consider in constructing
kite especially the principle of

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balance, this is also application for

triangle congruence, perpendicular
lines and angle bisector.
Absolutely that`s right Kaye.

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E. Lesson Proper

Today, we are going to discuss applying triangle

congruence to construct perpendicular lines and angle
bisectors.

We construct a Kite that we usually we use during

this summer season.

Class, I have question. What do you think will

happen if the right part of the kite is bigger or
smaller that its left part?
Ma’am
Yes?
Ma’am the kite will not fly
smoothly
Very Good.
Make sure that the kite is balance if the
corresponding sides and angle are congruent.

Let’s define Geometric Construction

Who want to read the definition of Geometric
Construction?

Yes, Marco? Ma’am

Geometric Construction is a process

of drawing geometric figures such
as lines, angles and other geometric
shapes using only a pencil,
compass, and straightedge.
Okay, thank you.

During your grade 7 days you are already did the

manual geometric construction using pencil,
compass and straightedge, do you still remember?

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Yes ma’am

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Good!

Today am going to teach you on how to construct

perpendicular lines and angle bisectors.

Before we construct, let’s defined what is

perpendicular segments or lines. Anyone?
(Students raise their hands)
Yes, Mane?
Lines that intersect at a common
point forming 90degree angle.
Very good

Two distinct lines intersecting each other at 90 ̊ or at

a right angle are called perpendicular lines.

The term ‘perpendicular’ originated from the Latin

word ‘perpendicularis,’ meaning a plumb line.

Example:

If two lines AB and CD are perpendicular, then we

can write them as AB CD. The symbol is used to
indicate the lines are perpendicular.

We can observe many perpendicular lines in real life

situation.

Class, Give some examples of perpendicular lines

that we used in our school, house and etc?

Yes, Zen? Ma’am

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Correct.
Corner of Blackboard and Window

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Some examples are: the sides of a set square, the

arms of a clock, the corners of the blackboard,
window and the Red Cross symbol.

Again, What is the symbol used to represent a

perpendicular line?

Yes, Christian? (Students raise their hands)

(Student draw on the board)

Very Good! “⊥”

Construction of Perpendicular Lines using two

congruent right triangle Example 1:

Step in constructing Perpendicular line

1. Given two congruent right triangles by LA

Theorem, determine the other corresponding parts
that are congruent.

Using Corresponding Parts of Congruent Triangles

are Congruent (CPCTC), What are the corresponding
parts are congruent in the given above? Anyone?

Yes, Riza?

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Ma’am

̅𝐴𝑋̅̅̅≅𝑋𝐷̅̅̅̅, ̅𝑌𝑋̅̅̅≅𝐶𝐷̅̅̅̅

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Okay, Correct

The following corresponding parts are congruent

∠AXY ≅ ∠XDG
𝐴𝑋̅̅̅̅≅𝑋𝐷̅̅̅̅
̅𝑌𝑋̅̅̅≅𝐶𝐷̅̅̅̅

We now proceed to next step. Who wants to read the

2nd step in constructing of Perpendicular Lines using
two congruent right triangles?
(Students raise their hands)
Yes, Jen?
Next step: Put the two triangles
side by side in such a way that the
vertices labeled with X coincide.

Last step: Determine the relationship of ∠𝐴𝑋𝑌 and

∠𝐶𝑋𝐷

Perpendicular lines are two distinct lines intersecting

each other, how many degree angle? Anyone?
Ma’am

Yes Mika?
90degree angle Ma’am

Very good!

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𝑚∠𝐴 + 𝑚∠𝐴𝑋𝑌 + 𝑚∠𝐴𝑌𝑋 = 180°

𝑚∠A + 𝑚∠AXY + 90 ° = 180 °
𝑚∠A + 𝑚∠AXY = 180 – 90 °
𝑚∠A + 𝑚∠AXY = 90 °
𝑚∠AXY = 90 ° − 𝑚∠A

Since ∠𝐴 ≅ ∠𝐶𝑋𝐷, then

𝑚∠𝐴𝑋𝑌 = 90° −𝑚∠𝐶𝑋𝐷 or 𝑚∠𝐴𝑋𝑌 + 𝑚∠𝐶𝑋𝐷 =
90°

Thus, ∠𝐴𝑋𝑌 and ∠𝐶𝑋𝐷 are complementary.

Since the sum of the measures of ∠𝐴𝑋𝑌 and ∠𝐶𝑋𝐷 is

equal to 90 degrees, then ̅𝐴𝑋̅̅̅ is perpendicular to
̅𝑋𝐷̅̅̅, or 𝑋𝐷̅̅̅̅ is perpendicular to 𝐴𝑋̅̅̅̅ by definition of
perpendicularity.
An angle bisector is a ray or a line
We now proceed to Angle Bisector What that divides an angle into two
Is an Angle Bisector? equal parts. The word “bisector”
implies division into two equal
parts

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In the following image, ABC is divided into two

equal parts by the angle bisector BD.

An angle bisector of a 60 ̊ angle will divide it into

two angles of 30 ̊ each. It divides an angle into two
congruent angles.

For example an angle bisector divides an angle of

80 ̊. What will be the measure of each angle?
Anyone to the class?
(student raise their hand)
Yes, Jade?

Very good! Ma`am 40 ̊

Properties of Angle Bisector. Who want to read the

Properties of Angle Bisector?

Yes, Michel?
(student raise their hand)

• An angle bisector divides an

angle into two angles of
equal measure.
• Any given point lying on the
angle bisector is at an equal
distance from the arms or
sides of the angle.
• The angle bisector in a
triangle divides the opposite
side in a ratio that is equal to
the ratio of the other two
sides

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Okay, thank you.

Angle bisector construction requires a ruler and a

compass. Let’s understand the steps to construct an
angle bisector using two congruent triangles.

Steps:

1. Given two congruent triangles by

SAS Postulate, determine the other
corresponding parts that are congruent.

(student raise their hand)

What are corresponding parts of two triangles that ∠1 ≅ ∠5

are congruent? ∠2 ≅ ∠6
̅𝐴𝐵̅̅̅≅𝐵𝐷̅̅̅̅
Yes, Hanz?

Very good!
We now proceed to the next step.
2. Put the two triangles together in such
a way that a pair of corresponding sides
coincide. See the thicker line.

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(student raise their hand)

3. Determine the common side or the

side shared by the triangles.

What is the common side of the two triangle?

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Yes, Maze?
The common side is ̅𝐵𝐶̅̅̅

Ma`am two angles are adjacent if

they have a common side and a
common vertex.

4. Determine the adjacent angles formed Before

that what is Adjacent Angles?

Okay very good!

Adjacent angles:
Pair 1: ∠1 and ∠5
Pair 2: ∠3 and ∠4

(student raise their hand)

Determine the relationship of the

adjacent angles. The two pairs of
adjacent angles are congruent as
they are corresponding parts of the
Next step, who want to read the step 5?
congruent triangles.
Yes?

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We now proceed to the last step.

Determine the relationship of any one of adjacent
angles to the sum of their measures.

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𝑚∠1 = 𝑚∠5
𝑚∠𝐴𝐵𝐷 = 𝑚∠1 + 𝑚∠5
𝑚∠𝐴𝐵𝐷 = 𝑚∠1 + 𝑚∠1
𝑚∠𝐴𝐵𝐷 = 2(𝑚∠1)
𝑚∠1 = ½ (𝑚∠𝐴𝐵𝐷)

𝑚∠3 = 𝑚∠4
𝑚∠𝐴𝐶𝐷 = 𝑚∠3 + 𝑚∠4
𝑚∠𝐴𝐶𝐷 = 𝑚∠3 + 𝑚∠3
𝑚∠𝐴𝐶𝐷 = 2(𝑚∠3)
𝑚∠3 = ½(𝑚∠𝐴𝐶𝐷)

Thus, one of the adjacent angles is half of the whole

angle. Since the two adjacent angles are congruent
and one of the angles is half of ∠𝐴𝐵𝐷 or half of
∠𝐴𝐶𝐷, it follows then that side BC divides both
∠𝐴𝐵𝐷 and ∠𝐴𝐶𝐷 congruently. Thus, ̅𝐵𝐶̅̅̅bisects both
∠𝐴𝐵𝐷 and ∠𝐴𝐶𝐷. Hence, ̅𝐵𝐶̅̅̅ is an angle bisector.

Is there any question, class?

None ma`am
F. Generalization
So Class, what is perpendicular lines?
(student raise their hand)
Yes, Mitch?
Perpendicular lines are two lines
that intersect at a right angle. This
is often shown by a right angle
symbol in the corner where the two
lines meet.

Very good
What else have you learned for today? Ma`am we learned how to
Construct Perpendicular lines and
Angle bisector.

In Angle bisector, if the given measure of the angle is

50 ̊. What will be the measure of each angle?

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Ma`am
Yes, Nic?

Each angle will measure 25 ̊. I

know that an angle bisector divides
an angle into two equal segments.

Very good!
So now, did you understand our topic for today?
Yes ma’am. We understand it well!

That is amazing. You are now ready to become Math

Wizards.

G. Application

Let’s apply what we have learned today in real life

situation. I will post pictures with statements on the
screen, raise the smiley face if you identify perpendicular
lines and sad face if not. Are the instruction clear?
Yes ma`am.

SMILEY FACE

1. The arms of a Clock.

2. Redcross symbol

SMILEY FACE

3. A Chair in a classroom
SAD FACE

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Good job class, give yourself three claps!

VI. Evaluation
Directions: Use the given two congruent triangles to answer the questions that

follow. Use one whole sheet of paper. A.

Given: ∆𝑯𝑱𝑰 ≅ ∆𝑯𝑲𝑰.

1. What triangle congruence postulate is illustrated in the figure?

2. What are the corresponding congruent sides?
3. What are the corresponding congruent angles?
4. If you put together the two triangles in such a way that side ̅𝐻𝐼̅̅̅ of ∆𝐻𝐽𝐼
coincides with the side ̅𝐻𝐼̅̅̅ of ∆𝐻𝐾𝐼, what new figure is formed?
5. Do the sides of the two triangles that coincide appear to be congruent? Why?
6. What are the pairs of adjacent angles?
7. How are the adjacent angles related to each other?
8. What does ̅𝐻𝐼̅̅̅ do to ∠𝐽𝐻𝐾 and ∠𝐽𝐼𝐾?
Answer:
1. SSS Congruence
Postulate

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2. ̅𝑯𝑱̅̅̅ ≅ ̅𝑯𝑲̅̅̅̅, 𝑱𝑰̅ ≅ ̅𝑲𝑰̅̅̅,

̅𝑯𝑰̅̅̅ ≅ ̅𝑯𝑰̅̅̅
3. ∠1 ≅ ∠4, ∠2 ≅ ∠5
and ∠3 ≅ ∠6
4. KITE
5. Yes, because it is the
common side
6. ∠1 ≅ ∠4 and ∠2 ≅
∠5
7. The pairs of
adjacent angles are
congruent as they
are corresponding
parts of congruent
triangles.
8. ̅𝑯𝑰̅̅̅ is an angle
bisector

VII. Additional Activities I

CAN DO IT!
Situation: Your barangay will hold a kite-making contest and submission of kite
design is required. On a clean bond paper, draw the design of the kite that you
want as an entry for the contest. The following rubric will be used to judge your
kite design.

Rubrics:
Categories and 5 4 3
Criteria Good Fair Poor
Design Geometry, Right Any two of None or only one
Angles, and geometry, right of geometry, right
Symmetry are angles and angles, and
properly symmetry are symmetry is
illustrated in the illustrated in the illustrated in the
design. design. design.
Creativity Design Design is artistic, Design is basic,
incorporates 50% functional, 20% functional,
artistic elements, and 50% original. and 20% original.
100% functional,
and 100%
original.

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