School Cayucay National High Grade Level 8

School
Teacher Rosemarie T. Bernal Learning Area MATHEMATICS
Teaching Dates MARCH 27, 2023 Quarter Third Quarter
DETAILED LESSON and Time 7:45-8:45 am and 1:00-
PLAN 2:00 pm

Quarter 3, Week 7
I. OBJECTIVES
The learner demonstrates understanding of key concepts of axiomatic structure of
A. Content standards
geometry and triangle congruence
The learner is able to
a. Formulate an organized plan to handle a real-life situation
b. Communicate mathematical thinking with coherence and clarity in
B. Performance Standards
formulating, investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and accurate
representations.
Most Essential Learning Competency: The learner proves statements on triangles.
(M8GE-IIIg-1)
C. Learning Competencies /
At the end of the discussion the learner should be able to:
Objectives
a. State the H-AA (Hypotenuse-Acute Angle) congruence theorem.
b. Illustrate the H-AA (Hypotenuse-Acute Angle) congruence theorem.
c. Proving statements on triangle congruence involving real life situation.

II. CONTENT

III. LEARNING RESOURCES

A. References
1. Teacher’s Guide pages
2. Learner’s Material pages ADM Mathematics 8 Quarter 3-Module 7, pp. 9-10
Exploring Mathematics 8, pp. 360-380; Geometry (Illinois, Prentice Mathematics),
3. Textbook pages
pp. 235-230
4. Additional Materials from
Learning Resource (LR) portal
B. Other Learning Resources
PowerPoint Presentation and Chalk and Board
C. Materials
IV. PROCEDURE

Review

You have proven triangles congruence using the SSS Postulate, SAS Postulate,
ASA Postulate, and SAA Theorem.

Let’s Reflexive Property of have a review of geometric

(Pre-Developmental) Congruence properties used to prove the triangles
A. Reviewing Previous Lesson Definition of Angle Bisector congruence postulate.
or Presenting the New Lesson Vertical Angle Theorem
PAIC

B. Establishing a purpose for

the lesson Drill:

Direction: Identify the images below. Hint: Terms in Geometry

+
+ - cycle
 Now that you have already an in depth understanding regarding Triangle
Congruence Postulate, let us now proceed with our new lesson, proving statements
on triangles. You will illustrate another theorem to prove that the triangles are
congruent, but the triangles you are going to encounter revolve around right
triangles alone.

Below is the definition and illustration of Right Triangle

The students will read and illustrate the definition of Right Triangle
(Developmental)
C. Presenting
Right Triangle
Examples/Instances of the New
Lesson (Literacy and Arts A
A right-angled triangle is a triangle
Integration)
with one of the angles as 90° . A 90-
degree angle is called a right angle,
and hence the triangle with a right
angle is called a right triangle. The Leg
longest side of the right triangle,
which is also the side opposite the
right angle, is the hypotenuse and the C
arms of the right are the legs or the B Leg
height and base.

In this lesson, you are going to prove statements on right triangle congruence
theorem. Today, you are going to focus on the H-AA (Hypotenuse-Acute Angle)
congruence theorem.

Below is the definition and illustration of H-AA (Hypotenuse-Acute Angle)

congruence theorem.

The H-A (The If ∠C and ∠ R are right

A
Hypotenuse-Acute angles, AB ≅ PQ , and
D. Discussing New Concepts
Angle Theorem) ∠ B ≅ ∠Q, then ∆ ABC
and Practicing New Skills #1
B ≅ ∆PQR
If hypotenuse and an C R
acute of one right
triangle are congruent to P
the corresponding
hypotenuse and an acute
angle of another, then R
Q

the two triangles are

congruent.

E. Discussing New
Concepts and Practicing Geometric Property
New Skills #2
Definition of Perpendicularity

Perpendicular If line
AB ⊥ AC , then
∠ BAC is a ¿ angle

A
P

Given: ∠C and ∠ R are ¿ angles .

B
C
Q R AB ≅ PQ ; ∠ B ≅ ∠Q
R
 Prove: ∆ ABC ≅ ∆PQR

Two column proof:

Statements Reasons
1. ∠C and ∠ R are ¿ angles . Given
2. ∠C≅ ∠ R Any two right angles are congruent
3. AB ≅ PQ Given
4. ∆ ABC ≅ ∆ PQR SAA theorem

Example 1: Use the H-AA (Hypotenuse -Acute Angle) Theorem to prove the
two triangles are congruent.

Use the two-column proof.

Given: ∠Y and ∠O are right angles

P O
PY ∥ ON
Prove: ∆ PYN≅ ∆ NOP

Y N

Proof:

Statements Reasons
1. ∠Y and ∠O are right angles Given
2. PY ∥ ON Given
3. ∠Y≅ ∠O Any two right angles are congruent
4. ∠YPN≅ ∠ONP PAIC
5. PN ≅ PN Reflexive Property of Congruence
6. ∆ PYN≅ ∆ NOP H-AA Congruence Theorem

Example 2: Use the H-AA (Hypotenuse-Acute Angle) Theorem to prove the

two triangles are congruent.
 Given: SN ≅ TN

SR ⊥ ℜ
TE ⊥ ℜ
S

Prove: ∆ SRN≅ ∆ TEN N E

R

Proof:

Statements Reasons
1. SN ≅ TN Given
2. SR ⊥ ℜ Given
3. TE ⊥ ℜ Given
4. ∠ R∧∠ E are right angles Definition of Perpendicularity
5. ∠ R ≅ ∠ E Any two right angles are congruent
6. ∠ SNR ≅ ∠ TNE Vertical Angle theorem
7. ∆ SRN≅ ∆ TEN H-AA Congruence Theorem

Activity: Prove two triangles are congruent using H-AA Congruence

Postulate.
L
1. Given: ⃗
LP bisects ∠ MLN
F. Developing Mastery PM ⊥⃗LM
PN ⊥⃗
ln M N
Prove: ∆ MLP≅ ∆ PNL
P

Proof:
Statements Reasons
 1. ⃗
LP bisects ∠ MLN Given
2. PM ⊥⃗ LM Given
3. PN ⊥⃗ln Given
4. ∠ MLP ≅ ∠ NLP Angle Bisector
5. ∠ M ∧∠ N are right angles Definition of Perpendicularity
6. ∠ M ≅∠ N Any two right angles are congruent
7. LP ≅ LP Reflexive Property of Congruence
8. ∆ MLP≅ ∆ PNL H-AA Congruence Theorem

Q R
2. Given: RS≅ TS
RQ ⊥ QU
TU ⊥ QU

Prove: ∆ RQS≅ ∆ TUS S

T U

Proof:
Statements Reasons
1. RS≅ TS Given
2. RQ ⊥ QU Given
3. TU ⊥ QU Given
4. ∠ Q∧∠ U are right angles Definition of Perpendicularity
5. ∠Q ≅ ∠U Any two right angles are congruent
6. ∠ RSQ ≅∠TSU Vertical Angle theorem
7. ∆ SRN≅ ∆ TEN H-AA Congruence Theorem

Activity:

When a light ray from an object meets mirror, it is reflected back to your eye.
For example, in the diagram, a light ray from point C is reflected at point D and
travels back to point A, ∠CDB , is congruent to the angle of reflection, ∠ ADB.

Given: ∠CDB ≅ ∠ ADB

DB ⊥ AC
AD ≅CD
(Post-Developmental)
G. Finding Practical Prove that ∆ ABD≅ ∆ CBD
Application and Skills
in Daily Living
(Science Integration)

Proof:
Statements Reasons
1. ∠ CDB ≅ ∠ ADB Given
2. DB ⊥ AC Given
3. AD ≅CD Given
4. ∠ ABD∧∠CBD are right Definition of Perpendicularity
angles
5. ∠ ABD ≅ ∠CBD Any two right angles are congruent
6. ∆ ABD≅ ∆ CBD H-AA Congruence Theorem

H. Making Generalization
 The teacher will ask the following questions

and Abstraction About 1. How can we prove that the two right triangles are congruent?
the Lessons 2. How do we use geometric properties in proving two right triangles are
congruent?

Prove two triangles are congruent using H-AA Congruence Theorem.

J
Given: KI≅ JI
KH ⊥ HL I
H
JL ⊥ LH L

Prove: ∆ KHI≅ ∆ JLI K

I. Evaluating Learning Proof:

Statements Reasons
1. KI≅ JI Given
2. KH ⊥ HL Given
3. JL ⊥ LH Given
4. ∠ H nad ∠ L are ¿ angles Definition of Perpendicularity
5. ∠ H ≅ ∠ L Any two right angles are congruent
6. ∠ KIH ≅ ∠ H LIJ Vertical Angle Theorem
7. ∆ KHI≅ ∆ JLI H-AA Congruence Postulate

J. Additional Activities for

Application or Have an advance study regarding H-Leg (Hypotenuse-Leg) Congruence Theorem
Remediation

V. REMARKS

VI. REFLECTION
No. of learners who earned 80%
in the evaluation

No. of learners who require

additional activities for
remediation who scored 80%
below
Did the remedial lessons work?
No. of learners who have caught
up with the lesson
No. of learners who continue to
require remediation
Which of my teaching strategies
worked well? Why did this work?
What difficulties did I encounter
which my principal or supervisor
can help me solve?

Prepared by: Checked by: Noted by:

ROSEMARIE T. BERNAL JHEREMY R. GARCIA REY B. PASCUA

Practice Teacher Cooperating Teacher Principal I