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Title: Q4 Math8 PROVING TRIANGLE CONGRUENCE
Description: COMPLETE NOTES OF PROVING TRIANGLE CONGRUENCE FOR QUARTER 4 MATH8

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ST
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PHOENIX Math for the 21st Century Learners: Second Edition, PHOENIX Publishing House Inc
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Prove two triangles are congruent applying the triangle congruence postulates
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These can now be used to prove the congruence of triangles using deductive reasoning
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𝐡
EXAMPLE 1: Given: Μ…Μ…Μ…Μ…
𝐴𝐡 β‰… Μ…Μ…Μ…Μ…
𝐴𝐷, Μ…Μ…Μ…Μ…
𝐡𝐢 β‰… Μ…Μ…Μ…Μ…
𝐷𝐢
Prove: βˆ†π΄π΅πΆ β‰… βˆ†π΄π·πΆ
𝐢
Proof:
STATEMENT
Μ…Μ…Μ…Μ… β‰… 𝑨𝑫
Μ…Μ…Μ…Μ…
1
...
𝑩π‘ͺ
Μ…Μ…Μ…Μ… β‰… 𝑨π‘ͺ
Μ…Μ…Μ…Μ…
3
...
βˆ†π΄π΅πΆ β‰… βˆ†π΄π·πΆ

𝐴

REASONS
1
...
Given
3
...
SSS Postulate

𝐷

The 𝑆𝐴𝑆 Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two
sides and the included angle of another triangle, then the triangles are congruent
...

EXAMPLE 2: Given: Μ…Μ…Μ…Μ…
𝐴𝐡 β‰… Μ…Μ…Μ…Μ…
𝐢𝐷, Μ…Μ…Μ…Μ…
𝐴𝐡 βŠ₯ 𝐡𝐷
𝐢𝐷 βŠ₯ Μ…Μ…Μ…Μ…
𝐡𝐷, 𝐸 is the midpoint of 𝐡𝐷
Prove: βˆ†ABE β‰… βˆ†CDE
𝐴

𝐡

Proof:

𝐢

𝐸

𝐷

STATEMENT
Μ…Μ…Μ…Μ… β‰… π‘ͺ𝑫
Μ…Μ…Μ…Μ…
1
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∠𝐡 is a right angle
Μ…Μ…Μ…Μ… βŠ₯ 𝐡𝐷
Μ…Μ…Μ…Μ…
3
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βˆ†π΄π΅πΆ β‰… βˆ†π΄π·πΆ
5
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Μ…Μ…Μ…Μ… and 𝐡𝐸
Μ…Μ…Μ…Μ…
...

𝐢𝐷

Μ…Μ…Μ…Μ…Μ…
6
...

7
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βˆ†ABE β‰… βˆ†CDE

REASONS
1
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Definition of Perpendicular
3
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Definition of Perpendicular
5
...

6
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Definition of midpoint
8
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𝐴
EXAMPLE 3: Given: ∠1 β‰… ∠2, 𝐷 is the midpoint of Μ…Μ…Μ…Μ…
𝐡𝐢
...

1

2

Prove: βˆ†π΄π·π΅ β‰… βˆ†π΄π·πΆ
𝐡
Proof:

STATEMENT

𝐷

𝐢

REASONS

1
...
∠𝐴𝐡𝐷 and ∠1 form a linear pair
∠𝐴𝐢𝐷 and ∠2 form a linear pair
3
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βˆ π‘¨π‘©π‘« β‰… βˆ π‘¨π‘ͺ𝑫
5
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1
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Μ…Μ…Μ…Μ…Μ…
𝑩𝑫 β‰… Μ…Μ…Μ…Μ…
π‘ͺ𝑫
7
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∠𝐴𝐷𝐢 is a right angle
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βˆ π‘¨π‘«π‘© β‰… βˆ π‘¨π‘«π‘ͺ

6
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βˆ†π΄π·π΅ β‰… βˆ†π΄π·πΆ

9
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Definition of a linear pair
3
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Supplements to congruent angles are congruent
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Given

7
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Any two right angles are congruent
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KINDLE YOUR IDEAS

ACTIVITY
Direction: Complete each proof
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CHOICES:
β€’ Any two right angles are congruent
...
Given:

Prove:
Proof:

Μ…Μ…Μ…Μ… β‰… Μ…Μ…Μ…Μ…
𝐴𝐢
𝐷𝐸
Μ…Μ…Μ…Μ… βŠ₯ 𝐢𝐡
Μ…Μ…Μ…Μ… , 𝐷𝐸
Μ…Μ…Μ…Μ… βŠ₯ 𝐸𝐡
Μ…Μ…Μ…Μ…
𝐴𝐢
Μ…Μ…Μ…Μ… β‰… 𝐸𝐡
Μ…Μ…Μ…Μ…
𝐢𝐡
βˆ†π΄πΆπ΅ β‰… βˆ†π·πΈπ΅

STATEMENT
Μ…Μ…Μ…Μ… β‰… 𝐷𝐸
Μ…Μ…Μ…Μ…
1
...
𝐴𝐢
3
...


𝐡
𝐴

𝐷

REASONS
1
...

3
...
∠𝐢 β‰… ∠𝐸
5
...


6
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CHOICES:
β€’ Alternate-Interior Angles are equal
β€’ Given
β€’ Definition of Vertical Angles
Μ…Μ…Μ…Μ…
β€’ Μ…Μ…Μ…Μ…
𝐴𝐡||𝐷𝐸
β€’ Angle-Side-Angle Postulate
β€’ βˆ†π΄π΅πΆ β‰… βˆ†π·πΈπΆ

𝐸

5
...
Given:
Prove:
Proof:

𝐡

STATEMENT

𝐷

REASONS

1
...
Given

Μ…Μ…Μ…Μ… β‰… Μ…Μ…Μ…Μ…
2
...


3
...


4
...


5
...


𝐢

𝐴

𝐸

ADDITIONAL NOTES TO REVIEW:
SPECIAL RIGHT TRIANGLES
In a right triangle, if one angle measures 30˚, the length of the hypotenuse is twice the length of the side
opposite the angle with measure 30˚
...


If one angle of a right triangle measures 30˚, the length of the side opposite the angle with measure 60˚ is
equal toβˆšπŸ‘ times the length of the side opposite the 30˚ angle
...


4π‘₯ + 6

8π‘₯ βˆ’ 18

Solve for the value of x
...

Then transpose to combine similar terms, 6 + 18 = 8π‘₯ βˆ’ 4π‘₯
...

24 4π‘₯
=
4
4
6=π‘₯
,


Title: Q4 Math8 PROVING TRIANGLE CONGRUENCE
Description: COMPLETE NOTES OF PROVING TRIANGLE CONGRUENCE FOR QUARTER 4 MATH8