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LC Math 2 Adv – Triangle Congruence: ASA and SSS Congruence (LT 7)
ANGLE-SIDE-ANGLE TRIANGLE CONGRUENCE CRITERIA (ASA): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′. If 𝑚∠𝐶𝐴𝐵 =
𝑚∠𝐶 ′ 𝐴′ 𝐵′ (Angle), 𝐴𝐵 = 𝐴′𝐵′ (Side), and 𝑚∠𝐶𝐵𝐴 = 𝑚∠𝐶 ′ 𝐵′ 𝐴′ (Angle), then the triangles are congruent.
Proof:
Assuming ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′ are distinct triangles where 𝒎∠𝑪𝑨𝑩 = 𝒎∠𝑪′ 𝑨′ 𝑩′ , 𝑨𝑩 = 𝑨′𝑩′, and 𝒎∠𝑪𝑩𝑨 =
𝒎∠𝑪′ 𝑩′ 𝑨′, what transformations are needed to yield the diagram to the right? Be
specific.
What is the final isometry needed to prove that ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐵𝐶′?
Do we know that 𝑟(𝐶′′′) = 𝐶? Explain.
We have shown that ∆____________ ≅ ∆____________. This means that two triangles that have a pair of congruent sides, and
a pair of congruent included angles must be congruent.
The ASA criteria implies the existence of…
SIDE-SIDE-SIDE TRIANGLE CONGRUENCE CRITERIA (SSS): Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′, if 𝐴𝐵 = 𝐴′𝐵′
(Side), 𝐴𝐶 = 𝐴′𝐶′ (Side), and 𝐵𝐶 = 𝐵′𝐶′ (Side), then the triangles are congruent.
Proof:
Assuming ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′ are distinct triangles where 𝑨𝑩 = 𝑨′𝑩′, 𝑨𝑪 = 𝑨′𝑪′, and 𝑩𝑪 = 𝑩′𝑪′ , what
transformations are needed to yield the diagram to the right? Be specific.
Without any information about the angles, we cannot perform a reflection like we have for SAS and ASA. So instead, we
̅̅̅̅̅ and label the angles formed as r, s, t and u.
will draw an auxiliary line 𝐵𝐵′
How can △ 𝑨𝑩𝐵′ and △ 𝐵𝑩′𝑪 be used to show that △ 𝑨𝑩𝑪 and △ 𝑨′𝑩′𝑪′?
(Hint: Can you use a previous congruence criteria?)
We have shown that ∆____________ ≅ ∆____________. This means that two triangles that have three pairs of congruent
sides must be congruent.
The SSS criteria implies the existence of…
Exercises
Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists
between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them.
1.
Given:
̅̅̅̅, 𝑚∠𝐻 = 𝑚∠𝑃
𝑀 is the midpoint of 𝐻𝑃
2.
Given:
Rectangle 𝐽𝐾𝐿𝑀 with diagonal ̅̅̅̅̅
𝐾𝑀
3.
Given:
𝑅𝑌 = 𝑅𝐵, 𝐴𝑅 = 𝑋𝑅
4.
Given:
𝑚∠𝐴 = 𝑚∠𝐷, 𝐴𝐸 = 𝐷𝐸
5.
Given:
𝐴𝐵 = 𝐴𝐶, 𝐵𝐷 = 4 𝐴𝐵, 𝐶𝐸 = 4 𝐴𝐶
6.
Given:
Circles with centers 𝐴 and 𝐵 intersect at 𝐶 and 𝐷
1
1