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SSS, SAS, ASA, AAS, and HL theorems
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Two triangles are congruent if all corresponding sides and angles are equal.
Symbol:
You don't need to show all 6 parts are equal. These shortcuts work:
If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, the triangles are congruent.
Can you prove if , , and ?
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If two angles and the included side are congruent, the triangles are congruent.
If two angles and a non-included side are congruent, the triangles are congruent.
Right triangles only: If the hypotenuse and one leg are congruent, the triangles are congruent.
AAA - Shows similarity, not congruence SSA - Not sufficient (ambiguous case)
Corresponding Parts of Congruent Triangles are Congruent
Once you prove triangles are congruent, you can conclude ALL corresponding parts are equal.
We have three pairs of congruent sides:
This satisfies SSS (Side-Side-Side).
Answer: Yes, by SSS postulate
Given: , , . Which congruence postulate proves ?
We have:
In right triangles and (right angles at and ), and . Are the triangles congruent? If so, by what theorem?
Both are right triangles.
Given:
The side is included between the two angles and .
This is ASA (Angle-Side-Angle).
Answer: ASA
We have:
This satisfies HL (Hypotenuse-Leg) for right triangles.
Answer: Yes, by HL theorem