Congruent triangles are triangles with identical sides and angles. The three sides of one are exactly equal in measure to the three sides of another. The three angles of one are each the same angle as the other.
Five ways are available for finding two triangles congruent:
An included angle lies between two named sides. In △CAT below, included ∠A is between sides t and c:
An included side lies between two named angles of the triangle.
A postulate is a statement taken to be true without proof. The SSS Postulate tells us,
Congruence of sides is shown with little hatch marks, like this: ∥. For two triangles, sides may be marked with one, two, and three hatch marks.
If △ACE has sides identical in measure to the three sides of △HUM, then the two triangles are congruent by SSS:
The SAS Postulate tells us,
△HUG and △LAB each have one angle measuring exactly 63°. Corresponding sides g and b are congruent. Sides h and l are congruent.
A side, an included angle, and a side on △HUG and on △LAB are congruent. So, by SAS, the two triangles are congruent.
This postulate says,
We have △MAC and △CHZ, with side m congruent to side c. ∠A is congruent to ∠H, while ∠C is congruent to ∠Z. By the ASA Postulate these two triangles are congruent.
We are given two angles and the non-included side, the side opposite one of the angles. The Angle Angle Side Theorem says,
Here are congruent △POT and △LID, with two measured angles of 56° and 52°, and a non-included side of 13 centimeters:
[construct as described]
By the AAS Theorem, these two triangles are congruent.
Exclusively for right triangles, the HL Postulate tells us,
The hypotenuse of a right triangle is the longest side. The other two sides are legs. Either leg can be congruent between the two triangles.
Here are right triangles △COW and △PIG, with hypotenuses of sides w and i congruent. Legs o and g are also congruent:
[insert congruent right triangles left-facing △COW and right facing △PIG]
So, by the HL Postulate, these two triangles are congruent, even if they are facing in different directions.
Given: △MAG and △ICG
MC ≅ AI
AG ≅ GI
Prove: △MAG ≅ △ICG
MC ≅ AI Given
AG ≅ GI
∠MGA ≅ ∠ IGC Vertical Angles are Congruent
△MAG ≅ △ICG Side Angle Side
If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
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