3.1: Congruence Criteria | Geometry Honors

How it works

Congruent triangles have matching side lengths and angle measures. SSS, SAS, ASA, AAS, and the right-triangle hypotenuse-leg criterion establish congruence. AAA establishes similarity; SSA is generally inconclusive.

\triangle ABC\cong\triangle DEF\ \Rightarrow\ AB=DE,\ BC=EF,\ AC=DF

Worked examples

Example 1

Triangles have respective side lengths $\displaystyle 3,4,5$ and $\displaystyle 3,4,5$ . Which criterion applies?

SSS proves congruence when corresponding sides are paired.

Example 2

Two sides and the included angle match. Is that enough?

Yes, SAS fixes the triangle. The angle must be between the matching sides.

Problems

Problem 3.1.1

Triangle $\displaystyle ABC$ has sides $\displaystyle AB=3$ , $\displaystyle BC=4$ , $\displaystyle AC=5$ . Triangle $\displaystyle DEF$ has $\displaystyle DE=3$ , $\displaystyle EF=4$ , $\displaystyle DF=5$ . State a valid congruence correspondence.

Triangle with vertices A at 0,3, B at 0,0, C at 4,0, and side lengths 3,4,5. — The right angle is at B; the hypotenuse AC has length 5.

Show worked solution

$\displaystyle \triangle ABC\cong\triangle DEF$ by SSS. In particular, $\displaystyle B$ corresponds to $\displaystyle E$ .

Triangle with vertices A at 0,3, B at 0,0, C at 4,0, and side lengths 3,4,5. The solution adds labeled relationships. — For angle C, opposite = 3, adjacent = 4, and hypotenuse = 5.

Problem 3.1.2

Why is AAA insufficient for congruence?

Show worked solution

Triangles with side lengths $\displaystyle 3,4,5$ and $\displaystyle 6,8,10$ have equal corresponding angles but different sizes.

Problem 3.1.3

Two right triangles have equal hypotenuses and one matching leg. Are they congruent?

Show worked solution

Yes, by hypotenuse-leg. Pythagoras also forces the remaining legs to match.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 3.1.1

Problem 3.1.2

Problem 3.1.3

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