### MATH 113: Triangle Congruence

This worksheet asks the question "when can we say that two triangles are congruent?"

Page 660 of the textbook tells us that "Two triangles are **congruent** if, and only if, there is a correspondence of vertices of the triangles such that the corresponding sides and corresponding angles are congruent."
In other words, two triangles are congruent if there's a way to "match up" their angles and sides.

We know from experience that if two triangles' side lengths match, then their angles must also match and they must be congruent.

We may also remember that even if all the angles of two triangles match, their sides may not match -- one triangle may be larger than the other.

What if two sides match, or two angles? How much information do we need to guarantee that two triangles are congruent? Work with a partner to answer that question for each scenario below. If you're not sure of an answer, try drawing two different triangles which match as described.

In each scenario, the first triangle has vertices ABC and the second has vertices DEF.

- If triangle 1 has side lengths AB = 3, BC = 4 and CA = 5 and triangle 2 has side lengths DE = 4, EF = 3 and FD = 5, are the two triangles congruent? If not, draw a counterexample.

- If the angle at vertex A measures 60 degrees, the angle at vertex B measures 40 degrees, the angle at vertex D measures 40 degrees and the angle at vertex F measures 60 degrees, must the two triangles be congruent? If not, draw a counterexample.

- If the angle at vertex A measures 60 degrees, the angle at vertex B measures 40 degrees, the angle at vertex D measures 40 degrees, the angle at vertex F measures 60 degrees, and AB = FD, must the two triangles be congruent? If not, draw a counterexample.

- If AB = DE, BC = EF and the measures of angles B and E are equal, must the two triangles be congruent? If not, draw a counterexample.

- If two side lengths of the two triangles match (AB = DE and BC = EF) then must the two triangles be congruent? If not, draw a counterexample.

- If AB = DE, BC = EF and the measures of angles A and D are equal, must the
two triangles be congruent? If not, draw a counterexample.

- If m ∠ A = m ∠ D, m ∠ B = m ∠ E and BC = EF, must the two triangles be congruent? If not, draw a counterexample.