Question

what must be true for two triangles to be congruent?
a. only the angles must be equal
b. all corresponding sides and angles must be equal
c. corresponding angles must be equal but sides can differ
d. only one pair of corresponding sides must be equal

prove that △pqr ≅ △xyz using asa congruence, where ∠p = ∠x = 40°, ∠q = ∠y = 60°, and pq = xy = 5. fill in the missing congruence criterion.

statement	reason
pq = xy = 5	given: corresponding sides are congruent
∠q = ∠y = 60°	given: corresponding angles are congruent
△pqr ≅ △xyz	_____

a. aas congruence criterion
b. sss congruence criterion
c. hl congruence criterion
d. asa congruence criterion

Explanation:

First Question (What must be true for two triangles to be congruent?):

To determine the correct option, we recall the definition of congruent triangles. Congruent triangles are triangles that have the same size and shape, which means all corresponding sides and angles are equal.

• Option a is incorrect because only equal angles would mean similar (not congruent) triangles if sides differ.
• Option c is incorrect as differing sides would not make them congruent (that's similar triangles).
• Option d is incorrect because only one pair of equal sides is insufficient for congruence.
• Option b correctly states that all corresponding sides and angles must be equal for congruence.

The ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

• In $\triangle PQR$ and $\triangle XYZ$, we have $\angle P=\angle X$ (first angle), $PQ = XY$ (included side), and $\angle Q=\angle Y$ (second angle). This matches the ASA congruence criterion.
• Option a (AAS) is for two angles and a non - included side, option b (SSS) is for three sides, and option c (HL) is for right - angled triangles (hypotenuse - leg). So the correct criterion is ASA.

Answer:

b. All corresponding sides and angles must be equal

Second Question (Prove that $\boldsymbol{\triangle PQR \cong \triangle XYZ}$ using ASA congruence...):