Question

question 1-3
triangles rst and xyz are shown below.
emma knows that ∠r ≅ ∠x and ∠t ≅ ∠z. she claims that triangles rst and xyz are congruent. as part of her reasoning, which criterion could she use? select all that apply.
☐ angle-angle (aa)
☐ hypotenuse-leg (hl)
☐ side-angle-side (sas)
☐ angle-side-angle (asa)
☐ side-side-side (sss)

Explanation:

1. First, identify the given congruent parts: $\angle R \cong \angle X$, $\angle T \cong \angle Z$, and from the diagram, side $RT \cong XZ$ (the sides between the given angles).
2. Match this to congruence criteria: two pairs of congruent angles and the included congruent side correspond to the Angle-Side-Angle (ASA) rule.
3. Additionally, since two pairs of angles are congruent, the third pair of angles is also congruent (by triangle angle sum theorem), making the triangles similar via AA. But since we also have a congruent side, ASA is the direct congruence criterion here. AA alone only proves similarity, not congruence, so it does not apply for congruence.
4. Eliminate other options: HL is for right triangles only, SAS requires two sides and included angle, SSS requires all three sides, none of which match the given information.

Answer:

Angle-Side-Angle (ASA)