Question

what additional congruence criteria is needed to prove the triangles are congruent by asa?

select one:
a. \\(\overline{qv} \cong \overline{tu}\\)
b. \\(\angle tvu \cong \angle quv\\)
c. \\(\overline{qu} \cong \overline{tv}\\)
d. \\(\angle t \cong \angle q\\)

Explanation:

⚡ Using what you learned: congruence criteria for triangles (sss, sas, asa, aas, hl)

Step 1: Identify given congruent parts

From the diagram, we have:

• One pair of congruent angles marked with red arcs:

$$ \angle TUV \cong \angle QVU $$

• A shared side between the two triangles \( \Delta TUV \) and \( \Delta QVU \):

$$ \overline{UV} \cong \overline{VU} \quad \text{(Reflexive Property)} $$

Step 2: Determine the requirement for ASA

The Angle-Side-Angle (ASA) congruence criterion requires two angles and the included side of one triangle to be congruent to two angles and the included side of another triangle.

For the shared side \( \overline{UV} \) to be the included side, we need the angles at both endpoints of this segment to be congruent:

• At endpoint \( U \): we already have \( \angle TUV \) and \( \angle QVU \).
• At endpoint \( V \): we need the other pair of angles sharing this side to be congruent:

$$ \angle TVU \cong \angle QUV $$

Answer:

b. \( \angle TVU \cong \angle QUV \)