Question

multiple choice 1 point
which of these statements, if true, is sufficient to prove that triangles str and pqr
are similar?

△pqr is isosceles
tq = \\(\frac{1}{2}\\)⋅qr
∠s ≅ ∠qpr
∠s ≅ ∠r

Explanation:

Step1: Identify shared angle

Triangles $STR$ and $PQR$ share $\angle R$.

Step2: Check AA similarity condition

For AA (Angle-Angle) similarity, we need one more pair of congruent angles. If $\angle S \cong \angle QPR$, we have two pairs of congruent angles: $\angle R \cong \angle R$ and $\angle S \cong \angle QPR$, which proves similarity.

Step3: Eliminate other options

• $\triangle PQR$ being isosceles gives no link to $\triangle STR$ angles/sides.
• $TQ=\frac{1}{2}QR$ does not confirm proportional sides for similarity.
• $\angle S \cong \angle R$ only gives one congruent angle in $\triangle STR$, not a pair across both triangles.

Answer:

$\angle S \cong \angle QPR$