Question

what are the corresponding parts of the figures below? select all that apply.
∠s ≅ ∠x
( overline {ts} cong overline {vw} )
∠r ≅ ∠x
( overline {sr} cong overline {xv} )

Explanation:

Step1: Analyze the first triangle

In the first triangle \( \triangle STR\), \(\angle T = 90^{\circ}\), \(\angle S=67^{\circ}\), so \(\angle R=180^{\circ}-90^{\circ}-67^{\circ} = 23^{\circ}\).

Step2: Analyze the second triangle

In the second triangle \( \triangle VWX\), \(\angle V = 90^{\circ}\), \(\angle W = 67^{\circ}\), so \(\angle X=180^{\circ}-90^{\circ}-67^{\circ}=23^{\circ}\).

Step3: Check angle - angle equality

Since \(\angle S = 67^{\circ}\) and \(\angle W = 67^{\circ}\), \(\angle S\cong\angle W\). Also, \(\angle R = 23^{\circ}\) and \(\angle X = 23^{\circ}\), \(\angle R\cong\angle X\). And \(\angle T=\angle V = 90^{\circ}\).
For the sides:

• In \(\triangle STR\) and \(\triangle VWX\), \(\overline{TS}\) and \(\overline{VW}\) are the sides adjacent to the \(67^{\circ}\) angle and opposite to the \(23^{\circ}\) angle.
• \(\overline{SR}\) and \(\overline{WX}\) are the hypotenuses of the right - angled triangles. \(\overline{TR}\) and \(\overline{VX}\) are the sides adjacent to the \(23^{\circ}\) angle and opposite to the \(67^{\circ}\) angle.

Answer:

\(\angle S\cong\angle X\) (Incorrect, \(\angle S = 67^{\circ}\), \(\angle X=23^{\circ}\)), \(\overline{TS}\cong\overline{VW}\) (Correct, as they are corresponding sides in similar right - angled triangles with the same set of angles), \(\angle R\cong\angle X\) (Correct, \(\angle R=\angle X = 23^{\circ}\)), \(\overline{SR}\cong\overline{XV}\) (Incorrect, \(\overline{SR}\) is the hypotenuse of \(\triangle STR\) and \(\overline{XV}\) is a non - hypotenuse side of \(\triangle VWX\)). So the correct ones are \(\overline{TS}\cong\overline{VW}\) and \(\angle R\cong\angle X\).