Question

angles ptq and str are vertical angles and congruent.
which arcs are congruent?
\\(\widehat{sp}\\) and \\(\widehat{sr}\\)
\\(\widehat{pq}\\) and \\(\widehat{sr}\\)
\\(\widehat{pq}\\) and \\(\widehat{qr}\\)
\\(\widehat{sp}\\) and \\(\widehat{pr}\\)

Explanation:

Identify given geometric relationships

We are given a circle with intersecting chords \(PR\) and \(QS\) intersecting at point \(T\).
The problem states that angles \(\angle PTQ\) and \(\angle STR\) are vertical angles and are congruent:

$$\angle PTQ \cong \angle STR$$

Relate vertical angles to intercepted arcs

Using the Circle Geometry knowledge point, the measure of an angle formed by two intersecting chords inside a circle is related to the measures of its intercepted arcs.
Specifically, the vertical angles \(\angle PTQ\) and \(\angle STR\) intercept arcs \(\widehat{PQ}\) and \(\widehat{SR}\) respectively.

Apply arc measure properties

Using the Arc Measure knowledge point, since the vertical angles \(\angle PTQ\) and \(\angle STR\) are congruent, their corresponding intercepted arcs must also have equal measures:

$$m\widehat{PQ} = m\widehat{SR}$$

Therefore, the arcs \(\widehat{PQ}\) and \(\widehat{SR}\) are congruent.

Match with the given options

We evaluate the multiple-choice options:

• \(\widehat{SP}\) and \(\widehat{SR}\): Incorrect.
• \(\widehat{PQ}\) and \(\widehat{SR}\): Correct, as they are the intercepted arcs of the congruent vertical angles.
• \(\widehat{PQ}\) and \(\widehat{QR}\): Incorrect.
• \(\widehat{SP}\) and \(\widehat{PR}\): Incorrect.

Answer:

• (A) \(\widehat{SP}\) and \(\widehat{SR}\)
• (B) \(\widehat{PQ}\) and \(\widehat{SR}\) (Correct answer)
• (C) \(\widehat{PQ}\) and \(\widehat{QR}\)
• (D) \(\widehat{SP}\) and \(\widehat{PR}\)