Question

Helppppp! Angles PTQ and STR are vertical angles and congruent. Circle T is shown. Line segments T P, T Q, T R, and T S are radii. Lines are drawn to connect the points on the circle and form secants P Q, Q R, R S, and S P. Angles P T Q and S T R are congruent. Which chords are congruent? QP and SR QR and PR and RS PR and PS

Answer 1

Answer:

$\overline{QP} = \overline{SR}$

Step-by-step explanation:

Given: ∠PTQ ≅ ∠STR and TP, TQ, TR and TS are radii of the circle.

If  two angle are equal, then they are equal in size and their measurements are equal.

IF ∠PTQ ≅ ∠STR then they are also equal in their measurements.

Now, In ΔPTQ and ΔSTR

∠PTQ ≅ ∠STR  (vertically opposite angle)

$\overline{TQ} \cong \overline{ST}$  (Radius of the circle are equal and congruent in length)

$\overline{PT} \cong \overline{TR}$  (Radius of the circle)

∴ ΔPTQ ≅ ΔSTR (SAS postulate)

∴ $\overline{QP}\cong \overline{SR}$ (corresponding sides of congruent triangle are equal  and congruent).

Answer 2

You can use the fact that a closed figure made by connecting the adjacent end points of two equal and intersecting line segment is a rectangle.

Option A:  $\overline{QP} \: \rm and \: \overline{SR}$correct choice since the line segment QP is and line segment SR are congruent.

Given that:

• Angles PTQ and STR are vertical angle and congruent.
• Circle T(centered at T).
• Line segments TP, TQ, TR, and TS are radii.

How to find which lines are intersecting?

Since angle  PTQ and angle STR are congruent, thus we have:

PR and SQ as straight intersecting lines.

Due to angle PTS and angle QTR are opposite angles made by intersecting lines, we have them as congruent or say of equal measurement.

$\angle PTS \cong \angle QTR$

Since $\overline{TP} = \overline{TQ} = \overline{TR} = \overline{TS}$, thus we have got:

$\overline{PR} = \overline{SQ}$ in measurement ( as PR is sum of PT and TR and same for SQ)

How is a rectangle made?

"A closed figure made by connecting the adjacent end points of two equal and intersecting line segment is a rectangle", we have PQRS as a rectangle.

As a rectangle have opposite sides parallel and equal, we have got:

$\overline{PQ} \cong \overline{SR}$

Thus, Option A: $\overline{QP} \: \rm and \: \overline{SR}$ is the correct choice since the line segment QP is and line segment SR are congruent.

Learn more about rectangles here:

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