Question

$\triangle prt$ is equilateral. complete the proof that $\triangle pqr \cong \triangle tsr$.

	statement	reason
2	$\angle qpr \cong \angle rts$	given
3	$\angle srt \cong \angle prq$
4	$\overline{rt} \cong \overline{pr}$
5	$\triangle pqr \cong \triangle tsr$

Explanation:

Step1: Analyze ∠SRT ≅ ∠PRQ

Vertical angles are equal. ∠SRT and ∠PRQ are vertical angles formed by the intersection of lines PQ and TS (or PR and RT? Wait, actually, the intersection of the two diagonals? Wait, in the diagram, the lines intersect at R? Wait, no, the triangles share the vertical angles at R. So ∠SRT and ∠PRQ are vertical angles, so they are congruent by the Vertical Angles Theorem.

Step2: Analyze \(\overline{RT} \cong \overline{PR}\)

Since \(\triangle PRT\) is equilateral, all sides are equal. So \(PR = RT = PT\), so \(\overline{RT} \cong \overline{PR}\) by the definition of an equilateral triangle (all sides congruent).

Step3: Analyze \(\triangle PQR \cong \triangle TSR\)

We have two angles and a side: ∠QPR ≅ ∠RTS (given), \(\overline{RT} \cong \overline{PR}\) (from step 2), and ∠SRT ≅ ∠PRQ (from step 1). So by the ASA (Angle - Side - Angle) Congruence Postulate, \(\triangle PQR \cong \triangle TSR\).

For statement 3, the reason is "Vertical Angles Theorem" (vertical angles are congruent). For statement 4, the reason is "Sides of an equilateral triangle are congruent" (or "Definition of equilateral triangle"). For statement 5, the reason is "ASA Congruence Postulate".

But let's focus on filling the reasons:

Statement 3: ∠SRT ≅ ∠PRQ, Reason: Vertical Angles Theorem (vertical angles are congruent).

Statement 4: \(\overline{RT} \cong \overline{PR}\), Reason: All sides of an equilateral triangle are congruent (since \(\triangle PRT\) is equilateral, \(PR = RT\)).

Statement 5: \(\triangle PQR \cong \triangle TSR\), Reason: ASA (Angle - Side - Angle) Congruence Postulate (we have ∠QPR ≅ ∠RTS, \(\overline{PR} \cong \overline{RT}\), and ∠PRQ ≅ ∠SRT, so ASA).

But let's answer the blanks:

For statement 3: Reason is "Vertical Angles are congruent" (or "Vertical Angles Theorem").

For statement 4: Reason is "Sides of an equilateral triangle are congruent" (since \(\triangle PRT\) is equilateral, \(PR = RT\)).

For statement 5: Reason is "ASA (Angle - Side - Angle) Congruence Postulate".

But let's check the steps:

1. Given: \(\triangle PRT\) is equilateral.

1. Given: ∠QPR ≅ ∠RTS.

1. ∠SRT ≅ ∠PRQ: Reason - Vertical Angles Theorem (vertical angles are congruent).

1. \(\overline{RT} \cong \overline{PR}\): Reason - In an equilateral triangle, all sides are congruent (so \(PR = RT\)).

1. \(\triangle PQR \cong \triangle TSR\): Reason - ASA (Angle - Side - Angle) Congruence Postulate (because we have two angles and the included side: ∠QPR ≅ ∠RTS, \(\overline{PR} \cong \overline{RT}\), and ∠PRQ ≅ ∠SRT).

So filling the reasons:

Statement 3 Reason: Vertical Angles Theorem

Statement 4 Reason: Sides of an equilateral triangle are congruent

Statement 5 Reason: ASA Congruence Postulate

But the question says "Complete the proof", so we need to fill the reasons for each statement (3,4,5). Let's assume we need to fill each:

For statement 3: ∠SRT ≅ ∠PRQ, Reason: Vertical Angles are congruent (Vertical Angles Theorem).

For statement 4: \(\overline{RT} \cong \overline{PR}\), Reason: Equilateral triangles have all sides congruent (so \(PR = RT\)).

For statement 5: \(\triangle PQR \cong \triangle TSR\), Reason: ASA (Angle - Side - Angle) Congruence.

Answer:

For statement 3: Vertical Angles Theorem
For statement 4: Sides of an equilateral triangle are congruent
For statement 5: ASA (Angle - Side - Angle) Congruence Postulate