Short Answer
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Supply the missing reasons to complete the proof.
Given: angleQ congruent to angleT and Line segment Q Rcongruent to segment T R
Prove: Segment P R is congruent to Segment S R
An incomplete 2 column proof is shown with a figure.
The figure shows Triangles Q P R and S T R with common vertex of R formed by the intersection of segments Q T and P S.
Proof:
Statement 1: angle Q is congruent to angle T and segment Q R is congruent to segment T R
Reason 1: Given
Statement 2: angle P R Q is congruent to angle S R T
Reason 2: vertical angles are congruent
Statement 3: Triangle P R Q is congruent to triangle S R T
Reason 3: Question Mark
Statement 4: Segment P R is congruent to Segment S R
Reason 4: Question Mark
(2 points)
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Reason 3: The two triangles have two pairs of congruent angles and a pair of congruent sides (angle-angle-side congruence).
Reason 4: Corresponding parts of congruent triangles are congruent (CPCTC).
Proof:
Statement 1: angle Q is congruent to angle T and segment Q R is congruent to segment T R
Reason 1: Given
Statement 2: angle P R Q is congruent to angle S R T
Reason 2: Vertical angles are congruent
Statement 3: Triangle P R Q is congruent to triangle S R T
Reason 3: By Angle-Side-Angle (ASA) congruence theorem, as angle P R Q is congruent to angle S R T (given) and angle Q is congruent to angle T (given), and segment Q R is congruent to segment T R (given).
Statement 4: Segment P R is congruent to Segment S R
Reason 4: Corresponding parts of congruent triangles are congruent (CPCTC).
To complete the missing reasons in the proof, we need to provide explanations for Statements 3 and 4.
To determine the reason for Statement 3, we can use the information that we have congruent angles and congruent corresponding sides. Therefore, we can use the Side-Angle-Side (SAS) congruence postulate to conclude that Triangle PRQ is congruent to Triangle SRT.
The reason for Statement 3 is: SAS congruence postulate
To determine the reason for Statement 4, we can use the fact that corresponding parts of congruent triangles are congruent. Since Triangle PRQ is congruent to Triangle SRT from Statement 3, we can conclude that the corresponding sides are congruent. Specifically, the corresponding side PR of Triangle PRQ is congruent to the corresponding side SR of Triangle SRT.
The reason for Statement 4 is: Corresponding parts of congruent triangles are congruent (CPCTC)