In order to find the value of x, we can write the following statements and reasons:
Triangle PQR:
1. Statement: PS = PS
Reason: Reflexive property of equality
2. Statement: angle PSQ is congruent to angle PSR
Reason: Given
3. Statement: SQ is equal to SR
Reason: Given:
4. Statement: Triangle PSQ is congruent to triangle PSR
Reason: Case SAS of congruency
5. Statement: PQ is congruent to PR, so PR = 8
Reason: Corresponding sides of congruent triangles
Quadrilateral VSTU:
1. Statement: VT = VT
Reason: Reflexive property of equality
2. Statement: angle SVT is congruent to angle UVT
Reason: Given
3. Statement: angle VST is congruent to angle VUT
Reason: Given
4. Statement: Triangle VST is congruent to triangle VUT
Reason: Case AAS of congruency
5. Statement: ST = TU
Reason: Corresponding sides of congruent triangles
So we have 5z = 2z + 3. Solving for z, we have z = 1, so ST = 5z = 5
