Question

complete the proof that qs ↔ ∥ tv ↔. 1. ∠sru ≅ ∠rut given 2. ∠prq ≅ ∠sru vertical angle theorem 3. ∠rut ≅ ∠prq 4. qs ↔ ∥ tv ↔ reasons: given, vertical angle theorem, corresponding angles theorem, properties of addition, subtraction, multiplication, and division, reflexive property of congruence, reflexive property of equality, substitution

Explanation:

Step1: Given angle - equality

$\angle SRU\cong\angle RUT$ is given.

Step2: Vertical - angle property

$\angle PRQ\cong\angle SRU$ by the Vertical Angle Theorem (vertical angles are congruent).

Step3: Transitive property of congruence

Since $\angle PRQ\cong\angle SRU$ and $\angle SRU\cong\angle RUT$, then $\angle RUT\cong\angle PRQ$ by the Transitive Property of Congruence.

Step4: Corresponding - angles imply parallel lines

$\angle RUT$ and $\angle PRQ$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. So, $\overleftrightarrow{QS}\parallel\overleftrightarrow{TV}$ by the Corresponding Angles Theorem.

Answer:

1. Given
2. Vertical Angle Theorem
3. Transitive Property of Congruence
4. Corresponding Angles Theorem