ralph wants to prove that his construction of an angle bisector of $ \angle rpq $ is correct. to do this, he needs to prove that $ \angle qps \cong \angle rps $. ralph extends his angle bisector to cross over a ray constructed from point $ q $ that is parallel to $ \overline{pr} $. the intersection of the bisector extension and the ray is labeled point $ t $ as shown in the diagram. ralph finds that $ \overline{qp} \cong \overline{qt} $.
based on the given information, which of the following angles must be congruent to qps? select all that apply.
$ \angle pqs $
$ \angle qts $
$ \angle tqs $
$ \angle prs $
$ \angle rps $

Question

Accepted Answer

# Explanation:
## Step1: Use properties of parallel lines
Since $QT\parallel PR$, by the alternate - interior angles theorem, $\angle QTS=\angle RPS$.
## Step2: Use isosceles triangle property
Given $\overline{QP}\cong\overline{QT}$, in $\triangle QPT$, by the isosceles triangle theorem ($QP = QT$), $\angle QPS=\angle QTS$.
## Step3: Transitive property
Since $\angle QPS=\angle QTS$ and $\angle QTS=\angle RPS$, by the transitive property of congruence, $\angle QPS=\angle RPS$.

# Answer:
$\angle QTS$, $\angle RPS$

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QUESTION IMAGE

Question

Explanation:

Step1: Use properties of parallel lines

Step2: Use isosceles triangle property

Step3: Transitive property

Answer: