Question

Co.plete the following proof.Given: right angle QPS is congruent to right angle TPRProve: right angle QPR is congruent to right angle TPSProof:

Answer 1

We know that

$m\angle QPS=m\angle TPR$

We also know that

$\begin{gathered} m\angle QPS=m\angle QPR+m\angle RPS \\ m\angle TPR=m\angle TPS+m\angle RPS \end{gathered}$

If we substitute the last two equations in the first one, then we have:

$\begin{gathered} m\angle QPS=m\angle TPR \\ m\angle QPR+m\angle RPS=m\angle TPS+m\angle RPS \end{gathered}$

We note that the angles m$\begin{gathered} m\angle QPR+m\angle RPS=m\angle TPS+m\angle RPS \\ m\angle QPR=m\angle TPS+m\angle RPS-m\angle RPS \\ m\angle QPR=m\angle TPS \end{gathered}$Therefore the angles QPR is congruent to the angle TPS.