Question

Complete the following proof.Given: right angle QPS is congruent to right angle TPRPrive: right angle QPR is contingent to right angle TPSProof:Then answer b, c, e, and f

Answer 1

a.

`\angle QPS\cong\angle TPR`

Which means that the two angles are congruent.

Reason: Given

b.

`m\measuredangle QPS=m\measuredangle TPR`

Which means that the measure of angle QPS is equal to the measure of angle TPR.

Reason: Congruent angles

The measure of congruent angles are equal.

c.

`\begin{gathered} m\measuredangle QPS=m\measuredangle QPR+m\measuredangle RPS \\ m\measuredangle TPR=m\measuredangle TPS+m\measuredangle RPS \end{gathered}`

It means that the sum of angle QPR and angle RPS equals angle QPS.

Reason: Angle Addition Postulate

When we have two adjacent angles the measure of the angle containing the two angles equals the sum of the measure of the two angles.

`m\measuredangle ABC=m\measuredangle ABD+m\measuredangle DBC`

d.

Substituting equation b into the equation c;

`\begin{gathered} \text{Substituting;} \\ m\measuredangle QPS=m\measuredangle TPR \\ \text{ into} \\ m\measuredangle QPS=m\measuredangle QPR+m\measuredangle RPS \\ m\measuredangle TPR=m\measuredangle TPS+m\measuredangle RPS \end{gathered}`

So, we have;

`\begin{gathered} m\measuredangle TPR=m\measuredangle QPR+m\measuredangle RPS \\ m\measuredangle TPR=m\measuredangle TPS+m\measuredangle RPS \end{gathered}`

Reason: Substitution

e.

since we now have two equations for angle TPR in d above. we can now equate the two equations;

`\begin{gathered} m\measuredangle QPR+m\measuredangle RPS=m\measuredangle TPS+m\measuredangle RPS \\ m\measuredangle QPR=m\measuredangle TPS \end{gathered}`

Reason: Equating and subtraction

We equate two expressions of the same angle and subtract angle RPS from both sides.

f.

`\angle QPR\cong\angle TPS`

Reason: Congruent Angles.

Since the measure of angle QPR is equal to the measure of angle TPS ( as shown in e above) then the two angles are congruent.

Proved!