Question

Use the information given in the diagram to prove that triangle PUX is congruent to triangle QSY. I have multiple photos I would like to upload on here.

Answer 1

SOLUTION

We want to use the information in the diagram to prove that

$\Delta PUX\cong\Delta\text{QSY}$

Now, we have been given for number 1

2.

$\begin{gathered} RS=VU \\ \text{Definition of congruent segments } \end{gathered}$

3.

$\begin{gathered} RU=RS+SU,VS=VU+SU \\ \text{Segment addition postulate } \end{gathered}$

4.

$\begin{gathered} VS=RS+SU \\ Substitution\text{ property of equality} \end{gathered}$

5.

$\begin{gathered} RU\cong VS \\ Transitive\text{ property of equality } \end{gathered}$

6.

$\begin{gathered} RU=VS \\ \text{Definition of congruent segments} \end{gathered}$

7.

$\begin{gathered} \Delta PUR\cong\Delta QSV \\ \text{ASA congruence theorem } \end{gathered}$

8.

$\begin{gathered} m\angle RUX\cong m\angle VSY \\ m\angle PUR\cong m\angle QSV \\ \text{Corresponding parts of congruent triangles are congruent } \end{gathered}$

9. and 10. is good (correct)

11.

$\begin{gathered} m\angle PUX=m\angle QSY \\ \text{Substitution property of equality} \end{gathered}$

12.

$\begin{gathered} m\angle QSY=m\angle PUR+m\angle RUX \\ \text{Transitive property of equality} \end{gathered}$

13.

$\begin{gathered} m\angle PUX=m\angle QSY \\ \text{Definition of congruent angles } \end{gathered}$

14.

$\begin{gathered} \Delta PUX\cong\Delta\text{QSY} \\ \text{ASA congruence theorem} \end{gathered}$