Chapter 5.4, Problem 42WE

To determine

To prove: $\Delta RYZ$ is a equilateral triangle.

Explanation of Solution

Given information: 

The given figure is:

  

Rectangle RSTW

  $\Delta WYT\text{ and }\Delta STZ$ are equilateral triangle.

  $\begin{array}{l}WT=RS=WY=YT\\ ST=RW=SZ=ZT\end{array}$

Proof: 

In $\Delta WYR\text{ and }\Delta RSZ$

  $\begin{array}{l}WR=SZ(GIVEN)\\ \angle YWR=\angle RSZ\text{ (as }\angle YWT+\angle TWR=\angle RST+\angle TSZ\text{)}\\ WY=RS\text{ (Given)}\\ \text{By SAS rule}\\ \Delta WYR\cong \Delta RSZ\\ By\text{ congurent property}\\ \text{RY}=\text{RZ}\mathrm{.....}\text{(1)}\end{array}$

In $\Delta RSZ\text{ and }\Delta YTZ$

  $\begin{array}{l}SR=YT(GIVEN)\\ \angle RSZ=\angle YTZ\\ WY=RS\text{ (Given)}\\ \text{By SAS rule}\\ \Delta RSZ\cong \Delta YTZ\\ By\text{ congurent property}\\ \text{RZ}=\text{YZ}\\ \because \text{RZ}=\text{RY}\\ \therefore \text{RY}=\text{RZ}=\text{YZ}\\ \text{So }\Delta \text{RYZ is a equilateral triangle}\text{.}\end{array}$