Question

Given: Rectangle RSTW; equilateral $\Delta \text{YWT}$ and STW.

State what is true of $\Delta \text{RYZ}$.

Write a paragraph proof.

Short answer

$\Delta \text{RYZ}$is an equilateral triangle.

Step-by-step solution

Step 1. Consider the diagram.

Here, RSTW is a rectangle and $\Delta WYT$ and $\Delta STZ$ are equilateral triangle.

Step 2. State the proof.

As RSTW is a rectangle and $\Delta WYT$and $\Delta STZ$are equilateral triangle.

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

In $\Delta WYR$and $\Delta RSZ$,

$WR=SZ$ (Given)

$\angle YWR=\angle RSZ$ (As $\angle YWT+\angle TWR=\angle RST+\angle TSZ$).

$WY=RS$ (Given)

By SAS rule,

$\Delta WYR\cong \Delta RSZ$

By congruent property,

$RY=RZ$ …… (1)

In $\Delta RSZ$and $\Delta YTZ$,

$SR=YT$ (Given)

$\angle RSZ=\angle YTZ$

$SZ=ZT$ (Given)

By SAS rule,

$\Delta RSZ\cong \Delta YTZ$

By congruent property,

$RZ=YZ$

Since, $RZ=RY$

Therefore, $RY=RZ=YZ$.

So, $\Delta RYZ$is an equilateral triangle.

Step 3. State the conclusion.

Therefore, $\Delta RYZ$is an equilateral triangle (proved).