Question

Quad. is a rhombus.

If $\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{25}$, find the measures of $\angle \mathbf{2}\mathbf{,}\angle \mathbf{3}\mathbf{,}\angle \mathbf{4}\mathbf{and}\angle \mathbf{5}$.

Short answer

The measures of $\angle 2,\angle 3,\angle 4\text{and}\angle 5$are $\mathbf{25}\mathbf{°}\mathbf{,}\mathbf{65}\mathbf{°}\mathbf{,}\mathbf{65}\mathbf{°}\text{and}\mathbf{90}\mathbf{°}$respectively.

Step-by-step solution

Step 1. Property of rhombus.

The diagonals of a rhombus are perpendicular.

In quad. SLTM, LM and ST are diagonals of rhombus then by property of rhombus, $m\angle 5=90°$.

Step 2. Description of step.

Diagonals of a rhombus bisects the vertex angles.

Here, $\angle LSM$is divided into $\angle 1\text{and}\angle \text{2}$, that is, $m\angle 1=m\angle 2$. As $m\angle 1=25$, $m\angle 2=25°$.

Step 3. Description of step.

Here, $\angle SLT$is divided into $\angle 3\text{and}\angle 4$, that is, $m\angle 3=m\angle 4$.

Step 4. Description of step.

Here, $\angle S$and$\angle L$ are supplementary.

Substitute $m\angle 3$for $m\angle 4$.

$\begin{array}{c}m\angle 1+m\angle 2+m\angle 3+m\angle 4=180\\ 25+25+m\angle 3+m\angle 3=180\\ 2m\angle 3=180-50\\ 2m\angle 3=130\\ m\angle 3=65°\end{array}$

Step 5. Description of step.

As $m\angle 3=65°$and $m\angle 3=m\angle 4$implies that $m\angle 4=65°$.

Therefore, the measures of $\angle 2,\angle 3,\angle 4\text{and}\angle 5$are $25°,65°,65°\text{and90}°$respectively.