Question

Quad. FLAT is a rectangle.

If $\mathbf{m}\angle \mathbf{1}\mathbf{=}\mathbf{18}$find the measures of$\angle \mathbf{2}\mathbf{,}\mathbf{}\angle \mathbf{3}$ and $\angle \mathbf{4}$.

Short answer

The measures of $\angle 2,\angle 3$and$\angle 4$ are $\mathbf{18}\mathbf{,}\mathbf{72}\text{and}\mathbf{72}$respectively.

Step-by-step solution

Step 1. Description of step.

The diagonals of the rectangle are equal in length and bisect each other.

Therefore, $OT=OA$. As $OT=OA$, by isosceles triangle theorem the angles opposite to these sides are equal, that is, $m\angle 1=m\angle 2$. Therefore $m\angle 2=18$.

Step 2. Description of step.

From the given figure it can be observed that $\angle 2$and $\angle 3$are complementary angles, such that,

$\begin{array}{c}m\angle 2+m\angle 3=90\\ 18+m\angle 3=90\\ m\angle 3=90-18\\ =72\end{array}$

Step 3. Description of step.

The diagonals of the rectangle are equal in length and bisect each other.

Therefore, $OA=OL$. As $OA=OL$, by isosceles triangle theorem the angles opposite to these sides are equal, that is, $m\angle 3=m\angle 4$. Therefore $m\angle 4=72$.

Therefore, the measures of $\angle 2,\angle 3$and $\angle 4$are $18,72\text{and72}$respectively.