Question

$\overline{\mathrm{GM}}$is a median of right $\Delta \mathrm{IRG}$

If $\mathrm{m}\angle 1=32$find the measures ofrole="math" localid="1637864336352" $\angle 2,\angle 3$ and $\angle 4$.

Short answer

The measures of $\angle 2,\angle 3\text{and}\angle 4$are $32,58\text{and}58$respectively.

Step-by-step solution

Step 1. Description of step.

The length of the median is equal to half the length of the hypotenuse.

Therefore, $RM=GM$.

Step 2. Description of step.

Consider $\Delta RMG$, in which $RM=GM$then by isosceles triangle theorem, $m\angle 1=m\angle 2=32$.

Consider $\Delta IRG$such that

$\begin{array}{c}m\angle IGR=m\angle 2+m\angle 3\\ =90\end{array}$

Step 3. Description of step.

Consider $\Delta IRG$then by an angle sum theorem,

$\begin{array}{c}m\angle 1+m\angle 2+m\angle 3+m\angle 4=180\\ 32+\left(m\angle 2+m\angle 3\right)+m\angle 4=180\\ 32+90+m\angle 4=180\\ m\angle 4=180-122\\ =58\end{array}$

Step 4. Description of step.

Substitute 32 for $m\angle 2$into $m\angle 2+m\angle 3=90$to find the value of $m\angle 3$.

$\begin{array}{c}m\angle 2+m\angle 3=90\\ 32+m\angle 3=90\\ m\angle 3=90-32\\ =58\end{array}$

Therefore, the measures of $\angle 2,\angle 3\text{and}\angle 4$are $32,58\text{and58}$respectively.