Question

use the given figure to answer the following. (the figure drawn to scale.) find b if a = 45° and α + β = 80°. b =

Explanation:

Step1: Recall triangle - angle sum property

In right - triangle $ACD$, $\angle ADC = 90^{\circ}$, and in $\triangle ABC$, the sum of interior angles is $180^{\circ}$. Also, in right - triangle $ACD$, $\angle A+\alpha = 90^{\circ}$. Given $\angle A = 45^{\circ}$, we can find $\alpha$.
Since $\angle A+\alpha=90^{\circ}$ and $\angle A = 45^{\circ}$, then $\alpha=90^{\circ}-\angle A=90^{\circ}-45^{\circ}=45^{\circ}$.

Step2: Find the value of $\beta$

Given $\alpha+\beta = 80^{\circ}$ and $\alpha = 45^{\circ}$, we can find $\beta$ by subtraction.
$\beta=80^{\circ}-\alpha$. Substituting $\alpha = 45^{\circ}$, we get $\beta = 80^{\circ}-45^{\circ}=35^{\circ}$.

Step3: Find angle $B$

In right - triangle $BCD$, $\angle BDC = 90^{\circ}$. Using the angle - sum property of a triangle ($\angle B+\angle BDC+\beta=180^{\circ}$ and $\angle BDC = 90^{\circ}$), we have $\angle B=90^{\circ}-\beta$.
Substituting $\beta = 35^{\circ}$, we get $\angle B = 55^{\circ}$.

Answer:

$55$