Question

in the figure below, the measure of ∠4 = 90°, the measure of ∠5 = 56°, and the measure of ∠6 = 34°. what are the measures of ∠1, ∠2, and ∠3? m∠1 = 90° m∠2 = 56° m∠3 = □°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles. Since $\angle4 = 90^{\circ}$, then $m\angle1=90^{\circ}$. $\angle2$ and $\angle5$ are vertical angles. Since $\angle5 = 56^{\circ}$, then $m\angle2 = 56^{\circ}$.

Step2: Use angle - sum property of a full - turn

The sum of angles around a point is $360^{\circ}$. Also, we know that $\angle1+\angle2+\angle3+\angle4+\angle5+\angle6 = 360^{\circ}$. Substitute the known values: $90^{\circ}+56^{\circ}+m\angle3 + 90^{\circ}+56^{\circ}+34^{\circ}=360^{\circ}$. Combine like - terms: $(90 + 90)+(56 + 56)+34+m\angle3=360^{\circ}$, $180^{\circ}+112^{\circ}+34^{\circ}+m\angle3=360^{\circ}$, $326^{\circ}+m\angle3=360^{\circ}$.

Step3: Solve for $\angle3$

Subtract $326^{\circ}$ from both sides of the equation: $m\angle3=360^{\circ}-326^{\circ}=34^{\circ}$.

Answer:

$34$