Question

find the measures of the missing angles. m∠1 = select m∠2 = select m∠3 = select m∠4 = select m∠5 = select

Explanation:

Step1: Find $m\angle3$

Vertical angles are equal. Since the angle opposite $\angle3$ is $55^{\circ}$, $m\angle3 = 55^{\circ}$.

Step2: Find $m\angle4$

Linear - pair angles sum to $180^{\circ}$. Since the adjacent angle to $\angle4$ is $105^{\circ}$, $m\angle4=180 - 105=75^{\circ}$.

Step3: Find $m\angle1$ and $m\angle2$

In the top - triangle, we know one angle is $55^{\circ}$ (the vertical angle). Let's use the angle - sum property of a triangle ($180^{\circ}$). In the triangle with $\angle1$, $\angle2$ and the $55^{\circ}$ angle, and since the two sides are equal (indicated by the marks), $\angle1=\angle2$. So $m\angle1 = m\angle2=\frac{180 - 55}{2}=62.5^{\circ}$.

Step4: Find $m\angle5$

In the bottom - triangle, we know two angles: one is $55^{\circ}$ (vertical angle) and another is $75^{\circ}$ ($\angle4$). Using the angle - sum property of a triangle ($180^{\circ}$), $m\angle5=180-(55 + 75)=50^{\circ}$.

Answer:

$m\angle1 = 62.5^{\circ}$
$m\angle2 = 62.5^{\circ}$
$m\angle3 = 55^{\circ}$
$m\angle4 = 75^{\circ}$
$m\angle5 = 50^{\circ}$