Question

1. find the measure of each missing angle.

1. find the value of x.

Explanation:

Step1: Find \(m\angle3\)

In a right - triangle, the two non - right angles are complementary. The large right - triangle has angles \(90^{\circ}\), \(35^{\circ}\), and \(\angle3\). Using the angle - sum property of a triangle (\(m\angle A+m\angle B + m\angle C=180^{\circ}\)), for the large right - triangle:
\(m\angle3=180^{\circ}-90^{\circ}-35^{\circ}=55^{\circ}\)

Step2: Find \(m\angle4\)

Since the segments are marked as equal (the mid - segment in the triangle), \(\angle4\) and \(\angle3\) are equal (isosceles triangle property). So \(m\angle4 = m\angle3 = 55^{\circ}\)

Step3: Find \(m\angle5\)

\(\angle4\) and \(\angle5\) are supplementary (\(m\angle4 + m\angle5=180^{\circ}\)). Then \(m\angle5=180^{\circ}-m\angle4=180^{\circ}-55^{\circ}=125^{\circ}\)

Step4: Find \(m\angle1\)

In the small right - triangle with right angle \(90^{\circ}\) and \(\angle5\) as an exterior angle. Using the exterior - angle property (\(m\angle5=m\angle1 + 90^{\circ}\)), then \(m\angle1=m\angle5 - 90^{\circ}=125^{\circ}-90^{\circ}=35^{\circ}\)

Step5: Find \(m\angle2\)

\(\angle1\) and \(\angle2\) are supplementary (\(m\angle1 + m\angle2=180^{\circ}\)). So \(m\angle2=180^{\circ}-m\angle1=180^{\circ}-35^{\circ}=145^{\circ}\)

Answer:

\(m\angle1 = 35^{\circ}\), \(m\angle2 = 145^{\circ}\), \(m\angle3 = 55^{\circ}\), \(m\angle4 = 55^{\circ}\), \(m\angle5 = 125^{\circ}\)