Question

1. find all missing angles. 9. m∠1=__ m∠2= m∠3= m∠4=__

Explanation:

Step1: Find \(m\angle1\)

In a right - triangle, the sum of angles is \(180^{\circ}\). For the larger right - triangle with one angle \(72^{\circ}\), \(m\angle1=180^{\circ}-90^{\circ}-72^{\circ}\)

$$m\angle1 = 18^{\circ}$$

Step2: Find \(m\angle2\)

Since the right - angle is \(90^{\circ}\) and one part is \(72^{\circ}\), \(m\angle2=90^{\circ}-72^{\circ}\)

$$m\angle2 = 18^{\circ}$$

Step3: Find \(m\angle4\)

In the smaller triangle, using the angle sum property of a triangle (\(180^{\circ}\)). Let's first note that in the larger concept of the figure (right - angled at the vertex with \(72^{\circ}\) and \(m\angle2\)). For the non - overlapping part of the two triangles (the smaller triangle with angles \(57^{\circ}\), \(m\angle3\), \(m\angle4\) and the relation to the right - angle).
We know that in the larger right - triangle, if we consider the non - \(72^{\circ}\) and non - \(m\angle1\) part. Also, using the angle sum property of a triangle for the smaller triangle: \(m\angle4=180^{\circ}-90^{\circ}-(57^{\circ}+m\angle2)\)
Substitute \(m\angle2 = 18^{\circ}\), then \(m\angle4=180^{\circ}-90^{\circ}-(57^{\circ}+18^{\circ})\)

$$m\angle4=15^{\circ}$$

Step4: Find \(m\angle3\)

Using the angle sum property of a triangle (\(180^{\circ}\)) for the triangle with angles \(57^{\circ}\), \(m\angle3\), \(m\angle4\)
\(m\angle3=180^{\circ}-57^{\circ}-m\angle4\)
Substitute \(m\angle4 = 15^{\circ}\)

$$m\angle3=108^{\circ}$$

Answer:

\(m\angle1 = 18^{\circ}\), \(m\angle2 = 18^{\circ}\), \(m\angle3 = 108^{\circ}\), \(m\angle4 = 15^{\circ}\)