Question

1. use the diagram to the right to answer the following questions. 2 pts each

$m\angle1 =$
$m\angle2 =$
$m\angle3 =$
$m\angle4 =$

Explanation:

Step1: Find \(m\angle1\)

Use the triangle angle - sum property. In the left - most triangle, \(\angle1+15^{\circ}+63^{\circ}=180^{\circ}\) (sum of angles in a triangle is \(180^{\circ}\)).
So, \(m\angle1 = 180^{\circ}-(15^{\circ}+63^{\circ})=180^{\circ}-78^{\circ}=102^{\circ}\)

Step2: Find \(m\angle2\)

\(\angle1\) and \(\angle2\) are supplementary (linear pair). So \(m\angle2=180^{\circ}-m\angle1\)
Since \(m\angle1 = 102^{\circ}\), then \(m\angle2=180^{\circ}-102^{\circ}=78^{\circ}\)

Step3: Find \(m\angle3\)

In the right - angled triangle (with \(\angle3\)), one angle is \(90^{\circ}\) and another is \(63^{\circ}\). Using the triangle angle - sum property \(m\angle3+63^{\circ}+90^{\circ}=180^{\circ}\)
\(m\angle3=180^{\circ}-(90^{\circ}+63^{\circ})=180^{\circ}-153^{\circ}=27^{\circ}\)

Step4: Find \(m\angle4\)

In the large triangle (with \(\angle4\)), we know two angles: \(15^{\circ}\) and \(m\angle3 = 27^{\circ}\)
Using the triangle angle - sum property \(m\angle4+15^{\circ}+(63^{\circ}+27^{\circ})=180^{\circ}\) (the angle adjacent to \(63^{\circ}\) and \(m\angle3\) is \(63^{\circ}+27^{\circ}\))
\(m\angle4=180^{\circ}-(15^{\circ}+90^{\circ})=75^{\circ}\)

Answer:

\(m\angle1 = 102^{\circ}\), \(m\angle2 = 78^{\circ}\), \(m\angle3 = 27^{\circ}\), \(m\angle4 = 75^{\circ}\)