Question

for the figure shown, find ( mangle 1 ) and ( mangle 2 ).
(the figure is not drawn to scale)
( mangle 1=square^{circ} )

Explanation:

Step1: Use the exterior angle theorem for the triangle with \(30^{\circ}\) and \(\angle1\)

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
Let's assume the triangle with one angle \(30^{\circ}\) and \(\angle1\). The exterior angle related to them (let's call it \(x\)) is \(96^{\circ}-\angle2\). But we can also use the other triangle.

Wait, better approach:
For the triangle with angles \(30^{\circ}\), \(\angle1\) and the angle adjacent to \(96^{\circ}\) (which is \(180 - 96=84^{\circ}\))
By the angle - sum property of a triangle (\(a + b + c=180^{\circ}\) for a triangle with angles \(a\), \(b\), \(c\))
\(30+\angle1 + 84 = 180\)
\(\angle1=180-(30 + 84)\)
\(\angle1 = 66^{\circ}\)

Step2: Use the linear pair property for \(\angle2\)

Since \(\angle2\) and \(96^{\circ}\) form a linear pair (\(a + b=180^{\circ}\) for a linear pair), \(\angle2=180 - 96\)
\(\angle2 = 84^{\circ}\)

Answer:

\(m\angle1 = 66^{\circ}\)