Question

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

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For the triangle with angle \(31^{\circ}\) and \(\angle1\), we have \(\angle1 = 31^{\circ}\) (because the non - adjacent interior angle to the exterior angle formed by the line is equal to the given angle, assuming the lines are parallel and using the alternate interior angle concept. But if we consider the basic triangle - related angle property where \(\angle1\) and \(31^{\circ}\) are alternate interior angles (if we assume a parallel line situation which is often implied in such angle - finding problems with a simple triangle - like figure).

Step2: Use the linear pair property

We know that \(\angle2\) and the \(64^{\circ}\) angle form a linear pair. A linear pair of angles is supplementary, so \(m\angle2=180^{\circ}-64^{\circ}\)

$$m\angle2 = 116^{\circ}$$

Answer:

\(m\angle1 = 31^{\circ}\), \(m\angle2=116^{\circ}\)