Question

complete each step to show that ( mangle3 + mangle2 = mangle4 ).

( mangle4 + mangle1 = 180^circ ) (quadleftarrowquad) ( angle4 ) and ( angle1 ) are supplementary.

( mangle3 + mangle2 + mangle1 = 180^circ ) (quadleftarrowquad) a triangle’s (\boldsymbol{?}) angles sum to ( 180^circ ).

Explanation:

Step1: Recall Triangle Angle Sum

The triangle angle sum theorem states that the sum of the interior angles of a triangle is \(180^\circ\). In the given triangle, \(\angle 3\), \(\angle 2\), and \(\angle 1\) are the interior angles. So the missing term is "interior".

Step2: Verify with Given Equations

We have two equations: \(m\angle 4 + m\angle 1 = 180^\circ\) (supplementary angles) and \(m\angle 3 + m\angle 2 + m\angle 1 = 180^\circ\) (triangle interior angles sum). By equating the two right - hand sides (since both equal \(180^\circ\)), we can show \(m\angle 3 + m\angle 2 = m\angle 4\), and the key here for the second equation is the sum of interior angles of a triangle.

Answer:

interior