Question

in the figure below, suppose m∠1 = 62° and m∠3 = 84°. complete the statements below. the sum of the interior angle measures of a triangle must be □°. so, m∠1 + m∠3 + m∠4 = 180°. we are given that m∠1 = 62°. so, m∠3 + m∠4 = 118°. from the figure, we can see that m∠1 + m∠2 = 242°. since m∠1 = 62°, it must be that m∠2 = 34°.

Explanation:

Step1: Recall triangle - angle sum property

The sum of interior angles of a triangle is always 180°.

Step2: Find \(m\angle4\)

Given \(m\angle1 = 62^{\circ}\) and \(m\angle3=84^{\circ}\), and \(m\angle1 + m\angle3+m\angle4 = 180^{\circ}\). Then \(m\angle4=180-(m\angle1 + m\angle3)=180-(62 + 84)=34^{\circ}\).

Step3: Analyze the linear - pair relationship

We know that \(\angle1\) and \(\angle2\) form a linear - pair. If \(m\angle1 + m\angle2 = 242^{\circ}\) and \(m\angle1 = 62^{\circ}\), then \(m\angle2=242 - 62=180^{\circ}\) (There is an error in the original problem statement where it says \(m\angle2 = 34^{\circ}\) based on the wrong equation \(m\angle1 + m\angle2 = 242^{\circ}\). In fact, \(\angle1\) and \(\angle2\) are supplementary, so \(m\angle2=180 - m\angle1=180 - 62 = 118^{\circ}\)).

Answer:

The sum of the interior angle measures of a triangle must be \(180^{\circ}\).
\(m\angle4 = 34^{\circ}\), \(m\angle2=118^{\circ}\)