Question

which statements regarding the diagram are true? check all that apply.
∠xfg is an interior angle of △efg.
∠efg is an interior angle of △efg.
∠fez is an exterior angle of △efg.
∠yge is an exterior angle of △efg.
∠egf and ∠fgy are supplementary angles.
∠feg and ∠fge are supplementary angles.

Explanation:

Step1: Recall the definition of interior and exterior angles of a triangle

An interior angle of a triangle is an angle formed inside the triangle by two of its sides. An exterior angle of a triangle is an angle formed outside the triangle by one of its sides and the extension of an adjacent side.

• For \(\angle XFG\): It is not an interior angle of \(\triangle EFG\) because it is not formed by two sides of \(\triangle EFG\) inside the triangle.
• For \(\angle EFG\): It is formed by two sides \(EF\) and \(FG\) of \(\triangle EFG\) inside the triangle, so it is an interior angle.
• For \(\angle FEZ\): It is not an exterior angle of \(\triangle EFG\).
• For \(\angle YGE\): It is formed by side \(EG\) of \(\triangle EFG\) and the extension of side \(FG\) (since \(GY\) is an extension), so it is an exterior angle.
• For \(\angle EGF\) and \(\angle FGY\): They form a linear - pair (a straight line), so \(\angle EGF+\angle FGY = 180^{\circ}\), and they are supplementary.
• For \(\angle FEG\) and \(\angle FGE\): In a triangle, the sum of interior angles \(\angle FEG+\angle FGE+\angle EFG=180^{\circ}\). They are not supplementary (sum to \(180^{\circ}\)) unless \(\angle EFG = 0^{\circ}\), which is not the case.

Answer:

\(\angle EFG\) is an interior angle of \(\triangle EFG\), \(\angle YGE\) is an exterior angle of \(\triangle EFG\), \(\angle EGF\) and \(\angle FGY\) are supplementary angles. So the correct statements are: \(\angle EFG\) is an interior angle of \(\triangle EFG\), \(\angle YGE\) is an exterior angle of \(\triangle EFG\), \(\angle EGF\) and \(\angle FGY\) are supplementary angles.