Question

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

Explanation:

Step1: Recall interior/exterior angle definitions

Interior angles of a triangle are formed by two sides of the triangle, inside the triangle. Exterior angles are adjacent to an interior angle and form a linear pair (sum to \(180^\circ\)) with it. Supplementary angles sum to \(180^\circ\).

Step2: Analyze each statement

- \(\angle XFG\): Formed by \(FX\) and \(FG\), outside \(\triangle EFG\) (since \(FX\) is an extension of \(EF\) beyond \(F\)), so not an interior angle.
- \(\angle EFG\): Formed by \(EF\) and \(FG\), inside \(\triangle EFG\) → interior angle.
- \(\angle FEZ\): Formed by \(FE\) and \(EZ\) (extension of \(EG\) beyond \(E\)), adjacent to \(\angle FEG\) (interior angle) and forms a linear pair → exterior angle.
- \(\angle YGE\): Formed by \(YG\) and \(GE\), but \(YG\) is an extension of \(FG\) beyond \(G\)? Wait, \(YG\) is along \(EG\)? No, \(EG\) and \(YG\) are a straight line? Wait, the diagram: \(E - G - Y\) is a straight line? Wait, \(E\), \(G\), \(Y\) are colinear? Wait, \(Z - E - G - Y\) is a line? Wait, no, \(Z - E - G\) and \(G - Y\) are a straight line? Wait, the triangle is \(EFG\). So \(\angle YGE\): Let's see, exterior angle of \(\triangle EFG\) at \(G\) would be adjacent to \(\angle EGF\). But \(\angle YGE\): Is \(Y\) on the extension of \(FG\)? Wait, the diagram: \(F - E - G\)? No, the triangle is \(E\), \(F\), \(G\). So sides: \(EF\), \(FG\), \(EG\). So at \(G\), the exterior angle would be adjacent to \(\angle EGF\), formed by extending \(FG\) or \(EG\). Wait, \(E - G - Y\) is a straight line? So \(EG\) and \(GY\) are a straight line. Then \(\angle YGE\): Wait, \(\angle EGF\) is at \(G\) between \(EG\) and \(FG\). \(\angle YGE\) is between \(YG\) (same as \(EG\) extended) and \(GE\)? No, that's a straight angle. Wait, maybe I misread. Wait, the options: \(\angle YGE\) – maybe not. Wait, \(\angle EGF\) and \(\angle FGY\): \(FG\) and \(GY\) – are \(F - G - Y\) colinear? If \(F\), \(G\), \(Y\) are colinear, then \(\angle EGF\) (at \(G\) between \(EG\) and \(FG\)) and \(\angle FGY\) (at \(G\) between \(FG\) and \(GY\)) form a linear pair → supplementary.
- \(\angle FEG\) and \(\angle FGE\): These are two interior angles of the triangle, their sum is not \(180^\circ\) (triangle angles sum to \(180^\circ\), so two angles sum to \(180^\circ - \) third angle, not \(180^\circ\) unless it's a right angle, but not necessarily). So they are not supplementary.

So correct statements: \(\angle EFG\) (interior), \(\angle FEZ\) (exterior), \(\angle EGF\) and \(\angle FGY\) (supplementary). Wait, let's recheck:

1. \(\angle XFG\): Outside the triangle → false.
2. \(\angle EFG\): Inside → true.
3. \(\angle FEZ\): Exterior (adjacent to \(\angle FEG\), linear pair) → true.
4. \(\angle YGE\): Let's see, the diagram: \(Y\) is on the extension of \(EG\)? No, \(Y\) is on the extension of \(FG\)? Wait, the triangle is \(E\), \(F\), \(G\). So vertices: \(E\), \(F\), \(G\). Sides: \(EF\), \(FG\), \(EG\). So at \(E\): sides \(EF\) and \(EG\). At \(F\): \(EF\) and \(FG\). At \(G\): \(FG\) and \(EG\). Exterior angle at \(E\): formed by extending \(EG\) (to \(Z\)) and \(EF\) → \(\angle FEZ\) (between \(FE\) and \(EZ\)) → yes, exterior. At \(F\): extending \(EF\) to \(X\), so \(\angle XFG\) is exterior at \(F\), but it's not an interior angle. At \(G\): extending \(FG\) to \(Y\), so \(\angle FGY\) is exterior at \(G\), and \(\angle EGF\) (interior) and \(\angle FGY\) are supplementary (linear pair). \(\angle YGE\): If \(Y\) is on \(EG\) extension, then \(\angle YGE\) is a straight angle, not an exterior angle of the triangle. So \(\angle YGE\) is not an exterior angle. Then:

True statements:
- \(\angle EFG\) is an interior angle.
- \(\angle FEZ\) is an exterior angle.
- \(\angle EGF\) and \(\angle FGY\) are supplementary.

Wait, let's check each option:

1. \(\angle XFG\) is interior? No (outside the triangle).
2. \(\angle EFG\) is interior? Yes.
3. \(\angle FEZ\) is exterior? Yes (adjacent to \(\angle FEG\), linear pair).
4. \(\angle YGE\) is exterior? No (if \(Y\) is on \(EG\) line, \(\angle YGE\) is a straight angle, not an exterior angle of the triangle).
5. \(\angle EGF\) and \(\angle FGY\) are supplementary? Yes (linear pair, sum to \(180^\circ\)).
6. \(\angle FEG\) and \(\angle FGE\) are supplementary? No (they are two interior angles, sum to \(180^\circ - \angle EFG\), not \(180^\circ\) unless \(\angle EFG = 0^\circ\), which is impossible).

So the correct options are:
- \(\boldsymbol{\angle EFG}\) is an interior angle of \(\triangle EFG\).
- \(\boldsymbol{\angle FEZ}\) is an exterior angle of \(\triangle EFG\).
- \(\boldsymbol{\angle EGF}\) and \(\boldsymbol{\angle FGY}\) are supplementary angles.

Wait, but the options given:

The checkboxes:

- \(\angle XFG\) is interior: ❌
- \(\angle EFG\) is interior: ✅
- \(\angle FEZ\) is exterior: ✅
- \(\angle YGE\) is exterior: ❌ (wait, maybe I misread the diagram. Maybe \(Y\) is on the extension of \(FG\), so \(F - G - Y\) is a straight line. Then \(\angle YGE\): no, \(\angle YGE\) is at \(G\) between \(YG\) (extension of \(FG\)) and \(GE\). So exterior angle at \(G\) would be \(\angle FGY\) (between \(FG\) and \(YG\)) and \(\angle EGF\) (between \(EG\) and \(FG\)) are supplementary. But \(\angle YGE\): if \(YG\) is along \(FG\) extension, then \(\angle YGE\) is between \(YG\) and \(GE\), which is \(\angle FGE\)? No, maybe the diagram has \(E - G - Y\) as a straight line, so \(EG\) and \(GY\) are colinear. Then \(\angle YGE\) is a straight angle, not an exterior angle. So \(\angle YGE\) is not an exterior angle.

So the correct ones are:

- \(\angle EFG\) (interior)
- \(\angle FEZ\) (exterior)
- \(\angle EGF\) and \(\angle FGY\) (supplementary)

So the checkboxes to mark are:

- \(\angle EFG\) is an interior angle of \(\triangle EFG\).
- \(\angle FEZ\) is an exterior angle of \(\triangle EFG\).
- \(\angle EGF\) and \(\angle FGY\) are supplementary angles.

Answer:

• \(\boldsymbol{\angle EFG}\) is an interior angle of \(\triangle EFG\).
• \(\boldsymbol{\angle FEZ}\) is an exterior angle of \(\triangle EFG\).
• \(\boldsymbol{\angle EGF}\) and \(\boldsymbol{\angle FGY}\) are supplementary angles.

(In the format: the checkboxes corresponding to these statements should be checked.)