Question

15.
in exercises 15 and 16, tell whether the information in the diagram allows you to conclude that eh bisects ∠feg. explain your reasoning. (see example 4.)
diagram for 15: a diagram with points e, f, g, h. eh is a line, eg and ef have markings, and there are right angles on eg and ef from h?
16.
diagram for 16: a diagram with points e, f, g, h. eh is a line, eg and ef have markings, no right angles?

Explanation:

Problem 15:

Step1: Identify Given Information

We have \( EH \) as a segment, \( EG \) and \( EF \) with perpendiculars (right angles) from \( H \) to \( EG \) and \( EF \), and the segments from \( H \) to \( EG \) and \( H \) to \( EF \) are congruent (marked with same tick marks). Also, \( EH \) is common.

Step2: Apply Angle Bisector Theorem

The Angle Bisector Theorem states that if a point is equidistant from the sides of an angle, then it lies on the angle bisector. Here, \( H \) is equidistant from \( EG \) and \( EF \) (since the perpendicular distances are equal, and \( EH \) is the common side, triangles \( EH G \) and \( EH F \) are congruent by HL (Hypotenuse - Leg) criterion). So, \( \angle GEH=\angle FEH \), meaning \( EH \) bisects \( \angle FEG \).

Problem 16:

Step1: Identify Given Information

We have \( EH \) as a segment, \( EG \) and \( EF \) with segments from \( H \) to \( G \) and \( H \) to \( F \) congruent (marked with same tick marks), and \( EH \) is common. But there are no right angles marked.

Step2: Analyze for Angle Bisector

The Angle Bisector Theorem requires equidistant (perpendicular distance) from the sides. Here, we only know \( HG = HF \), not that they are perpendicular to \( EG \) and \( EF \). So, we can't conclude \( EH \) bisects \( \angle FEG \) because congruent non - perpendicular segments don't imply angle bisector.

Final Answers:

• For Problem 15: Yes, \( EH \) bisects \( \angle FEG \) (by Angle Bisector Theorem, \( H \) is equidistant from \( EG \) and \( EF \) as the perpendiculars are equal, so \( \angle GEH=\angle FEH \)).
• For Problem 16: No, we can't conclude \( EH \) bisects \( \angle FEG \) (only \( HG = HF \), no perpendicularity, so can't apply Angle Bisector Theorem).

Answer:

Step1: Identify Given Information

We have \( EH \) as a segment, \( EG \) and \( EF \) with segments from \( H \) to \( G \) and \( H \) to \( F \) congruent (marked with same tick marks), and \( EH \) is common. But there are no right angles marked.

Step2: Analyze for Angle Bisector

The Angle Bisector Theorem requires equidistant (perpendicular distance) from the sides. Here, we only know \( HG = HF \), not that they are perpendicular to \( EG \) and \( EF \). So, we can't conclude \( EH \) bisects \( \angle FEG \) because congruent non - perpendicular segments don't imply angle bisector.

Final Answers:

• For Problem 15: Yes, \( EH \) bisects \( \angle FEG \) (by Angle Bisector Theorem, \( H \) is equidistant from \( EG \) and \( EF \) as the perpendiculars are equal, so \( \angle GEH=\angle FEH \)).
• For Problem 16: No, we can't conclude \( EH \) bisects \( \angle FEG \) (only \( HG = HF \), no perpendicularity, so can't apply Angle Bisector Theorem).