1. **Stating the problem:** You asked why we assume that the lines EA and EC bisect the angles, and whether only perpendicular bisectors divide angles into two equal parts. 2. **Clarifying angle bisectors vs perpendicular bisectors:** - An **angle bisector** is a line or ray that divides an angle into two equal smaller angles. - A **perpendicular bisector** is a line that divides a segment into two equal parts at a 90-degree angle. 3. **Key difference:** - Angle bisectors split angles into two equal angles. - Perpendicular bisectors split line segments into two equal lengths and are perpendicular to the segment. 4. **Why we assume EA and EC bisect the angles:** - The problem states that EA and EC bisect angles FEB and BED respectively. - This means EA divides angle FEB into two equal angles, and EC divides angle BED into two equal angles. - This is a property of angle bisectors, not perpendicular bisectors. 5. **Summary:** - Perpendicular bisectors and angle bisectors are different concepts. - Only angle bisectors divide angles into two equal parts. - Perpendicular bisectors divide segments into two equal parts at right angles. **Final answer:** Lines EA and EC are angle bisectors by definition in the problem, so they divide the angles into two equal parts. Perpendicular bisectors do not bisect angles; they bisect segments perpendicularly.