Question

Given: AABC = AADC
Prove: AC bisects _BAD and AC bisects BCD.

Answer 1

To prove that AC bisects angles ∠BAD and ∠BCD, we need to show that the two adjacent angles formed by the intersection of AC and each angle are equal. By applying the angle addition property and the angle bisector theorem, we can demonstrate that AC bisects both angles.

Step-by-step explanation:

To prove that AC bisects angles ∠BAD and ∠BCD, we need to show that the two adjacent angles formed by the intersection of AC and each angle are equal.

Let's start by proving that AC bisects ∠BAD.

1. Draw a line segment BD that intersects AC at a point E.
2. By the angle addition property, ∠AAB' + ∠B'AD = ∠AAD.
3. Since AABC = AADC, we know that ∠AAB' = ∠AAD. So, ∠AAB' + ∠B'AD = 2∠AAB' = ∠AAD.
4. By the angle bisector theorem, AC bisects ∠BAD.

Next, let's prove that AC bisects ∠BCD.

1. Draw a line segment BD that intersects AC at a point F.
2. By the angle addition property, ∠CBD + ∠BDA = ∠ACD.
3. Since AABC = AADC, we know that ∠CBD = ∠ACD. So, ∠CBD + ∠BDA = 2∠CBD = ∠ACD.
4. By the angle bisector theorem, AC bisects ∠BCD.