Math / Geometry

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Given: $\overline{D E}$ bisects $\angle B D C$ and $\overline{D E}$ bisects $\angle B E C$ . Prove: $\angle A B E \cong \angle A C E$ .
Note: the segment $A E$ is a straight segment. Answer Attempt 1 out of 2

Studdy Solution

STEP 1

1. Line segment $\overline{DE}$ bisects $\angle BDC$ .
2. Line segment $\overline{DE}$ bisects $\angle BEC$ .
3. Segment $\overline{AE}$ is a straight line.
4. We need to prove that $\angle ABE \cong \angle ACE$ .

STEP 2

1. Understand the properties of angle bisectors.
2. Apply the Angle Bisector Theorem.
3. Use congruent angles and supplementary angles to prove $\angle ABE \cong \angle ACE$ .

STEP 3

Understand the properties of angle bisectors: - Since $\overline{DE}$ bisects $\angle BDC$ , it divides $\angle BDC$ into two equal angles: $\angle BDE$ and $\angle CDE$ . - Since $\overline{DE}$ bisects $\angle BEC$ , it divides $\angle BEC$ into two equal angles: $\angle BED$ and $\angle CED$ .

STEP 4

Apply the Angle Bisector Theorem: - Since $\overline{DE}$ is the angle bisector of $\angle BDC$ , we have $\angle BDE = \angle CDE$ . - Since $\overline{DE}$ is the angle bisector of $\angle BEC$ , we have $\angle BED = \angle CED$ .

STEP 5

Use congruent angles and supplementary angles to prove $\angle ABE \cong \angle ACE$ : - Since $\overline{AE}$ is a straight line, $\angle ABE + \angle BED = 180^\circ$ and $\angle ACE + \angle CED = 180^\circ$ . - From Step 2, $\angle BED = \angle CED$ . - Therefore, $\angle ABE = 180^\circ - \angle BED$ and $\angle ACE = 180^\circ - \angle CED$ . - Since $\angle BED = \angle CED$ , it follows that $\angle ABE = \angle ACE$ .
The proof is complete, and we have shown that:
$\angle ABE \cong \angle ACE$

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