Math  /  Geometry

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

Studdy Solution

STEP 1

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

STEP 3

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

STEP 4

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

STEP 5

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

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