Question

Given: m∠AEB = 45°
∠AEC is a right angle.

Prove: Ray E B bisects ∠AEC.

3 lines extend from point E. A horizontal line extends to the left to point A and a vertical line extends up to point C. Another line extends halfway between the other 2 lines to point B. Lines A E and E C form a right angle.
Proof:
We are given that m∠AEB = 45° and ∠AEC is a right angle. The measure of ∠AEC is 90° by the definition of a right angle. Applying the
gives m∠AEB + m∠BEC = m∠AEC. Applying the substitution property gives 45° + m∠BEC = 90°. The subtraction property can be used to find m∠BEC = 45°, so ∠BEC ≅ ∠AEB because they have the same measure. Since Ray E B divides ∠AEC into two congruent angles, it is the angle bisector.

Answer 1

Fill in the blank with the Angle Addition Postulate.

Complete proof (with reasons):

Given: m∠AEB = 45°, ∠AEC is a right angle.
To prove: Ray EB bisects ∠AEC.

1. ∠AEC is a right angle. — Given
2. m∠AEC = 90°. — Definition of right angle
3. m∠AEB + m∠BEC = m∠AEC. — Angle Addition Postulate
4. 45° + m∠BEC = 90°. — Substitution (use m∠AEB = 45°)
5. m∠BEC = 45°. — Subtraction property (subtract 45° from both sides)
6. ∠AEB ≅ ∠BEC. — Congruence by equal measures (both 45°)
7. Ray EB bisects ∠AEC. — Definition of angle bisector (it divides the angle into two congruent angles)

Thus EB bisects ∠AEC.