Question

given: bisects ∠aec. ∠aed is a straight angle. prove: m∠aeb = 45° complete the paragraph proof. we are given that bisects ∠aec. from the diagram, ∠ced is a right angle, which measures degrees. since the measure of a straight angle is 180°, the measure of angle must also be 90° by the . a bisector cuts the angle measure in half. m∠aeb is 45°.

Explanation:

Step 1: Recall right angle measure

A right angle, by definition, measures \( 90^\circ \). So \( \angle CED \), being a right angle, measures \( 90^\circ \).

Step 2: Identify supplementary angle

Since \( \angle AED \) is a straight angle (\( 180^\circ \)), and \( \angle CED = 90^\circ \), then \( \angle AEC \) must also be \( 90^\circ \) by the definition of supplementary angles (because \( \angle AEC + \angle CED=\angle AED = 180^\circ \), so \( \angle AEC=180^\circ - \angle CED = 90^\circ \)).

Step 3: Apply angle bisector

Given that \( \overline{EB} \) bisects \( \angle AEC \), it divides \( \angle AEC \) into two equal angles. So \( m\angle AEB=\frac{1}{2}m\angle AEC \). Since \( m\angle AEC = 90^\circ \), then \( m\angle AEB=\frac{90^\circ}{2}=45^\circ \).

Answer:

First blank: \( 90 \)
Second blank: \( \angle AEC \)
Third blank: definition of supplementary angles (or linear pair postulate, as \( \angle AEC \) and \( \angle CED \) form a linear pair)