Question

given: ( mangle aeb = 45^circ )
( angle aec ) is a right angle.
prove: ( overrightarrow{eb} ) bisects ( angle aec ).

proof:
we are given that ( mangle aeb = 45^circ ) and ( angle aec ) is a right angle. the measure of ( angle aec ) is ( 90^circ ) by the definition of a right angle. applying the
( \boxed{} ) gives ( mangle aeb + mangle bec = mangle aec ). applying the substitution property gives ( 45^circ + mangle bec = 90^circ ). the subtraction property can be used to find ( mangle bec = 45^circ ), so ( angle bec cong angle aeb ) because they have the same measure. since ( overrightarrow{eb} ) divides ( angle aec ) into two congruent angles, it is the angle bisector.

Explanation:

Step1: Identify the Angle Addition Postulate

The Angle Addition Postulate states that if a point lies in the interior of an angle, then the sum of the measures of the two smaller angles formed is equal to the measure of the original angle. In this case, point \( B \) is in the interior of \( \angle AEC \), so \( m\angle AEB + m\angle BEC = m\angle AEC \) follows the Angle Addition Postulate.

Step2: Confirm the Application

We know \( m\angle AEB = 45^\circ \) and \( m\angle AEC = 90^\circ \) (since \( \angle AEC \) is a right angle). Using the Angle Addition Postulate, we set up the equation \( 45^\circ + m\angle BEC = 90^\circ \). Solving for \( m\angle BEC \) (by subtracting \( 45^\circ \) from both sides), we get \( m\angle BEC = 45^\circ \). Since \( m\angle AEB = m\angle BEC = 45^\circ \), \( \overrightarrow{EB} \) divides \( \angle AEC \) into two congruent angles, so it is the angle bisector.

Answer:

The missing term is the "Angle Addition Postulate".