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 ( ) gives ( mangle aeb) the substitution property subtraction property segment addition property ( 45^{circ}) so angle addition postulate ave the same measure reflexive property congruent angles, it is transitive property

Explanation:

The angle addition postulate states that if we have two adjacent angles, the measure of the larger angle formed is the sum of the measures of the two smaller angles. Here, since \(\angle AEC\) is a right - angle (\(m\angle AEC = 90^{\circ}\)) and \(\angle AEB=45^{\circ}\), we can use the angle addition postulate (\(m\angle AEC=m\angle AEB + m\angle BEC\)) to find \(m\angle BEC\). By substitution (\(90^{\circ}=45^{\circ}+m\angle BEC\)), we get \(m\angle BEC = 45^{\circ}\). Since \(m\angle AEB=m\angle BEC = 45^{\circ}\), \(\overrightarrow{EB}\) bisects \(\angle AEC\). The angle addition postulate is the key property used in the relationship between the angles \(\angle AEC\), \(\angle AEB\), and \(\angle BEC\).

Answer:

angle addition postulate