Question

proof #4

statement
reason

m∠aeb = m∠ced
1.

m∠bec = m∠bec
2.

m∠aeb + m∠bec = m∠ced + m∠bec
3.

m∠aec = m∠aeb + m∠bec
4.

m∠bed = m∠ced + m∠bec
5.

m∠aec = m∠bed
6.

Explanation:

1. Vertical angles are congruent, so \(m\angle AEB = m\angle CED\).
2. Any angle is congruent to itself (reflexive property of equality), so \(m\angle BEC = m\angle BEC\).
3. If \(a = b\) and \(c = c\), then \(a + c=b + c\) (addition property of equality). Here \(a=m\angle AEB\), \(b = m\angle CED\), \(c=m\angle BEC\).
4. By the angle - addition postulate, if point \(E\) is in the interior of \(\angle AEC\), then \(m\angle AEC=m\angle AEB + m\angle BEC\).
5. By the angle - addition postulate, if point \(E\) is in the interior of \(\angle BED\), then \(m\angle BED=m\angle CED + m\angle BEC\).
6. If \(x=y\) and \(y = z\), then \(x = z\) (transitive property of equality). Here \(x=m\angle AEC\), \(y=m\angle AEB + m\angle BEC=m\angle CED + m\angle BEC\), \(z=m\angle BED\).

Answer:

1. Vertical angles are congruent
2. Reflexive property of equality
3. Addition property of equality
4. Angle - addition postulate
5. Angle - addition postulate
6. Transitive property of equality