Question

given: $overline{ab}paralleloverline{fe}$; $overline{bc}$ bisects $angle abe$; $overline{de}$ bisects $angle bef$. prove: $mangle abc = mangle def$ 3. $mangle abe = mangle bef$ 4. multiplication property 5. $mangle abc = mangle def$

Explanation:

Step1: Identify given information

We are given $\overline{AB}\parallel\overline{FE}$, $\overline{BC}$ bisects $\angle ABE$, and $\overline{DE}$ bisects $\angle BEF$.

Step2: Use parallel - line property

Since $\overline{AB}\parallel\overline{FE}$, by the alternate - interior angles theorem, $\angle ABE$ and $\angle BEF$ are equal in measure.

Step3: Apply angle - bisector definition

By the definition of an angle bisector, if $\overline{BC}$ bisects $\angle ABE$, then $m\angle ABC=\frac{1}{2}m\angle ABE$, and if $\overline{DE}$ bisects $\angle BEF$, then $m\angle DEF=\frac{1}{2}m\angle BEF$.

Step4: Use multiplication property

Since $m\angle ABE = m\angle BEF$, multiplying both sides by $\frac{1}{2}$ gives $\frac{1}{2}m\angle ABE=\frac{1}{2}m\angle BEF$.

Step5: Use substitution property

Substitute $m\angle ABC=\frac{1}{2}m\angle ABE$ and $m\angle DEF=\frac{1}{2}m\angle BEF$ into $\frac{1}{2}m\angle ABE=\frac{1}{2}m\angle BEF$ to get $m\angle ABC = m\angle DEF$.

Answer:

1. Statement: $\overline{AB}\parallel\overline{FE}$; Reason: Given
2. Statement: $\angle ABE=\angle BEF$ (alternate - interior angles are equal); Reason: If two parallel lines are cut by a transversal, then alternate - interior angles are congruent
3. Statement: $\overline{BC}$ bisects $\angle ABE$ and $\overline{DE}$ bisects $\angle BEF$; Reason: Given
4. Statement: $m\angle ABC=\frac{1}{2}m\angle ABE$ and $m\angle DEF = \frac{1}{2}m\angle BEF$; Reason: Definition of angle - bisector
5. Statement: Since $m\angle ABE=m\angle BEF$, then $\frac{1}{2}m\angle ABE=\frac{1}{2}m\angle BEF$; Reason: Multiplication Property (multiplying both sides of the equation $m\angle ABE = m\angle BEF$ by $\frac{1}{2}$)
6. Statement: $m\angle ABC=m\angle DEF$; Reason: Substitution Property (substituting $m\angle ABC=\frac{1}{2}m\angle ABE$ and $m\angle DEF=\frac{1}{2}m\angle BEF$ into $\frac{1}{2}m\angle ABE=\frac{1}{2}m\angle BEF$)

For the blanks in the given proof:

• For the third step (where it says $m\angle ABE = m\angle BEF$), the reason is "If two parallel lines ($\overline{AB}$ and $\overline{FE}$) are cut by a transversal, then alternate - interior angles are congruent"
• For the fourth step, the statement is $m\angle ABC=\frac{1}{2}m\angle ABE$ and $m\angle DEF=\frac{1}{2}m\angle BEF$
• For the fifth step, the reason is "Substitution Property"