opposite sides of a parallelogram are parallel. prove that opposite angles of a parallelogram are congruent. given: ( abcd ) is a parallelogram. prove: ( angle a cong angle c ), ( angle b cong angle d ). diagram of parallelogram ( abcd ) with vertices ( a, b, c, d ). steps: 2) ( mangle a + mangle d = 180 ) 3) ( bc parallel ad ) 4) ( mangle a + mangle b = 180 ); ( mangle c + mangle d = 180 ) 5) ( mangle a + mangle b = mangle b + mangle c ) 6) blank. reasons (right): 3) definition of parallelogram 4) same side interior angles postulate

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Accepted Answer

# Explanation:
## Step1: Analyze Step 5
We have $ m\angle A + m\angle B = m\angle B + m\angle C $. To show $ \angle A \cong \angle C $, we can use the Subtraction Property of Equality. Subtract $ m\angle B $ from both sides of the equation.
$$
\begin{align*}
m\angle A + m\angle B - m\angle B&= m\angle B + m\angle C - m\angle B\
m\angle A&= m\angle C
\end{align*}
$$
Since the measures of $ \angle A $ and $ \angle C $ are equal, $ \angle A \cong \angle C $ (by the definition of congruent angles: if two angles have equal measures, they are congruent).

## Step2: Prove $ \angle B \cong \angle D $
Similarly, from the equations $ m\angle A + m\angle D = 180^\circ $ (from same - side interior angles, since $ AB\parallel CD $) and $ m\angle A + m\angle B = 180^\circ $ (from same - side interior angles, since $ AD\parallel BC $), we can set $ m\angle A + m\angle D=m\angle A + m\angle B $. Then, by the Subtraction Property of Equality (subtract $ m\angle A $ from both sides), we get $ m\angle D = m\angle B $, so $ \angle B\cong\angle D $.

# Answer:
To fill in step 5's reason (for showing $ \angle A\cong\angle C $): Subtraction Property of Equality (and for $ \angle B\cong\angle D $ similarly). The key step for $ \angle A\cong\angle C $ is using the Subtraction Property of Equality on $ m\angle A + m\angle B = m\angle B + m\angle C $ to get $ m\angle A=m\angle C $, hence $ \angle A\cong\angle C $, and similarly for $ \angle B\cong\angle D $.

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze Step 5

Step2: Prove \( \angle B \cong \angle D \)

Answer: