Question

opposite sides of a parallelogram are parallel. prove that opposite angles of a parallelogram are congruent. given: abcd is a parallelogram. prove: ∠a ≅ ∠c, ∠b ≅ ∠d
diagram of parallelogram with vertices b, c, a, d (in order)

m∠a + m∠d = 180

1. ( overline{bc} parallel overline{ad} )
2. definition of parallelogram
3. ( mangle a + mangle b = 180 )

( mangle c + mangle d = 180 )

1. same - side interior angles postulate
2. ( mangle a + mangle b = mangle b + mangle c )
3. blank dropdown menu

Explanation:

Step1: Recall angle sum relations

From steps 4, we have \( m\angle A + m\angle B = 180^\circ \) (from \( AB \parallel CD \), same - side interior angles) and \( m\angle C + m\angle D = 180^\circ \), also from step 2 (or similar parallel side relation) \( m\angle A + m\angle D = 180^\circ \) and from step 4 (for \( BC\parallel AD \)) \( m\angle A + m\angle B = 180^\circ \), \( m\angle C + m\angle D = 180^\circ \). Also, from the parallel sides, we know that \( m\angle A + m\angle B=180^\circ \) and \( m\angle B + m\angle C = 180^\circ \) (because \( AB\parallel CD \) and \( BC \) is a transversal, same - side interior angles for \( \angle B \) and \( \angle C \)).

Step2: Use substitution property

We know that \( m\angle A + m\angle B=180^\circ \) (from \( AD\parallel BC \), same - side interior angles postulate) and \( m\angle B + m\angle C = 180^\circ \) (from \( AB\parallel CD \), same - side interior angles postulate). So, by the substitution property (if \( a + b=c + b \), then \( a = c \)), since \( m\angle A + m\angle B=180^\circ \) and \( m\angle B + m\angle C = 180^\circ \), we can substitute \( 180^\circ\) in the first equation with \( m\angle B + m\angle C \) (because both equal \( 180^\circ\)). So \( m\angle A + m\angle B=m\angle B + m\angle C \) is obtained by the substitution property (or transitive property of equality, since \( m\angle A + m\angle B = 180^\circ\) and \( m\angle B + m\angle C=180^\circ\), so \( m\angle A + m\angle B=m\angle B + m\angle C\)).

To justify step 5: \( m\angle A + m\angle B=m\angle B + m\angle C \), we can use the Substitution Property of Equality (or Transitive Property of Equality, since \( m\angle A + m\angle B = 180^\circ\) and \( m\angle B + m\angle C = 180^\circ\), so \( m\angle A + m\angle B=m\angle B + m\angle C\)). Another way: From step 4, \( m\angle A + m\angle B = 180^\circ\) and also \( m\angle B + m\angle C=180^\circ\) (because \( AB\parallel CD \) and \( BC \) is a transversal, same - side interior angles). So by transitive property (\( a = b\) and \( b = c\) implies \( a = c\)), where \( a=m\angle A + m\angle B\), \( b = 180^\circ\), \( c=m\angle B + m\angle C\)), we get \( m\angle A + m\angle B=m\angle B + m\angle C\).

Then, if we subtract \( m\angle B \) from both sides (Subtraction Property of Equality), we get \( m\angle A=m\angle C \), so \( \angle A\cong\angle C \). Similarly, we can prove \( \angle B\cong\angle D \).

For step 5, the justification is the Transitive Property of Equality (or Substitution Property, since both \( m\angle A + m\angle B \) and \( m\angle B + m\angle C \) equal \( 180^\circ\)) or more precisely, since \( m\angle A + m\angle B = 180^\circ\) (from \( AD\parallel BC \), same - side interior angles) and \( m\angle B + m\angle C=180^\circ\) (from \( AB\parallel CD \), same - side interior angles), by transitive property, \( m\angle A + m\angle B=m\angle B + m\angle C\).

Answer:

The justification for step 5 (\( m\angle A + m\angle B=m\angle B + m\angle C \)) is the Transitive Property of Equality (or "Substitution Property of Equality" as both \( m\angle A + m\angle B \) and \( m\angle B + m\angle C \) equal \( 180^\circ\)) or more specifically, since \( m\angle A + m\angle B = 180^\circ\) and \( m\angle B + m\angle C = 180^\circ\), by transitive property \( m\angle A + m\angle B=m\angle B + m\angle C\). And then, by subtracting \( m\angle B \) from both sides (Subtraction Property of Equality), we get \( m\angle A=m\angle C \), so \( \angle A\cong\angle C \). Similarly, \( \angle B\cong\angle D \) can be proven.

If we are only asked for the justification of step 5, the answer is the Transitive Property of Equality (or "Substitution Property" based on the equal sums).