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 abcd with vertices b, c, d, a
statements

1. ( overline{ab} parallel overline{cd} )
2. ( mangle b + mangle c = 180 )

( mangle a + mangle d = 180 )
reasons

1. dropdown
2. dropdown

Explanation:

Step1: Reason for \(\overline{AB} \parallel \overline{CD}\)

In a parallelogram, by definition, opposite sides are parallel. So the reason for \(\overline{AB} \parallel \overline{CD}\) is "Definition of a parallelogram".

Step2: Reason for \(m\angle B + m\angle C = 180^{\circ}\) and \(m\angle A + m\angle D = 180^{\circ}\)

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Here, \(\overline{AB} \parallel \overline{CD}\) and \(\overline{BC}\) (or \(\overline{AD}\)) is the transversal. So the reason is "Consecutive Interior Angles Theorem (Supplementary Angles)".

To complete the proof (beyond the given steps):

Step3: Also, \(\overline{AD} \parallel \overline{BC}\) (Definition of parallelogram)

Step4: \(m\angle A + m\angle B = 180^{\circ}\), \(m\angle C + m\angle D = 180^{\circ}\) (Consecutive Interior Angles Theorem)

Step5: From \(m\angle A + m\angle B = 180^{\circ}\) and \(m\angle A + m\angle D = 180^{\circ}\), we get \(m\angle B = m\angle D\) (Subtraction Property of Equality, since \(m\angle A\) is common)

Step6: Similarly, from \(m\angle B + m\angle C = 180^{\circ}\) and \(m\angle A + m\angle B = 180^{\circ}\), we get \(m\angle A = m\angle C\) (Subtraction Property of Equality, since \(m\angle B\) is common)

Step7: So, \(\angle A \cong \angle C\) and \(\angle B \cong \angle D\) (Definition of congruent angles, as their measures are equal)

Answer:

1. Reason: Definition of a parallelogram
2. Reason: Consecutive Interior Angles Theorem (Supplementary Angles)

(And the full proof shows \(\angle A \cong \angle C\), \(\angle B \cong \angle D\))