Question

quadrilateral abcd is a parallelogram. complete the statements to prove that \\( \overline { a b } \cong \overline { c d } \\) and \\( \overline { b c } \cong \overline { a d } \\).

Explanation:

For the first blank (reason for \( \angle CAB \cong \angle ACD \) and \( \angle BCA \cong \angle CAD \)):

Step 1: Recall the Alternate Interior Angles Theorem

The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Here, \( \overline{AB} \parallel \overline{CD} \) and \( \overline{BC} \parallel \overline{AD} \), and \( \overline{AC} \) is the transversal. So for \( \overline{AB} \parallel \overline{CD} \) with transversal \( \overline{AC} \), \( \angle CAB \) and \( \angle ACD \) are alternate interior angles. For \( \overline{BC} \parallel \overline{AD} \) with transversal \( \overline{AC} \), \( \angle BCA \) and \( \angle CAD \) are alternate interior angles. Thus, by the Alternate Interior Angles Theorem, these angles are congruent.

Step 2: For the second blank (reason for \( m\angle CAB = m\angle ACD \) and \( m\angle BCA = m\angle CAD \)):

Congruent angles have equal measures by definition. So if \( \angle CAB \cong \angle ACD \) and \( \angle BCA \cong \angle CAD \), then their measures are equal.

Answer:

s:

• First blank reason: Alternate Interior Angles Theorem
• Second blank reason: Congruent angles have equal measures