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 dropdown (reason for \( \angle CAB \cong \angle ACD \) and \( \angle BCA \cong \angle CAD \)):

Step1: Recall the theorem for parallel lines and transversal.

When a transversal intersects two parallel lines, alternate interior angles are congruent. Here, \( \overline{AB} \parallel \overline{CD} \) and \( \overline{AC} \) is a transversal, so \( \angle CAB \) and \( \angle ACD \) are alternate interior angles. Similarly, \( \overline{BC} \parallel \overline{AD} \) and \( \overline{AC} \) is a transversal, so \( \angle BCA \) and \( \angle CAD \) are alternate interior angles. So the reason is the Alternate Interior Angles Theorem.

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

Step2: Recall the property of congruent angles.

If two angles are congruent, their measures are equal. So if \( \angle CAB \cong \angle ACD \), then \( m\angle CAB = m\angle ACD \) and similarly for the other pair. So the reason is "Congruent angles have equal measures".

For the triangle congruence ( \( \triangle ABC \cong \triangle CDA \)):

Step3: Identify the congruence criterion.

We have two angles and the included side. \( \angle CAB \cong \angle ACD \), \( \overline{AC} \cong \overline{AC} \) (reflexive property), and \( \angle BCA \cong \angle CAD \). So by ASA (Angle - Side - Angle) congruence criterion, \( \triangle ABC \cong \triangle CDA \).

For the final statement ( \( \overline{AB} \cong \overline{CD} \) and \( \overline{BC} \cong \overline{AD} \)):

Step4: Recall the property of congruent triangles.

Corresponding parts of congruent triangles are congruent (CPCTC). Since \( \triangle ABC \cong \triangle CDA \), their corresponding sides \( \overline{AB} \) and \( \overline{CD} \), \( \overline{BC} \) and \( \overline{AD} \) are congruent.

Answer:

s:

• For \( \angle CAB \cong \angle ACD \) and \( \angle BCA \cong \angle CAD \): Alternate Interior Angles Theorem
• For \( m\angle CAB = m\angle ACD \) and \( m\angle BCA = m\angle CAD \): Congruent angles have equal measures
• For \( \triangle ABC \cong \triangle CDA \): ASA (Angle - Side - Angle) Congruence Criterion (implied by the previous steps)
• For \( \overline{AB} \cong \overline{CD} \) and \( \overline{BC} \cong \overline{AD} \): Corresponding Parts of Congruent Triangles are Congruent (CPCTC)