Question

1. xavier is given a parallelogram and tries to prove that its opposite angles are congruent. he draws a parallelogram and a diagonal, as shown.

complete the proof given.
statement\treason
abcd is a parallelogram\tgiven
bc ≅ ad and ab ≅ dc\t
ac ≅ ca\treflexive property of congruency
△abc ≅ △cda\t
∠b ≅ ∠d\tcpctc

Explanation:

Step1: Identify properties of parallelogram

In a parallelogram \(ABCD\), opposite sides are congruent. So \(BC \cong AD\) and \(AB \cong DC\) (by definition of parallelogram: opposite sides of parallelogram are equal).

Step2: Identify common side

\(AC \cong CA\) (reflexive property of congruence, a side is congruent to itself).

Step3: Prove triangle congruence

By SSS (Side - Side - Side) congruence criterion, \(\triangle ABC \cong \triangle CDA\) (since \(BC \cong AD\), \(AB \cong DC\), \(AC \cong CA\)).

Step4: Use CPCTC

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle B \cong \angle D\) (corresponding angles of congruent triangles \(\triangle ABC\) and \(\triangle CDA\)).

Answer:

The missing reasons and statements are filled as follows:

• For \(BC \cong AD\) and \(AB \cong DC\): Reason is "Opposite sides of a parallelogram are congruent".
• For \(\triangle ABC \cong \triangle CDA\): Reason is "SSS (Side - Side - Side) Congruence Criterion".
• The final conclusion \(\angle B \cong \angle D\) is by "CPCTC" as corresponding parts of congruent triangles \(\triangle ABC\) and \(\triangle CDA\) are congruent.