Question

Prove that opposite angles of a parallelogram are congruent. Be sure to create and name the appropriate geometric figures.

Answer 1

A parallelogram is a quadrilateral whose opposite sides are parallel. ABCD is a parallelogram; AB II CD and AC II BD

Theorem: opposite angles of a parallelogram are congruent
Given: Parallelogram PQRS
Prove: ∠CAB = ∠BDC and ∠ABD= ∠DCA

Refer to the attached picture to follow.

Create an extended line beyond and name it as point E and point F.
Proof:
   ∠CDB = ∠EBD - alternate interior angle
   ∠EBD = ∠CAB - corresponding angle
      This makes ∠CDB congruent to ∠CAB.
   ∠ACD = ∠FAC - alternate interior angle
   ∠FAC = ∠ABD - corresponding angle
      This makes ∠ACD congruent to ∠ABD

Proving that ∠CAB = ∠BDC and ∠ABD= ∠DCA or ∠CAB = ∠CDB and ∠ABD = ∠ACD