To prove: ABCD is a parallelogram. Given information: AB―∥CD―, AB―=∼DC―.

Proof: /&gt; It is given that, AB―=∼CD―, AD―=∼BC―. If lines are parallel, alternate interior angles are congruent. ∠CDB=∼∠ABD ∠DCA=∼∠BAC Two triangles are congruent by ASA congruence rule. △ABX=∼△CDX Corresponding parts of congruent triangles are congruent. AX―=∼XC― BX―=∼XD― Vertical angles are congruent. ∠AXD=∼∠BXC Two triangles are congruent by SAS congruence rule. △AXD=∼△CXB Corresponding parts of congruent triangles are congruent. BC―=∼AD― If both pairs of opposite sides are congruent, then a quadrilateral is a parallelogram. ABCD is a parallelogram.

Best solutions to - "To prove: ABCD is a parallelogram. Given information:..."

Burhan Hopper

Answered question

2021-07-26

To prove: ABCD is a parallelogram.
Given information:
$\overset{―}{A B} ∥ \overset{―}{C D}$ ,
$\overset{―}{A B} \tilde{=} \overset{―}{D C}$ .

Answer & Explanation

cheekabooy

Skilled2021-07-27Added 83 answers

Proof:

/>
It is given that,
$\overset{―}{A B} \tilde{=} \overset{―}{C D}$ ,
$\overset{―}{A D} \tilde{=} \overset{―}{B C}$ .
If lines are parallel, alternate interior angles are congruent.
$∠ C D B \tilde{=} ∠ A B D$
$∠ D C A \tilde{=} ∠ B A C$
Two triangles are congruent by ASA congruence rule.
$△ A B X \tilde{=} △ C D X$
Corresponding parts of congruent triangles are congruent.
$\overset{―}{A X} \tilde{=} \overset{―}{X C}$
$\overset{―}{B X} \tilde{=} \overset{―}{X D}$
Vertical angles are congruent.
$∠ A X D \tilde{=} ∠ B X C$
Two triangles are congruent by SAS congruence rule.
$△ A X D \tilde{=} △ C X B$
Corresponding parts of congruent triangles are congruent.
$\overset{―}{B C} \tilde{=} \overset{―}{A D}$
If both pairs of opposite sides are congruent, then a quadrilateral is a parallelogram.
ABCD is a parallelogram.

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