To prove: ABCD is a parallelogram. Given information: AB―paral≤lCD―, AB―=∼DC―.

Proof: 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.

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Burhan Hopper

Answered question

2021-07-26

To prove: ABCD is a parallelogram.
Given information:

\overset{―}{A B} p a r a l \leq l \overset{―}{C D}

\overset{―}{A B} \tilde{=} \overset{―}{D C}

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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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Prove that ABCD is a parallelogram \overline{AB}\parallel \overline{CD}, \overline{AB}\cong \overline{DC}

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