Expert Answer to "To prove: If the convex quadrilateral ABCD has..."

vazelinahS

Answered question

2021-08-02

To prove: If the convex quadrilateral ABCD has opposite sides congruent, then the opposide sides are parallel.
Given:
The given figure is,

Figure-(1)

Answer & Explanation

hesgidiauE

Skilled2021-08-03Added 106 answers

Approach:
For proving the statement, use construction and congruency properties.
Calculation:
$\overset{―}{A B} \tilde{=} \overset{―}{D C}$ (Given)
$\overset{―}{B C} \tilde{=} \overset{―}{A D}$ (Given)
$\overset{―}{B D} \tilde{=} \overset{―}{B D}$ (Same side)
Therefore, using the SSS (side, side, side) congruence, it can be said that $△ A B D \tilde{=} △ C D B$ .
So, if $△ A B D \tilde{=} △ C D B$ ,
Then, $∠ C B D = ∠ A D B and ∠ A B D = ∠ B D C$ since all the angles and sides are equal in congruent triangles. These are alternate interior angles, and if alternate interior angles are equal, then the sides containing them are parallel.
Therefore, from the above conditions, it can be said that $B C ∥ A D and A B ∥ C D$ .

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