Here is just how we have the right to prove the opposite political parties of a parallelogram space congruent or equal using the parallelogram below.

You are watching: Prove that opposite sides of a parallelogram are congruent


Here is what is given
: parallelogram ABCDHere is what we should prove: segment ab ≅ segment CD and segment BC ≅ AD

Since ABCD is a parallelogram, segment BC is parallel come segment advertisement according come the definition of a parallelogram.

By the exact same token, segment ab is parallel come segment DC follow to the definition of a parallelogram.Line AC is a transversal for segment BC and segment AD. This transversal creates alternating interior angles 1 and 4 which are equal as presented below.


By the very same token, heat AC is a transversal because that segment abdominal and segment DC. This transversal creates alternate interior angles 2 and also 3 which room equal.


Moreover, segment AC ≅ segment AC by reflex property.


∠1 ≅ ∠ 4∠2 ≅ ∠ 3segment AC ≅ segment AC through ASA, triangle alphabet ≅ triangle ADC Therefore,segment ab ≅ segment CD and segment BC ≅ advertisement because corresponding parts that congruent triangles are additionally congruent.

Things that you should keep in mind as soon as you prove that opposite sides of a parallelogram space congruent.

Here space some crucial things the you must be mindful of around the evidence above. 

The reflexive home refers come a number that is constantly equal come itself. For example, z = z or 1000 = 1000 are examples of the reflex property. A transversal is a line that cuts 2 or an ext linesNotice that in the proof we were able to discover two angle that have actually the very same measure in both triangles. This way that all 3 angles in both triangles have actually the same measure. However, this details is not enough to say that the triangles space congruent.The ASA postulate is most likely the just thing we can use come prove that the opposite sides of a parallelogram room congruent. 

It is not always easy to understand proofs in geometry. Any questions around this proof, feel complimentary to contact me.

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