In order come prove that the diagonals of an isosceles trapezoid are congruent, think about the isosceles trapezoid shown below. In this lesson, us will display you two different ways you can do the exact same proof making use of the exact same trapezoid.

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The very first way is to present that triangle abc is congruent come triangle DCB

Given: Isosceles trapezoid ABCD v segment ab ≅ segment DCProve: segment AC ≅ segment BD
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Since the trapezoid is isosceles, the base angles room congruent.Therefore, ∠CBA ≅ ∠BCD Furthermore, segment BC ≅ to segment BC through the reflexive residential property of congruence.By SAS Postulate, triangle alphabet ≅ triangle DCB.Therefore, segment AC ≅ segment BD

Things the you have to keep in mind as soon as you prove the the diagonals of one isosceles trapezoid space congruent.

Here room some things that you should know around the proof above. 

The declare if a trapezoid is isosceles, then the basic angles room congruent requires additionally a proof. However, we will certainly not prove here. The political parties that space equal in one isosceles trapezoid are always the political parties that space not parallel. In the trapezoid above, we present these sides through the red marks.The reflexive residential or commercial property refers come a number the is constantly equal come itself. For example, 9 = 9 or y = y are examples of the reflex property. In order come prove that the diagonals of one isosceles trapezoid space congruent, you can have additionally used triangle ABD and triangle DCA. 

Another great way to prove the the diagonals of an isosceles trapezoid space congruent. This time we present that triangle bad is congruent to triangle CDA

Given
: Isosceles trapezoid ABCD through segment abdominal muscle ≅ segment DCProve: segment AC ≅ segment BDSince the trapezoid is isosceles, the basic angles space congruentTherefore, ∠BAD ≅ ∠CDA

Notice that this time, we are not using the very same base angles as before. The base angle we are using now is regarded the base on height or segment AD.

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Furthermore, segment advertisement ≅ come segment ad by the reflexive residential property of congruence.By SAS Postulate, triangle poor ≅ triangle CDATherefore, segment AC ≅ segment BD
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