# Geometry: Writing a Paragraph Proof ? ?

How would I write a paragraph proof to show that the base angles of an isosceles trapezoid are congruent.

Here is the link to view the trapezoid:

http://img98.imageshack.us/img98/3546/181111ps4.pn...

** Any help would be appreciated, I am unsure of how I should start the proof and what info. to put into it.

Relevance

Draw AH┴DC and BK┴DC, we will get AH//BK and ABKH will be a parallelogram (actually a rectangle). Therefore AH = BK

Since ABCD is an isosceles trapezoid (of bases AB and DC), its other sides are congruent: AD = BC

Two right-angled rectangles ADH and BCK have congruent hypothenuses (AD = BC) and and congruent legs (AH = BK), therefore they are congruent.

Hence: angle ADC = angle BCD (q.e.d.)

It can be deduced easily that it is the same for the top pair of base angles: angle DAB = angle CBA

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• It is given that ABCD is an isosceles trapezoid with AB parallel to CD and AD equal to BC. To prove that angle ADC is equal to angle BCD.

Through A draw a line parallel to BC and cutting the side DC at E. Since AE and BC are parallel, angle AED is equal to angle BCD (I forgot the wording of the theorem, but it is interior exterior angles formed by a transversal with two parallel lines are equal.) ABCE is a parallelogram since The opposite pair of sides are parallel. AE is equal to BC since opposite sides of a parallelogram are equal. Therefore since they are both equal to BC, AE is equal to AD. Triangle AED is an isosceles triangle. Therefore angle ADE is equal to angle AED. Therefore the base angles (ADC and BCD) are equal since they are both equal to angle AED.

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