Question

prove that the diagonals of trapezium are equal

Answer 1

Isosceles Trapezoids
Irene has just bought a house and is very excited about the backyard. She's a bit of math nerd, and plans to create a garden in the shape of an isosceles trapezoid. She paints the lawn white where her future raised garden bed will be.

In geometry, a trapezoid is a quadrilateral that has at least one pair of parallel sides. A trapezoid in which non-parallel sides are equal is called an isosceles trapezoid.

ABCD is an isosceles trapezoid with AB parallel to CD (we write it like this in math: AB||CD), and line AD is equal to line BC.


Isosceles Trapezoid
Isosceles Trapezoid

But how can Irene prove her garden is an isosceles trapezoid? Let's see some important theorems related to isosceles trapezoid to help her out.

Base Angles
The angles formed between non-parallel sides and parallel sides, called base angles, are equal in an isosceles trapezoid. In Irene's lawn trapezoid ABCD, angles C and D are equal.


Base angles are equal

To prove this theorem, let's let's draw a line CE parallel to AD such that ADCE becomes a parallelogram.


Parallelogram ADCE
Parallelogram ADCE

In this parallelogram, we know that line AD = line CE. We also know that line AD = line BC, so we also know that line BC = line CE.

Now, as line BC and line CE are equal, the triangle BCE becomes an isosceles triangle. Therefore, angles CBE and CEB are equal.




We understand that line AD and line CE are parallel, and the line AE is the transversal. So, the sum of interior angles on the same side, angle DAE and angle CEA is 180 degrees

So,




Therefore, the angles DAB and CBA are equal.

Next, we know that ADCE is a parallelogram, so the opposite angles would be equal.



Now, angles CBE and BCD would be equal because they are alternate interior angles for the parallel lines AE and CD.




We already know that the angles CEB and CEB are equal. Therefore,




Thus, it is proved that the base angles in an isosceles trapezoid are equal.

Answer 2

hey friend... check out the attachment above... hope it'll help...

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