Proving that a Trapezoid is Isosceles | The Congruent Non-Parallel Sides Proof

Proving Isosceles Trapezoids

To prove that a trapezoid is isosceles, we need to show that its non-parallel sides are congruent

To prove that a trapezoid is isosceles, we need to show that its non-parallel sides are congruent. To do this, we can use various properties and theorems related to trapezoids and congruent triangles.

Let’s consider a trapezoid ABCD, where AB || CD. We want to prove that AD = BC.

Proof:
Step 1: Show that the diagonals are congruent.
First, let’s draw the diagonal AC, dividing the trapezoid into two triangles, triangle ABC and triangle ACD.

Step 2: Show that triangle ABC and triangle ACD are congruent.
To show that the triangles are congruent, we need to prove that two pairs of corresponding sides and angles are congruent.

Step 3: Prove congruent corresponding sides and angles.
a) AC = AC (Common side)
b) AB = CD (Opposite sides of a trapezoid are parallel and, therefore, congruent)
c) Angle A = Angle D (Opposite angles of a trapezoid are congruent)

Step 4: Use congruence to show congruent corresponding parts.
Using the congruence of triangle ABC and triangle ACD, we can conclude that their corresponding parts are congruent.

Step 5: Conclude that AD = BC.
Since corresponding sides are congruent, we can conclude that AD = BC.

Hence, we have proved that the non-parallel sides AD and BC of the trapezoid ABCD are congruent, making it an isosceles trapezoid.

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