Question

Need help, on level geometry 35 pts!

Let ABCD be a rectangle with A on top left, B on top right, D on bottom left and C on the bottom right. Prove that a rectangle has congruent diagonals.

Answer 1

A rectangle has congruent diagonals

Explanation:

* Lets explain how to solve the problem

- In any rectangle each two opposite sides are parallel and equal

- All the angles of a rectangles are right angles

- To prove that the diagonals of a rectangle are congruent, we will use

the SAS case of congruent

- SAS ⇒ 2 sides and including angle in the 1st Δ ≅ 2 sides and

including angle in the 2nd Δ

* Lets solve the problem

∵ ABCD is a rectangle

∴ AD = BC

∴ AB = CD

∴ m∠A = m∠B = m∠C = m∠D = 90°

∵ AC and BD are the diagonals of the rectangle

- In the 2 triangles ADC and BCD

∵ AD = BC ⇒ opposite sides in a rectangle

∵ m∠ADC = m∠BCD ⇒ all angles are equal in the rectangle

∵ DC = CD ⇒ common side in the two triangles

∴ ΔADC ≅ ΔBCD ⇒ SAS

- From congruent

∴ AC = BD

∵ AC and BD are the diagonals of the rectangle

∴ The diagonals of the rectangle are congruent

* A rectangle has congruent diagonals