prove that each angle of a rectangle is a right angle - Maths - Triangles

Question

prove that each angle of a rectangle is a right angle ?

Vimala · Accepted Answer

Consider the following figure of rectangle

The diagonals AC and BD intersect at O Thus the vertical angles $∠ A O D = ∠ B O C$ Since the line segments AB and CD are parallel lines, and AC is the transversal, then the alternate interior angles $∠ O C B = ∠ O A D$ Since the line segments AB and CD are parallel lines, and BD is the transversal, then the alternate interior angles $∠ O B C = ∠ O D A$ The sides AD = BC Therefore, by Angle-Side-Angle congruence, $Δ A O D = Δ B O C$ AB and CD are parallel, and BD is a transversal, therefore, the angles $∠ C D B = ∠ A B D$ Now, $∠ A D C = ∠ O D A + ∠ B D C$ and $∠ A B C = ∠ O B C + ∠ A B D$ Thus, $∠ A D C = ∠ A B C$ We also know that in a parallelogram, the sum of opposite angles are 180 degrees Also $∠ A D C + ∠ A B C = 180^{\circ}$ Since $∠ A D C = ∠ A B C$ , they have to be equal to 90 degrees each Thus, in a rectangle all the angles are 90 degrees