Sarthaks Test
0 votes
in Mathematics by (29.7k points)

If the diagonals of a parallelogram are equal, then show that it is a rectangle

1 Answer

0 votes
by (128k points)
selected by
Best answer

Given : ABCD is a parallelogram in which AC = BD. 

To Prove : ABCD is a rectangle. 

Proof : In ∆ABC and ∆ABD 

AB = AB [Common] 

BC = AD [Opposite sides of a parallelogram] 

AC = BD [Given] 

∴ ∆ABC ≅ ∆BAD [SSS congruence] 

∠ABC = ∠BAD …(i) [CPCT] 

Since, ABCD is a parallelogram, thus, 

∠ABC + ∠BAD = 180° …(ii) [Consecutive interior angles] 

∠ABC + ∠ABC = 180° 

∴ 2∠ABC = 180° [From (i) and (ii)] 

⇒ ∠ABC = ∠BAD = 90° 

This shows that ABCD is a parallelogram one of whose angle is 90°. Hence, ABCD is a rectangle. Proved.

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.