# Theorem 3 - In a Parallelogram Opposite Angles are Equal

**Given :** A quadrilateral ABCD is a parallelogram.

**To Prove : ** âˆ A = âˆ C and âˆ B = âˆ D

**Proof** **:**

AB Ã¯Ã¯ DC and AD is a transversal

.^{.}.Â Â âˆ A + âˆ D = 180Â° ...... (Consecutive Interior angles)...(i)

Again, AD Ã¯Ã¯ BC and DC is a transversal

.^{.}.Â Â âˆ D + âˆ C = 180Â° ...... (Consecutive Interior angles)Â ...(ii)

Comparing eqns (i) and (ii), we get

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ A + âˆ D = âˆ D + âˆ C

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ A = âˆ C

SimilarlyÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ B = âˆ D

Hence âˆ A = âˆ C and âˆ B = âˆ D

Â

ABCD is a parallelogram and line segments DX and BY bisect the angles D and B respectively (see Fig.).Â Show thatÂ DX Ã¯Ã¯ BY.** **

Since opposite angles are equal in a parallelogram,

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â âˆ D = âˆ B

.^{.}.Â Â Â Â Â Â Â Â Â Â Â âˆ D =âˆ B

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â âˆ 1 = âˆ 2 ......(i)

Now AB Ã¯Ã¯ DC and BY is the transversal,

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â âˆ 2 = âˆ 3 ...... (Alternative Interior angles)...(ii)

Comparing equations (i) and (ii), we get

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â âˆ 1 = âˆ 3

But âˆ 1 and âˆ 3 are the corresponding angles of lines DX and BY.

.^{.}.Â Â Â Â Â Â Â Â Â Â Â Â Â DX Ã¯Ã¯ BY