Lesson Opposite angles of a parallelogram

Opposite angles of a parallelogram

Theorem 1
In a parallelogram, the opposite angles are congruent.

Proof We have the parallelogram ABCD (Figure 1).                                                   We need to prove that the opposite angles are congruent: L A = L C and L B = L D. Let us draw the straight line AE as the continuation of the side AB of the parallelogram ABCD (Figure 2). Then the angles DAB and CBE are congruent as the corresponding angles at the parallel lines AD and BC and the transverse AE (see the lesson Parallel lines under the topic Angles, complementary, supplementary angles of the section Geometry in this site).

Figure 1. To the Theorem 1

Figure 2. To the proof of the Theorem 1

The angles CBE and BCD are congruent as the alternate interior angles at the parallel lines DC and AE and the transverse BC.
Hence, the angles DAB and BCD are congruent, or L A = L C.

Regarding the angles L B and L D, let us draw the straight line DF as the continuation of the side DC of the parallelogram ABCD (Figure 2).
Then the angles ADC and BCF are congruent as the corresponding angles at the parallel lines AD and BC and the transverse DF.
The angles BCF and ABC are congruent as the alternate interior angles at the parallellines DF and AB and the transverse BC.
Hence, the angles ADC and ABC are congruent, or L B = L D.

The proof is completed.