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# Proving a Quadrilateral is parallelogram

- Question
- In the figure above, D is a point on AB. Draw a line passing thought point D and parallel to BC intersects AC at point E. Draw another line passing though point E and parallel to AB intersects BC at point F. Prove the quadrilateral DEFB is a parallelogram.

- Proof
- since DE // BC (Given)
- so angle 1 = angle 4 (corresponding angles are equal)
- since EF // AB (Given)
- so angle 1 = angle 2 (alternate interior angles are equal)
- since angle 1 = angle 4 and angle 1 = angle 2 (equal quantity substitution)
- so angle 2 = angle 4
- note: angle 2 and angle 4 are a pair of opposite angles of the quadrilateral DEFB
- since DE // BC,
- so angle 2 = angle 3 (alternate interior angles are equal)
- since angle 1 = angle 2 and angle 2 = angle 3
- so angle 1 = angle 3 (equal quantity substitution)
- angle 5 = 180
^{o}- angle 1 - angle 6 = 180
^{o}- angle 3 - so angle 5 = angle 6
- note: angle 5 ang angle 6 are a pair of opposite angles of the quadrilateral DEFB
- since angle 2 = angle 4 and angle 5 = angle 6
- so quadrilateral DEFB is a parallelogram.
- Theorem: A quadrilateral is a parallelogram if opposite angles are congruent.