PARALLELOGRAM Parallelogram — a quadrilateral

PARALLELOGRAM

• Parallelogram - a quadrilateral whose opposite sides are parallel (lie on parallel lines).

• The basic properties of a parallelogram 1) Opposite sides of the parallelogram are equal: AB = CD, BC = AD 2) The opposite angles of a parallelogram are equal: ∠ABC = ∠CDA, ∠BCD = ∠DAB 3) Opposite sides of the parallelogram parallel: AB||CD, BC||AD 4) The sum of the angles of a parallelogram is equal to 360 °: ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360° 5) The sum of the angles of the parallelogram adjacent to either side is 180 °: ∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180° 6) Each diagonal divides the parallelogram into two congruent triangles. 7) Two diagonal divide the parallelogram into two pairs of congruent triangles

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DIAGONAL OF THE PARALLELOGRAM • Diagonal of the parallelogram is any segment connecting two vertices of the opposite angles of a parallelogram. • The parallelogram has two diagonally - long d 1, and a short – d 2

1. Formula diagonals of a parallelogram through the sides and the cosine of the angle β (for the cosine). d 1 = √a 2 + b 2 - 2 ab·cosβ d 2 = √a 2 + b 2 + 2 ab·cosβ 2. Formula diagonals of a parallelogram through the sides and the cosine of the angle α (for the cosine) d 1 = √a 2 + b 2 + 2 ab·cosα d 2 = √a 2 + b 2 - 2 ab·cosα

3. Formula diagonal of a parallelogram via the two sides and other well-known diagonal: d 1 = √ 2 a 2 + 2 b 2 - d 22 d 2 = √ 2 a 2 + 2 b 2 - d 12 4. Formula diagonally across the area of a parallelogram, and a certain diagonal angle between the diagonals: