4
The parallelogram and its properties
Textbook: pp. 14–16
GoalKnow the definition, properties and altitude of a parallelogram.
New words
parallelogram · parallelogrammopposite sides · qarama-qarshi tomonlardiagonal · diagonalaltitude of a parallelogram · parallelogramm balandligi
Explanation
If both pairs of opposite sides of a quadrilateral are parallel, we call it a parallelogram. Any diagonal splits a parallelogram into two congruent triangles; from this, both the opposite sides and the opposite angles are equal in pairs. The two diagonals meet at one point, which is the midpoint of each diagonal. The two angles next to one side make 180° together, because they are interior angles on the same side of parallel lines. A perpendicular from any point of one side to the line of the opposite side is called an altitude; a parallelogram usually has two different altitudes.
Worked examples
Adjacent angles of a parallelogram are x and 3x: 4x = 180°, x = 45°; the angles are 45°, 135°, 45°, 135°.
The sides of a parallelogram are in the ratio 3 : 5 and the perimeter is 64 cm. 2(3x + 5x) = 64, x = 4; the sides are 12 cm and 20 cm.
Class activity
“Build a parallelogram”: draw one using two pairs of parallel ruler edges or notebook lines, then measure its sides and angles.
Practice
1
One angle of a parallelogram is 72°. How many degrees is an adjacent angle?
108
2
The diagonals AC = 18 cm and BD = 26 cm of a parallelogram meet at O. How many cm is AO + BO?
22
3
The adjacent sides of a parallelogram are 14 cm and 23 cm. What is the perimeter (cm)?
74
4
Why are the opposite angles of a parallelogram equal?
Since A + B = 180° and B + C = 180°, ∠A = ∠C; in the same way ∠B = ∠D.