5
Tests for a parallelogram
Textbook: pp. 17–19
GoalKnow the four tests that show a quadrilateral is a parallelogram.
New words
test for a parallelogram · parallelogramm alomatiequal and parallel sides · teng va parallel tomonlarintersection of diagonals · diagonallarning kesishishiopposite angles · qarama-qarshi burchaklar
Explanation
A test is a condition that decides whether a quadrilateral is a parallelogram. Test 1: if two opposite sides are equal and parallel, it is a parallelogram. Test 2: if the diagonals intersect and bisect each other, it is a parallelogram. Test 3: if the opposite sides are equal in pairs, it is a parallelogram. Test 4: if the opposite angles are equal in pairs, it is a parallelogram. The tests are the converses of the properties, and they help us to prove things and to construct parallel lines.
Worked examples
The angles of a quadrilateral in order are 70°, 110°, 70°, 110°. The opposite angles are equal in pairs, so by test 4 it is a parallelogram.
The diagonals of quadrilateral ABCD meet at O with AO = OC = 5 cm and BO = OD = 7 cm. By test 2, ABCD is a parallelogram.
Class activity
“Pick the test”: the teacher states a fact about a quadrilateral and students name the test it matches or say it is not enough.
Practice
1
In quadrilateral ABCD, AB ∥ CD, AB = CD = 7 cm and BC = 9 cm. How long is AD (cm)?
9
2
The sides of a quadrilateral in order are 4 cm, 4 cm, 7 cm, 7 cm. Is it certainly a parallelogram?
No: its opposite sides (4 and 7) are not equal.
3
The diagonals of a quadrilateral meet at O and AO = OC = 6 cm. For it to be a parallelogram with OD = 8 cm, how long must BO be (cm)?
8
4
Why must the equal sides in test 1 also be parallel?
Equality alone is not enough: if two equal segments point in different directions and we join their ends, the result is not a parallelogram.