Everything Maths Grade 10

We have already shownA^ 2 =C^ 3 andA^ 1 =C^ 4. Therefore,

A^=A^ 1 +A^ 2 =C^ 3 +C^ 4 =C^

Furthermore,

B^=D^ (△ABC△CDA)

Therefore opposite angles of a parallelogram are equal.

Summary of the properties of a parallelogram:

• Both pairs of opposite sides are parallel.


• Both pairs of opposite sides are equal in length.


• Both pairs of opposite angles are equal.


• Both diagonals bisect each other.

A B

䐀 C

Worked example 4: Proving a quadrilateral is a parallelogram

QUESTION

Prove that if both pairs of opposite angles in a quadrilateral are equal, the quadrilateral is a parallelogram.

Z Y

圀 X

x

y x

y

SOLUTION

Step 1: Find the relationship betweenx^and^y

InW XY Z:

W^ =Y^ = ^y (given)

Z^=X^ = ^x (given)

W^+X^+Y^+Z^ = 360° (sum of\s in a quad)

)2^x+ 2^y = 360°

)x^+ ^y = 180°

W^+Z^ = ^x+ ^y

= 180°

But these are co-interior angles between linesW XandZY. ThereforeW X∥ZY.

Step 2: Find parallel lines

SimilarlyW^+X^= 180°. These are co-interior angles between linesXYandW Z. ThereforeXY∥W Z.

Both pairs of opposite sides of the quadrilateral are parallel, thereforeW XY Zis a parallelogram.

Chapter 7. Euclidean geometry 253