The following Venn diagram shows how the diff erent quadrilaterals relate to each other.
Quadrilaterals Parallelograms
Rectangles Squares Rhombuses
A square, rhombus and rectangle are also parallelograms.
A square also fi ts the defi nition of a rectangle (all angles are 90°) and a rhombus (all sides are equal length).
A polygon is equilateral if all its sides are equal. A square and a rhombus are equilateral polygons.
A regular polygon is a polygon where all sides are equal and all angles are equal, e.g. square, equilateral triangle.
Discuss and discover
Investigate the relationship between the opposite angles in a parallelogram.
(i) With a classmate, discuss what you already know about parallelograms.
(ii) Draw the parallelograms on the right.
(iii) Using a protractor, fi nd the measure of each of the four interior angles. Find the sum of these angles.
(iv) What do you notice about opposite angles? (v) What can you conclude about the opposite angles in a parallelogram?
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Theorem 9: In a parallelogram, opposite sides are equal and opposite angles are equal
In the diagram on the right: |AB| = |DC| |AD| = |BC|
and: |∠BAD| = |∠BCD| |∠ABC| = |∠ADC|
Proof of Theorem 9
The proof uses the alternate angle theorem to show that a diagonal divides the parallelogram into two congruent (identical) triangles. This gives opposite sides and (one pair of) opposite angles equal.
A B
The full formal proofs are not examinable for Junior Cycle Mathematics, but you should be able to display understanding of the proofs.