SSS - all corresponding sides are equal.
SAS - 2 pairs of corresponding sides are equal and the included angles are equal.
AAS - 2 angles are equal and one pair of corresponding sides is equal.
RHS - both triangles have right angles, hypotenuses are equal and one pair of corresponding sides are equal.

In this chapter, we are going to extend that knowledge and look at how we can join two congruent triangles to make a quadrilateral, and hence, how we can make deductions about these four-sided shapes. To do this, we need to look at the properties of some of these quadrilaterals. Here is a recap.

Parallelogram

Opposite sides in a parallelogram are parallel
Opposite angles in a parallelogram are equal
Opposite sides in a parallelogram are equal
Diagonals of a parallelogram bisect each other

Rectangle

Opposite sides in a rectangle are parallel
Opposite sides in a rectangle are equal
Diagonals of a rectangle bisect each other
Diagonals in a rectangle are equal
All angles are right angles

Square

All sides of a square are equal
Opposite sides in a square are parallel
Diagonals of a square are perpendicular to each other (cross at 90°)
Diagonals of a square bisect the angles at the vertices (makes them 45°)
Diagonals of a square bisect each other
Diagonals of a square are equal

Rhombus

Opposite angles of a rhombus are equal
Opposite sides in a rhombus are parallel
All sides of a rhombus are equal
Diagonals of a rhombus bisect each other at 90 degrees
Diagonals of a rhombus bisect corner angles
Diagonals of a rhombus bisect each other

Trapezium

An isosceles trapezium (trapezoid) has 2 pairs of adjacent angles equal
A trapezium (trapezoid) has one pair of opposite sides parallel
An isosceles trapezium (trapezoid) has one pair of opposites sides equal
Diagonals of an isosceles trapezium (trapezoid) are equal

Kite

A kite has 2 pairs of adjacent sides equal
A kite has 1 pair of opposite angles equal
The longest diagonal of a kite bisects the angles through which it passes
Diagonals of a kite are perpendicular to each other
The longest diagonal of a kite bisects the shorter diagonal

Now let's look through some examples to see this process in action.

Examples

Question 1

The following kite is formed by joining two congruent triangles. What type of triangles are they?

A) scalene B) isosceles C) equilateral

Think: What are the properties of these different types of triangles.

Do: The picture does not indicate that any of the side lengths are the same. So these triangles are scalene.

Question 2

$WXYZ$WXYZ is a rectangle. $\triangle WXZ$△WXZ is congruent to $\triangle XYW$△XYW.

a) Deduce the side that is equal in length to $WY$WY.

b) Then, what can we deduce about the diagonals of a rectangle?

Question 3

Two identical isosceles triangles are joined along the unequal edge. What kind of quadrilateral is formed?

Outcomes

9.G.T.1

Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

Congruence in Quadrilaterals