Congruence and Similarity

NZ Level 6 (NZC) Level 1 (NCEA)

Introduction to Similarity

Lesson

Consider the following pairs of shapes:

Pair A: Similar Triangles

Note:

- Since all sides in the smaller triangle are equal, all angles are equal. This is an equilateral triangle.
- Each side in the smaller triangle is multiplied by 3 to give the side length of the larger triangle.
- This makes the sides of the larger triangle all equal. So the larger triangle is also equilateral.

**All the corresponding sides between the two shapes are in the same ratio and all corresponding angles between the two shapes are equal.**

Pair B: Non-Similar Triangles

Note:

- Since all sides in the smaller triangle are equal, all angles are equal. This is an equilateral triangle.
- To make the larger triangle, two of the sides of the smaller triangle are enlarged by a factor of $3$3 and one of the sides is enlarged by a factor of $2$2.
- This makes the sides of the larger triangle unequal. So the larger triangle is not equilateral, and all angles are not $60^\circ$60°.

**The corresponding sides between the two shapes are NOT in the same ratio and all corresponding angles between the two shapes are NOT equal.**

Pair C: Non-Similar Rectangles

Note:

- Both shapes are rectangles
- To make the larger rectangle, two of the sides of the smaller rectangle are enlarged by a factor of 2 and the other two sides are not changed.

**Even though all corresponding angles between the two shapes are equal, the corresponding sides between the two shapes are NOT in the same ratio. That is, the same enlargement factor has not been applied to each side of the smaller rectangle.**

Pair D: Similar Quadrilaterals

Note:

- To make the larger quadrilateral, each side length of the smaller quadrilateral has been doubled.

**There are four pairs of equal corresponding angles, and all corresponding sides are in the same ratio.**

If the only difference between two shapes is the size, (one is an enlargement of the other) then the two shapes are similar.

In similar shapes

- All corresponding sides are proportional
- All corresponding angles are equal.

Did you know?

Sometimes a shape can be rotated, translated or reflected but still be similar.

Some of our standard shapes create similarity because of the nature of their properties:

All circles are similar because every circle is an enlargement of another. That is, one circle can create any other circle by enlarging or reducing its radius.

All squares a similar because every square is an enlargement of another.

How can a square of side lengths $4$4 cm create a similar square of side length $10$10 cm?

Solution: Multiply each side of the smaller square by $2.5$2.5.

In fact, all **REGULAR **shapes create similarity (shapes that have all side lengths of equal value).

Explore this interactive to create similar polygons.

Let's have a look at these worked examples.

Consider the following shapes:

Are the two shapes similar?

Yes

ANo

BYes

ANo

BGive a reason for your answer.

All sides are in the same ratio but not all matching angles are equal.

AAll matching angles are not equal and all sides are not in the same ratio

BAll matching angles are equal but all sides are not in the same ratio

CAll sides are in the same ratio but not all matching angles are equal.

AAll matching angles are not equal and all sides are not in the same ratio

BAll matching angles are equal but all sides are not in the same ratio

C

Consider the shapes attached:

Are the 2 shapes similar?

Yes

ANo

BYes

ANo

BFind the enlargement factor.

$6$6

A$5$5

B$7$7

C$6$6

A$5$5

B$7$7

C

Consider the following triangles:

Which of these triangles are similar?

ABCDABCDGive a suitable reason for their similarity.

All corresponding angles are equal.

ATwo angles are equal and one side is a multiple of the corresponding side of the other.

BAll corresponding sides are in the same ratio.

CAll corresponding angles are equal.

ATwo angles are equal and one side is a multiple of the corresponding side of the other.

BAll corresponding sides are in the same ratio.

C

Recognise when shapes are similar and use proportional reasoning to find an unknown length

Apply geometric reasoning in solving problems