Congruence and Similarity

Congruence and Similarity

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In Geometry, congruent figures are two figures (2D) that have the same shape and size. If two figures are congruent, one of the figures can be rotated, reflected and/or translated (shifted up, down, left and/or right) to match the other figure exactly. 

If triangle DEF (ΔDEF) is congruent to triangle RST (ΔRST), we write ΔDEF ≅ΔRST (triangle DEF is congruent to triangle RST). This is called a congruence statement. 

Corresponding sides and angles refer to the pairs of matching sides or angles that are in the same position in two different shapes. 

Corresponding sides

DE = RS

EF = ST

DF = RT 

Corresponding angles

∠DEF = ∠RST

∠EFD = ∠STR

∠EDF = ∠SRT

Testing for congruence

There are four congruence tests used for triangles. If two triangles satisfy one of the following tests, the two triangles are proved to be congruent. 

SSS: side-side-side

Two triangles are congruent if all corresponding sides are equal. 

SAS: side-angle-side

Two triangles are congruent if two pairs of corresponding sides and the angle included between them is equal. 

ASA: angle-side-angle

Two triangles are congruent if two pairs of corresponding angles and a pair of corresponding sides are equal. 

RHS: right angle-hypotenuse-side

Two right-angled triangles are congruent if the hypotenuses and one pair of corresponding sides are equal.

Multiple Choice

Example

Which congruence test would be used to show that the following pairs of triangles are congruent?

• SSS

• SAS

• ASA

• RHS

Multiple Choice

Example

Which congruence test would be used to show that the following pairs of triangles are congruent?

• SSS

• SAS

• ASA

• RHS

Fill in the Blank

Find the unknown values of the following congruent pairs of triangles.

Answer: x=___,y=___,z=___ (no spaces, no degree symbol)

Multiple Choice

Example

Using the given information only, which congruence test would be used to show that the following pairs of triangles are congruent?

• SSS

• SAS

• ASA

• RHS

Fill in the Blank

Find the unknown values of the following congruent pairs of triangles.

Answer: x=___,y=___,z=___ (no spaces, no degree symbol)

Multiple Select

Select a pair of congruent shapes from the following set.

Multiple Choice

Are these two shapes congruent

• Yes

• No

Multiple Choice

The two triangles in the image are congruent. Pick the test that proves this.

• SSS

• SAS

• ASA

• RHS

Multiple Choice

Is this pair of trianlges congruent?

• Yes

• No

Multiple Choice

The value of y is

• 76

• 180

• 4.2

• 180-76-4.2

Similar Figures

Similar figures have the same shape, but not necessarily the same size.

Examples of similar shapes

Similar shapes must have:

• Corresponding sides are proportional

• Corresponding angles are equal

Similar figures must have:

1. Corresponding sides are proportional (have a constant ratio, called the scale factor)

In the diagram: from the smaller triangle to larger triangle, the scale factor is 3

Similar figures must have:

2. Corresponding angles are equal. 

Multiple Choice

Are the two shapes in the diagram similar?

• Yes

• No

Multiple Choice

Are the two shapes in the diagram similar

?

• Yes

• No

Multiple Choice

Are the pair of shapes in the diagram similar?

• Yes

• No

Calculating scale factor

Fill in the Blank

The two images are similar. If the original shape is the larger rectagle, and the image is the smaller rectangle, calculate the scale factor

Write your answer as a decimal (number only)

Fill in the Blank

The two images are similar. If the original shape is the larger rectangle, and the image is the smaller rectangle, find the value of a

Write your answer as 'a=___' no spaces,no units

Multiple Choice

Are the following pair of triangles similar?

• Yes

• No

Fill in the Blank

To make the pair of triangles similar, what number should be in place of 27?

Answer - just number

Multiple Choice

What is the scale factor of this pair of similar figures?

• 6

• 1/6

• 18/3

Multiple Choice

What is the value of x if these triangles are similar?

• 4

• 6

• 18/3

Similarity tests for triangles

There are four tests that we can use to test similarity between two triangles. If any one of these tests is satisfied then the two triangles must be similar.

Test for similarity

• AAA: Three pairs of equal angles

• SSS: Three pairs of sides in the same ratio

• SAS: Two pairs of sides in the same ratio and an equal included angle

• RHS: Both have right angles, and the hypotenuses and another pair of sides are in the same ratio

Multiple Choice

Are △ABC and △DEF similar?

• Yes

• No

Multiple Choice

The pair of triangles are similar. Which test is satisfied?

• SSS

• SAS

• ASA

• RHS

Multiple Select

Which two of the following triangles are similar?