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Middle Years

7.07 Similar triangle tests

Lesson

Two triangles are considered to be similar if one of them can be scaled up or down in size, and then rotated and/or reflected to match the other.

 

Sides in fixed ratio

We can think of similarity as a weaker version of congruency, where corresponding distances do not need to be equal but instead are in some fixed ratio.

The fixed ratio of distances between two similar figures can be referred to as the scale factor.

 

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. These four tests are:

  • 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

These similarity tests can be proved to work in the same way that the congruency tests work, except with sides in fixed ratio rather than needing to be equal.

It is worth noting that, since all triangles have a fixed angle sum of $180^\circ$180°, having two matching angles is equivalent to having three matching angles.

 

Corresponding sides and angles

To determine which sides are corresponding in two potentially similar triangles, we can use their positions with respect to any matching angles. If there is a common sized angle in both triangles, then the sides opposite those angles will be corresponding.

In this case, the sides labelled $x$x and $y$y are corresponding since they are opposite the same sized angle.

Similarly, if side lengths are given and two pairs of sides are in a fixed ratio between the two triangles, the angles between these pairs of sides will be corresponding.

In this case, since the side pairs $5$5 & $8$8 and $10$10 & $16$16 are in the fixed ratio of $1:2$1:2, the angles between these pairs are corresponding and therefore equal.

 

Practice questions

Question 1

Consider the two triangles in the diagram below:

  1. Are $\triangle ABC$ABC and $\triangle DEF$DEF similar?

    Yes, they satisfy AAA.

    A

    Yes, they satisfy SSS.

    B

    Yes, they satisfy SAS.

    C

    Yes, they satisfy RHS.

    D

    No, they are definitely not similar.

    E

    Unknown, there is not enough information.

    F

    Yes, they satisfy AAA.

    A

    Yes, they satisfy SSS.

    B

    Yes, they satisfy SAS.

    C

    Yes, they satisfy RHS.

    D

    No, they are definitely not similar.

    E

    Unknown, there is not enough information.

    F
Question 2

Consider the following:

  1. Which two of the following triangles are similar?

    A

    B

    C

    D

    A

    B

    C

    D
  2. What similarity test does this pair satisfy?

    AAA

    A

    SSS

    B

    SAS

    C

    RHS

    D

    AAA

    A

    SSS

    B

    SAS

    C

    RHS

    D
Question 3

Consider the following diagram:

  1. Are the triangles $\triangle ABC$ABC and $\triangle ADE$ADE definitely similar?

    Yes

    A

    No

    B

    Yes

    A

    No

    B
  2. What similarity test does this pair satisfy?

    AAA

    A

    SSS

    B

    SAS

    C

    RHS

    D

    AAA

    A

    SSS

    B

    SAS

    C

    RHS

    D
  3. Given that $DE=7$DE=7, what will the length of $BC$BC be?

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