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# Similar Triangles – Explanation & Examples

Now that we are done with the congruent triangles, we can move on to another concept called **similar triangles.**

In this article, we will learn about similar triangles, features of similar triangles, how to use postulates and theorems to identify similar triangles, and lastly, how to solve similar triangle problems.

## What are Similar Triangles?

The concept of similar triangles and congruent triangles are two different terms that are closely related. **Similar triangles are two or more triangles with the same shape, equal pair of corresponding angles, and the same ratio of the corresponding sides.**

*Illustration of similar triangles:*

Consider the three triangles below. If:

- The ratio of their corresponding sides is equal.

AB/PQ = AC/PR= BC= QR, AB/XY= AC/XZ= BC/YZ

- ∠ A= ∠ P=∠X, ∠B = ∠Q= ∠Y, ∠C= ∠R =∠Z

Therefore, ΔABC ~ΔPQR~ΔXYZ

### Comparison between similar triangles and congruent triangles

Features | Congruent triangles | Similar Triangles |

Shape and size | same size and shape | Same shape but different size |

Symbol | ≅ | ~ |

Corresponding side lengths | The ratio of corresponding sides is congruent triangles is always equal to a constant number 1. | The ratio of all the corresponding sides in similar triangles is consistent. |

Corresponding angles | All corresponding angles are equal. | Each pair of corresponding angles are equal. |

### How to identify similar triangles?

We can prove similarities in triangles by applying similar triangle theorems. These are postulates or the rules used to check for similar triangles.

There are **three rules for checking similar triangles: AA** rule, SAS rule, or SSS rule.

**Angle-Angle (AA) rule:**With the AA rule, two triangles are said to be similar if two angles in one particular triangle are equal to two angles of another triangle.

**Side-Angle-Side (SAS) rule:**

The SAS rule states that two triangles are similar if the ratio of their corresponding two sides is equal and also, the angle formed by the two sides is equal.

**Side-Side-Side (SSS) rule:**

Two triangles are similar if all the corresponding three sides of the given triangles are in the same proportion.

## How to Solve Similar Triangles?

There are **two types of similar triangle problems**; these are problems that require you to prove whether a given set of triangles are similar and those that require you to calculate the missing angles and side lengths of similar triangles.

*Let’s take a look at the following examples:*

*Example 1 *

Check whether the following triangles are similar

__Solution__

Sum of interior angles in a triangle = 180°

Therefore, by considering Δ PQR

∠P + ∠Q + ∠R = 180°

60° + 70° + ∠R = 180°

130° + ∠R = 180°

Subtract both sides by 130°.

∠ R= 50°

Consider Δ XYZ

∠X + ∠Y + ∠Z = 180°

∠60° + ∠Y + ∠50°= 180°

∠ 110° + ∠Y = 180 °

Subtract both sides by 110°

∠ Y = 70°

Hence;

- By Angle-Angle (AA) rule, ΔPQR~ΔXYZ.
- ∠Q = ∠ Y = 70° and ∠Z = ∠ R= 50°

*Example 2*

Find the value of x in the following triangles if, ΔWXY~ΔPOR.

__Solution__

Given that the two triangles are similar, then;

WY/QR = WX/PR

30/15 = 36/x

Cross multiply

30x = 15 * 36

Divide both side by 30.

x = (15 * 36)/30

x = 18

Therefore, PR = 18

Let’s check if the proportions of the corresponding two sides of the triangles are equal.

WY/QR = WX/PR

30/15 = 36/18

2 = 2 (RHS = LHS)

*Example 3*

Check whether the two triangles shown below are similar and calculate the value k.

__Solution__

By Side-Angle-Side (SAS) rule, the two triangles are similar.

Proof:

8/ 4 = 20/10 (LHS = RHS)

2 = 2

Now calculate the value of k

12/k = 8/4

12/k = 2

Multiply both sides by k.

12 = 2k

Divide both sides by 2

12/2 = 2k/2

k = 6.

*Example 4*

Determine the value of x in the following diagram.

__Solution__

Let triangle ABD and ECD be similar triangles.

Apply the Side-Angle-Side (SAS) rule, where A = 90 degrees.

AE/EC= BD/CD

x/1.8 = (24 + 12)/12

x/1.8 = 36/12

Cross multiply

12x = 36 * 1.8

Divide both sides by 12.

x = (36 * 1.8)/12

= 5.4

Therefore, the value of x is 5.4 mm.