What is corresponding parts of similar triangles?

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:

1. The ratio of their corresponding sides is equal.

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

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

Therefore, ΔABC ~ΔPQR~ΔXYZ

Comparison between similar triangles and congruent triangles

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.

If the measures of the corresponding sides of two triangles are proportional then the triangles are similar. Likewise if the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the including angles are congruent then the triangles are similar.

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$

If a line is drawn in a triangle so that it is parallel to one of the sides and it intersects the other two sides then the segments are of proportional lengths:

$$\frac{AD}{DB}=\frac{EC}{BE}$$

Parts of two triangles can be proportional; if two triangles are known to be similar then the perimeters are proportional to the measures of corresponding sides.

Continuing, if two triangles are known to be similar then the measures of the corresponding altitudes are proportional to the corresponding sides.

Lastly, if two triangles are known to be similar then the measures of the corresponding angle bisectors or the corresponding medians are proportional to the measures of the corresponding sides.

The bisector of an angle in a triangle separates the opposite side into two segments that have the same ratio as the other two sides:

$$\frac{AD}{DC}=\frac{AB}{BC}$$

Video lesson

Find the value of x in the triangle