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Similar Triangles




In this lesson, we will learn

  • the properties of similar triangles
  • how to tell if two triangles are similar using the similar triangle theorem: AA rule, SAS rule or SSS rule
  • how to solve problems using similar triangles

Related Topics: Congruent Triangles

Properties of Similar Triangles

Similar triangles have the following properties:

  • They have the same shape but not the same size.
  • Each corresponding pair of angles is equal.
  • The ratio of any pair of corresponding sides is the same.
similar big triangle similar small triangle

If the above two triangles are similar then

similar triangle ratios

When the ratio is 1 then the similar triangles become congruent triangles (same shape and size).

How to tell if two triangles are similar

We can tell whether two triangles are similar without testing all the sides and all the angles of the two triangles. There are three rules or theorems to check for similar triangles. They are called the AA rule, SAS rule and SSS rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are similar.

AA Rule

The Angle-Angle (AA) rule states that

If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

AA big AA small

This is also sometimes called the AAA rule because equality of two corresponding pairs of angles would imply that the third corresponding pair of angles are also equal.

Example 1: Given the following triangles, find the length of s

example 1  big example 1 small

Solution:

Step 1: The triangles are similar because of the AA rule

Step 2: The ratios of the lengths are equal.

ratios

Step 3: Cross multiplying: 6s = 18 Þ s = 3

Answer:  The length of s is 3


SAS Rule

The Side-Angle-Side (SAS) rule states that

If the angle of one triangle is the same as the angle of another triangle and the sides containing these angles are in the same ratio, then the triangles are similar.

RAR big RAR small

Example 2: Given the following triangles, find the length of s

Solution:

Step 1: The triangles are similar because of the RAR rule

Step 2: The ratios of the lengths are equal.

Answer:  The length of s is 3


SSS Rule

The Side-Side-Side (SSS) rule states that

If two triangles have their corresponding sides in the same ratio, then they are similar.

similar triangles 



Similar Triangles AA SAS SSS
The following video shows the AA, SAS and SSS similarity theorem and how to use them.



 



The following videos will investigate the properties of similar triangles




The following videos will introduce the concept of similar triangles.




Solving Problems using Similar Triangles

The following videos give more examples of how to solve problems using similar triangles.



Using similar triangles to solve shadow problems




 



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