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.

If the above two triangles are similar then

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

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

AA Rule

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

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

Solution:

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

Step 2: The ratios of the lengths are equal.

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

Answer:  The length of s is 3

RAR Rule

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.

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

The following videos give more examples of the use of similar traingles.

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