Similar triangles



Similar triangles are triangles whose corresponding angles are equal. Corresponding sides are not equal. If the sides are also equal, we say that the triangles are congruent.

Triangles ABC and triangles EFG are similar for the following reasons:

The corresponding angles are equal:

Angles A and angle E shown in green are equal.

Angles B and angle F shown in red are equal.

Angles C and angle G shown in dark red are equal.


Similar-triangles



If you are looking at two or more triangles, the triangles don't have the same size, yet the corresponding angles of the triangles are equal, then the triangles are similar


You don't have to have the measure of all 3 corresponding angles to conclude that triangles are similar.


Angle Angle similarity postulate or AA similarity postulate:

If two angles of a triangle have the same measures as two angles of another triangle, then the triangles are similar


Why is it so?

The angles in a triangle must add up to 180 degrees

Consider this situation:

Triangle #1:

Angle #1 = 30 degrees. Angle #2 = 80 degrees

Triangle #2:

Angle #1 = 80 degrees. Angle #2 = 30 degrees

The last angle must be the same

Triangle #1: last angle = 180 - 30 - 80 = 70

Triangle #2: last angle = 180 - 80 - 30 = 70

All 3 angles are the same, so the triangles are similar


Look at the triangles below. According to the AA similarity postulate, they are similar

Similar-triangles


We know they are similar because angle A = angle D and angle C = angle F

We don't really care about the measure of the last angle

We can put the small triangle on the left inside the triangle on the right

Similar-triangles


The traingles are still similar.

The only difference this time is that angle C coincides with angle F.


A special figure that always gives similar triangles.

Similar-triangles


There are 3 triangles in the figure above

Triangle ABC, ABG, and AGC.

All 3 triangles are similar to one another.

The easiest to see why they are similar triangles is to plug in some numbers.

Let's say that angle B = 50°

Some important calculations:

In triangle ABG, the angle in green = 40° since 40° + 50° + 90° = 180°

The angle in brown in triangle AGC = 50° since 40° + 50° = 90°

Angle C = 40° since 50° + 40° + 90° = 180°

The result is summarized in the figure below:


Similar-triangles



Remember all we need to prove similarity is to find 2 corresponding angles that are equal

Show triangle ABC is similar to triangle ABG

For triangle ABC, angle A = 90° and angle B = 50°

For triangle ABG, angle G = 90° and angle B = 50°

Triangle ABC is similar to triangle ABG

Show triangle ABC is similar to triangle AGC

For triangle ABC, angle A = 90° and angle B = 50°

For triangle AGC, angle G = 90 ° and angle in brown = 50°

Triangle ABC is similar to triangle AGC

Show triangle ABG is similar to triangle AGC

For triangle ABG, angle B = 50° and angle in green = 40°

For triangle AGC, angle in brown = 50 ° and angle C = 40°

Triangle ABG is similar to triangle AGC








Page copy protected against web site content infringement by Copyscape

Recent Articles

  1. Average age word problem

    Aug 09, 16 01:40 PM

    Average age of Dipu and Apu is 22 years. Average age of Dipu and Tipu is 24 years. Age of Dipu is 21 years. What are the ages of Apu, and tipu ? Let

    Read More



New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.