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# Congruence and similarity | Lesson

## What do congruent and similar mean?

Congruent triangles have both the same shape and the same size. In the figure below, triangles start color #11accd, A, B, C, end color #11accd and start color #ca337c, D, E, F, end color #ca337c are congruent; they have the same angle measures and the same side lengths.
Similar triangles have the same shape, but not necessarily the same size. In the figure below, triangles start color #11accd, A, B, C, end color #11accd and start color #aa87ff, X, Y, Z, end color #aa87ff are similar, but not congruent; they have the same angle measures, but not the same side lengths.
Note: If two objects are congruent, then they are also similar.

### What skills are tested?

• Determining whether two triangles are congruent
• Determining whether two triangles are similar
• Using similarity to find a missing side length

## What are the triangle congruence criteria?

Two triangles are congruent if they meet one of the following criteria.
SSS
: All three pairs of corresponding sides are equal.
SAS
: Two pairs of corresponding sides and the corresponding angles between them are equal.
ASA
: Two pairs of corresponding angles and the corresponding sides between them are equal.
AAS
: Two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are equal.
HL
: The pair of
hypotenuses
and another pair of corresponding sides are equal in two right triangles.

## What are the triangle similarity criteria?

Two triangles are similar if they meet one of the following criteria.
AA
: Two pairs of corresponding angles are equal.
SSS
: Three pairs of corresponding sides are proportional.
SAS
: Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.

## Finding missing side lengths in similar triangles

The SSS similarity criterion allows us to calculate missing side lengths in similar triangles. For similar triangles A, B, C and X, Y, Z shown below:
\begin{aligned} \purpleC{XY}&=k(\blueD{AB}) \\\\ \purpleC{YZ} &=k(\blueD{BC}) \\\\ \purpleC{XZ} &=k(\blueD{AC}) \\\\ \dfrac{\purpleC{XY}}{\blueD{AB}}&=\dfrac{\purpleC{YZ}}{\blueD{BC}}=\dfrac{\purpleC{XZ}}{\blueD{AC}}=k \end{aligned}
To calculate a missing side length, we:
1. Write a proportional relationship using two pairs of corresponding sides.
2. Plug in known side lengths. We need to know 3 of the 4 side lengths to solve for the missing side length.
3. Solve for the missing side length.

TRY: PROPERTIES OF CONGRUENT TRIANGLES
Triangles A, B, C and D, E, F are shown above. If the two triangles are congruent, which of the following statements must be true?

TRY: CONGRUENCE CRITERIA
For triangles L, M, N and P, Q, R, L, M, equals, P, Q, and angle, M has the same measure as angle, Q. Which of the following statements, if true, is sufficient to show that the two triangles are congruent?

TRY: IDENTIFYING SIMILAR TRIANGLES
Triangle A, B, C is shown above. Which of the following triangles are similar to Triangle A, B, C ?

TRY: CALCULATING MISSING SIDE LENGTH
Triangles R, S, T and X, Y, Z are similar triangles. What is the length of R, T ?

## Things to remember

Congruent triangles have the same corresponding angle measures and side lengths. The triangle congruence criteria are:
• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• HL (Hypotenuse-Leg, right triangle only)
Similar triangles have the same corresponding angle measures and proportional side lengths. The triangle similarity criteria are:
• AA (Angle-Angle)
• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
If triangles A, B, C and X, Y, Z are similar, then their corresponding side lengths have the same ratio:
start fraction, X, Y, divided by, A, B, end fraction, equals, start fraction, Y, Z, divided by, B, C, end fraction, equals, start fraction, X, Z, divided by, A, C, end fraction, equals, k

## Want to join the conversation?

• Thanks a lot really happed Geometry might not be straight forward, but once you focus you can really get it down.
• I was having trouble with this today TT but I am so glad I decided to do extra practice and get help, it really helps a lot! :)
• Is the smaller triangle a 3-4-5 triangle?
• How does the last problem equal to 20? I get it up to singling the RT by itself but how is 30/12 x 8 equaling to 20? These answers should show the work vs. skipping the steps necessary to make everyone at any level understand.
• The first step is always to find the scale factor: the number you multiply the length of one side by to get the length of the corresponding side in the other triangle (assuming of course that the triangles are congruent).

In this case you have to find the scale factor from 12 to 30 (what you have to multiply 12 by to get to 30), so that you can multiply 8 by the same number to get to the length of RT. I like to figure out the equation by saying it in my head then writing it out:

'12 times the scale factor is 30'

12 * SF = 30

If you solve it algebraically (30/12) you get:

SF = 2.5

Now that we know the scale factor we can multiply 8 by it and get the length of RT:

8 * 2.5 = RT
RT = 20

If you want to know how this relates to the disjointed explanation above, 30/12 is like the ratio of the two known side lengths, and the other ratio would be RT/8. You can then equate these ratios and solve for the unknown side, RT.

30/12 = RT/8
2.5 = RT/8
RT = 20

OR

30/12 = RT/8
30 * 8 = 12 * RT
240 = 12 * RT
RT = 20

I hope that this isn't too late and that my explanation has helped rather than made things more confusing.
• examples of congruent shapes
• If you have a problem, like "SAS" and you have to find the missing side, what would you do?