Why must the angle in SAS be the included angle?

The two sides and the angle between them fix the shape completely. If the angle is somewhere else (SSA), two different triangles can fit the same data, so you cannot conclude that the triangles are similar.

When should I use AA, SSS or SAS?

Look at what is given. Two pairs of congruent angles: AA. Three pairs of side lengths: SSS. Two pairs of side lengths and the angle between them: SAS. In nested triangles a shared angle often gives you the angle for AA or SAS for free.

Does it matter which triangle I divide by?

Only for the value of $k$. Big over small gives $k > 1$ (enlargement), small over big gives $k < 1$ (reduction). Both work, as long as you divide in the same direction for every pair.

Teacher Resources

SSS and SAS Similarity

Q: How is SSS Similarity different from SSS Congruence?

SSS Congruence needs three pairs of **equal** sides. SSS Similarity only needs three pairs of sides in the **same ratio**. Congruence is the special case $k = 1$.

Prove triangles similar from side ratios (SSS) or two side ratios and the included angle (SAS), then use the scale factor for sides, perimeter and area.

Advanced30 minLesson

Printable worksheet

Definition

In AA Similarity you proved triangles similar with angles only. Now you will do it with sides. Remember: similar triangles have the same shape, their corresponding sides are in the same ratio, the scale factor

k

SSS Similarity (Side-Side-Side): if all three pairs of corresponding sides are in the same ratio, the triangles are similar.

\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k \quad\Rightarrow\quad \triangle ABC \sim \triangle DEF

SAS Similarity (Side-Angle-Side): if two pairs of corresponding sides are in the same ratio and the included angle (the angle between those two sides) is congruent, the triangles are similar.

\frac{DE}{AB} = \frac{EF}{BC} = k \text{ and } \angle B = \angle E \quad\Rightarrow\quad \triangle ABC \sim \triangle DEF

How to match the sides: when you only know the lengths, sort the sides of each triangle from shortest to longest and pair them in that order: shortest with shortest, middle with middle, longest with longest.

Once you know the triangles are similar:

every side of the image is $k$ times the matching side
the perimeter ratio is $k$
the area ratio is $k^2$

Compare with congruence: SSS and SAS congruence need equal sides (

k = 1

), SSS and SAS similarity only need proportional sides.

Try it now

Which triangle is similar to a triangle with sides

2

3

and

4

Worked Examples

\triangle ABC

has sides

AB = 4

BC = 6

CA = 8

\triangle DEF

has sides

DE = 6

EF = 9

FD = 12

. Are the triangles similar?

Sort each triangle's sides from shortest to longest

\triangle ABC

4, 6, 8

\triangle DEF

6, 9, 12

→ Pairs:

4 \leftrightarrow 6

6 \leftrightarrow 9

8 \leftrightarrow 12

Divide each side of $\triangle DEF$ by the matching side of $\triangle ABC$

\frac{6}{4} = 1.5

\frac{9}{6} = 1.5

\frac{12}{8} = 1.5

→ All three ratios are

1.5

Apply SSS Similarity

Three equal ratios, so the sides are proportional →

\triangle ABC \sim \triangle DEF

k = 1.5

Answer: Yes, $\triangle ABC \sim \triangle DEF$ by SSS Similarity with scale factor $k = 1.5$ .

Common Mistakes

Pairing the sides in the order they are listed, for example $4, 6, 8$ with $12, 9, 6$ , and getting ratios $3$ , $1.5$ , $0.75$

Why it's wrong: The order in which side lengths are written says nothing about which sides correspond. Wrong pairs give different ratios even for similar triangles.

Correct: Sort both triangles from shortest to longest first: $4, 6, 8$ and $6, 9, 12$ . Then all ratios are $1.5$ .

Using SAS with an angle that is not between the two sides

Why it's wrong: Two proportional sides and a non-included angle (SSA) do not fix the shape: two different triangles can have the same two sides and the same angle opposite one of them.

Correct: Check that the congruent angle is formed by exactly the two sides you compared. Otherwise SAS Similarity does not apply.

Thinking the area ratio is also $k$

Why it's wrong: Area is length times length, so both factors grow by $k$ : the area grows by $k \cdot k = k^2$ .

Correct: Perimeter ratio $= k$ , area ratio $= k^2$ . With $k = 3$ , the area is $9$ times as large, not $3$ times.

Dividing big by small for one pair and small by big for another, for example $\frac{6}{4} = 1.5$ but $\frac{6}{9} \approx 0.67$

Why it's wrong: The ratios then look different although the sides are proportional.

Correct: Always divide in the same direction: image side over original side (or the other way round for every pair).

Interactive Visual

Similar triangles

Scale factor k = 1.5

SSSAll three pairs of sides are in the same ratio k.

DE : AB = 6 : 4 = 1.5
EF : BC = 9 : 6 = 1.5
FD : CA = 12 : 8 = 1.5

Move the slider to change the scale factor k. The angles stay the same, every side is multiplied by k.

Similar triangles

Scale factor k = 2

SASTwo pairs of sides are in the same ratio k and the angles between them are equal.

XY : PQ = 10 : 5 = 2
YZ : QR = 16 : 8 = 2
∠Q = ∠Y = 60°

Move the slider to change the scale factor k. The angles stay the same, every side is multiplied by k.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which triangle is similar to a triangle with sides

2

3

and

4

Why It Matters

Often you can measure lengths more easily than angles, and then SSS and SAS similarity are the tools you need:

Surveying: to find the width of a lake you cannot cross, surveyors stake out a small triangle that shares an angle with a big one and use SAS similarity
Engineering: a model of a roof truss or a bridge is built with all lengths scaled by the same factor, so it has exactly the same shape as the real one
Printing and design: when a logo is enlarged, every length is multiplied by $k$ but the ink or material needed grows by $k^2$
Checking a drawing: a carpenter can check that a triangular frame matches a plan by comparing three length ratios, no protractor needed

Knowing that area grows with

k^2

(not

k

) also prevents costly mistakes, like ordering twice the paint for a wall that is twice as wide and twice as tall.

Real World Applications

Enlarging a Logo

Designers enlarge logos so that every length grows by the same scale factor. The enlarged triangle is similar to the original by SSS Similarity, so it keeps exactly the same shape.

Example:

A triangular logo with sides $2$ cm, $3$ cm and $4$ cm is printed with $k = 5$ on a poster: the sides become $10$ cm, $15$ cm and $20$ cm.

1Try It Yourself

A triangular logo has sides $3$ cm, $4$ cm and $6$ cm. On a banner its shortest side must be $7.5$ cm, and the logo must keep its shape.

How long is the longest side of the logo on the banner (in cm)?

Step 1: Write the mathematical expression

Multiply the longest side by the scale factor:

Measuring Across a Lake

Surveyors cannot stretch a tape measure across water. They mark two points on lines leading away from a corner point and use the small triangle that shares the corner angle with the big one.

Example:

From point $A$ two straight paths lead past the lake. A small triangle with sides $\frac{1}{4}$ as long as the big one along both paths shares the angle at $A$ , so the width of the lake is $4$ times the short distance measured on land.

2Try It Yourself

From point $A$ , $AD = 20$ m and $AB = 60$ m along one path, $AE = 25$ m and $AC = 75$ m along the other. $B$ and $C$ lie on opposite shores of the lake. On land, $DE = 18$ m.

How wide is the lake, the distance $BC$ (in m)?

Step 1: Write the mathematical expression

Multiply the measured distance $DE$ by the scale factor:

Fabric for a Bigger Sail

Sailmakers scale a proven sail design up or down. The new sail is similar to the old one, but the fabric needed grows with the area ratio $k^2$, not with $k$.

Example:

A sail that is $1.5$ times as long on every side needs $1.5^2 = 2.25$ times as much fabric.

3Try It Yourself

A small triangular sail has sides $3$ m, $4$ m and $5$ m and an area of $6$ m². A new sail has sides $6$ m, $8$ m and $10$ m.

How much fabric (in m²) does the new sail need?

Step 1: Write the mathematical expression

Multiply the old area by the area ratio:

Key Takeaways

1SSS Similarity: if all three pairs of corresponding sides have the same ratio $k$ , the triangles are similar
2SAS Similarity: two pairs of sides with the same ratio $k$ and a congruent included angle make the triangles similar
3When only lengths are given, sort the sides from shortest to longest and pair them in that order
4The scale factor $k$ is image side divided by original side; every missing side is $k$ times its partner
5Perimeter ratio $= k$ , area ratio $= k^2$
6Choose the criterion from what you know: two angles means AA, three sides means SSS, two sides and the angle between them means SAS

Frequently Asked Questions

SSS Congruence needs three pairs of equal sides. SSS Similarity only needs three pairs of sides in the same ratio. Congruence is the special case

k = 1

Glossary

SSS Similarity: If all three pairs of corresponding sides of two triangles are in the same ratio, the triangles are similar
SAS Similarity: If two pairs of corresponding sides are in the same ratio and the included angles are congruent, the triangles are similar
Included angle: The angle formed by two given sides of a triangle, at the vertex where they meet
Scale factor: The ratio $k$ of a side of one similar figure to the corresponding side of the other
Corresponding sides: Sides that are in the same position in two similar or congruent figures
Area ratio: The ratio of the areas of two similar figures, equal to $k^2$

Formula Card

SSS Similarity

\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k \Rightarrow \triangle ABC \sim \triangle DEF

All three side ratios are equal (sides sorted from shortest to longest)

SAS Similarity

\frac{DE}{AB} = \frac{EF}{BC} = k,\ \angle B = \angle E \Rightarrow \triangle ABC \sim \triangle DEF

Two side ratios are equal and the included angles are congruent

Perimeter ratio

\frac{P_2}{P_1} = k

Perimeters grow by the same factor as the sides

Area ratio

\frac{A_2}{A_1} = k^2

Areas grow by the square of the scale factor

Today's challengeOne problem, three tries

SSS and SAS Similarity

Definition

Worked Examples

Common Mistakes

Interactive Visual

Similar triangles

Similar triangles

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Enlarging a Logo

Measuring Across a Lake

Fabric for a Bigger Sail

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

SSS Similarity

SAS Similarity

Perimeter ratio

Area ratio

More in This Topic

Introduction to Congruence

Introduction to Similarity

Congruent Triangles

AA Similarity