What's the difference between similar triangles and congruent triangles?

Similar triangles have the same shape and the same angles, but different sizes. Congruent triangles are identical in both shape and size. With congruent triangles, the ratio of corresponding sides is always 1:1, making them a special case of similar triangles.

How do I know which sides correspond between two triangles?

Corresponding sides are determined by which angles they are opposite to. If angle A in triangle 1 equals angle D in triangle 2, then the side opposite angle A corresponds to the side opposite angle D. You can also match sides that are between equal angles.

Can I use similar triangles if I only know two sides of each triangle?

Yes, if you know that two angles of one triangle equal two angles of another triangle (AA similarity), the triangles are automatically similar even without knowing all three sides. You can then use the proportion to find the missing sides.

What if I accidentally use non-corresponding sides in my proportion?

Your calculated ratio will be incorrect, and any side lengths you find using that wrong ratio will also be wrong. Always verify that all three ratios are equal; if one differs, you've likely used non-corresponding sides.

Similar Triangles

When two triangles have the same shape but different sizes, their corresponding sides are proportional — use this to find missing side lengths.

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

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What each symbol means

What Similar Triangles takes

Symbol	Meaning
$a_i$	The length of side $a$ in triangle $i$ (where $i$ is 1 or 2), measured in linear units like inches or centimeters; misidentifying non-corresponding sides as side $a$ breaks the proportion.
$b_i$	The length of side $b$ in triangle $i$ (where $i$ is 1 or 2), measured in linear units; this must correspond to side $b$ in the other triangle, not just any side of that triangle.
$c_i$	The length of side $c$ in triangle $i$ (where $i$ is 1 or 2), measured in linear units; confusing it with a side that doesn't correspond to side $c$ in the other triangle makes the ratio incorrect.

When to use it

Use similar triangles when you need to find unknown side lengths in geometric figures that have the same shape but different sizes.

Level

Usually taught in: Geometry · Appears on: SAT, ACT

Worked examples

1. Finding missing sides with a known scale factor

Problem

Triangle ABC has sides 4, 6, and 8 inches. Triangle DEF is similar to ABC, with the side corresponding to the 4-inch side being 12 inches. Find the other sides of DEF.

$\frac{4}{12} = \frac{6}{b_2} = \frac{8}{c_2}$
Set up the similar triangles proportion using corresponding sides.
$\frac{4}{12} = \frac{1}{3}$
Simplify the first ratio to find the scale factor is $1:3$ .
$b_2 = 3 \cdot 6 = 18, \quad c_2 = 3 \cdot 8 = 24$
Multiply each side of triangle ABC by the scale factor 3.

Answer: $b_2 = 18 \text{ inches, } c_2 = 24 \text{ inches}$

Since the triangles are similar, all corresponding sides have the same ratio. Once we find the scale factor from one pair of sides, we can use it to find the remaining sides. The ratio of $1:3$ means each side of ABC is one-third the length of the corresponding side in DEF.

2. Verifying similarity and finding a missing side

Problem

Triangle PQR has sides 5, 12, and 13. Triangle STU has two sides of length 10 and 24. If these triangles are similar, what is the length of the third side of STU?

$\frac{5}{10} = \frac{1}{2}, \quad \frac{12}{24} = \frac{1}{2}$
Check the known corresponding sides: both simplify to $\frac{1}{2}$ .
$\frac{1}{2} = \frac{13}{x}$
If the triangles are similar, the third pair must have the same ratio.
$1 \cdot x = 2 \cdot 13, \quad x = 26$
Cross-multiply and solve for the unknown side.

Answer: $x = 26$

We verified similarity by confirming that two pairs of corresponding sides have equal ratios, then used that ratio to find the third side. This shows PQR is half the size of STU, and all three ratios equal $1:2$ .

3. Similar triangles in game shield scaling

Problem

In a role-playing game, two characters have similar-shaped shield designs. At level 5, Player A's shield has sides 8, 15, and 17 inches. Player B is also at level 5 with a similar shield whose longest side is 25.5 inches. What are the dimensions of Player B's shield?

$\frac{8}{a_2} = \frac{15}{b_2} = \frac{17}{25.5}$
Set up the proportion where 25.5 is Player B's side corresponding to Player A's side of 17.
$\frac{17}{25.5} = \frac{17}{51/2} = \frac{17 \cdot 2}{51} = \frac{34}{51} = \frac{2}{3}$
Simplify the known ratio to find the scale factor is $2:3$ .
$\frac{2}{3} = \frac{8}{a_2} \implies a_2 = \frac{8 \cdot 3}{2} = 12$
Cross-multiply to solve for the first unknown side: $a_2 = 12$ inches.
$\frac{2}{3} = \frac{15}{b_2} \implies b_2 = \frac{15 \cdot 3}{2} = 22.5$
Apply the same ratio to find the second unknown side: $b_2 = 22.5$ inches.

Answer: $\text{Player B's shield has sides } 12, 22.5, \text{ and } 25.5 \text{ inches}$

Similar triangles maintain the same ratio for all pairs of corresponding sides. Once we calculated the scale factor from the given side lengths, we could find the remaining dimensions without additional measurements. This illustrates how similar triangles apply in real-world scaling scenarios, from game design to architecture and engineering.

Common mistakes

Where Similar Triangles usually goes wrong

The mistake	Why it is wrong	The fix
Using sides that don't correspond, like the longest side of triangle 1 with the shortest side of triangle 2, to set up the proportion.	Corresponding sides are determined by matching angles, not by relative size; using non-corresponding sides violates the definition of similar triangles and gives an incorrect ratio.	Identify corresponding sides by matching equal angles in the triangles, then pair the side opposite angle A with the side opposite the equal angle in the other triangle.
Checking only two ratios, like $\frac{a_1}{a_2} = \frac{b_1}{b_2}$ , and assuming the third is equal without verifying.	For triangles to be similar, all three pairs of corresponding sides must have exactly the same ratio; if even one differs, the triangles are not similar.	Always check all three ratios $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ before claiming similarity or using the proportion to find missing sides.
Setting up the proportion backwards by swapping the triangles, writing $\frac{a_2}{a_1}$ instead of $\frac{a_1}{a_2}$ .	Reversing the fraction gives you the reciprocal of the correct ratio, causing you to either solve for the wrong variable or get an answer that is inverted.	Be consistent: decide which triangle is triangle 1 and which is triangle 2, then always use triangle 1 in the numerator and triangle 2 in the denominator.

Tips and when to use something else

Identify corresponding sides by matching equal angles: if angle A of triangle 1 equals angle D of triangle 2, then the sides opposite those angles correspond.
Use any one pair of corresponding sides to find the scale factor, then apply that factor to find all other unknown sides.
If you only know two angles of each triangle are equal (AA similarity criterion), the triangles are automatically similar — you don't need all three sides.
For right triangles, if the acute angles match, the triangles are similar; consider using the Pythagorean Theorem instead if you need to find a side quickly.

Related formulas

Pythagorean TheoremFind the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.Special Right TrianglesSpecial Right Triangles give you the exact side ratios for 30-60-90 and 45-45-90 triangles, so you can find missing sides without the Pythagorean Theorem.RatioA ratio compares two quantities to show their relationship, and can be written as a fraction or with a colon to help you scale or find missing values.ProportionUse Proportion when two ratios are equal to find a missing value by cross-multiplying to solve equations with fractions.Area of a TriangleCalculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.

Frequently asked questions

What's the difference between similar triangles and congruent triangles?: Similar triangles have the same shape and the same angles, but different sizes. Congruent triangles are identical in both shape and size. With congruent triangles, the ratio of corresponding sides is always $1:1$ , making them a special case of similar triangles.
How do I know which sides correspond between two triangles?: Corresponding sides are determined by which angles they are opposite to. If angle A in triangle 1 equals angle D in triangle 2, then the side opposite angle A corresponds to the side opposite angle D. You can also match sides that are between equal angles.
Can I use similar triangles if I only know two sides of each triangle?: Yes, if you know that two angles of one triangle equal two angles of another triangle (AA similarity), the triangles are automatically similar even without knowing all three sides. You can then use the proportion to find the missing sides.
What if I accidentally use non-corresponding sides in my proportion?: Your calculated ratio will be incorrect, and any side lengths you find using that wrong ratio will also be wrong. Always verify that all three ratios are equal; if one differs, you've likely used non-corresponding sides.

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Reviewed 2026-09-18