Using Similarity Criteria to Determine Corresponding Side Relationships

Hello! In the previous lesson, you practised solving quadratic equations by factorisation and the quadratic formula. Both methods required a reliable routine: identify the information, choose a valid method, and keep each algebraic step organised. We will use the same approach in geometry.

This lesson introduces similar triangles. You will learn how to prove that two triangles are similar using AA, SSS, or SAS, write the vertices in the correct corresponding order, and use that order to form side ratios and find missing lengths.

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Similar means same shape, possibly a different size

Two triangles are similar when:

• their corresponding angles are congruent;
• their corresponding side lengths are proportional.

“Proportional” means every corresponding side has been enlarged or reduced by the same scale factor.

For example, a triangle with sides $4$, $6$, and $8$ is similar to one with sides $6$, $9$, and $12$. Each side of the first triangle has been multiplied by $\frac{3}{2}$.

Similarity is written using $\sim$. For instance,

$$△ABC\sim △DEF$$

is not just a statement that the triangles are similar. The order gives essential information:

Vertex in first triangle	Corresponding vertex in second triangle
$A$	$D$
$B$	$E$
$C$	$F$

Therefore, the corresponding sides are:

$$AB\leftrightarrow DE$$ $$BC\leftrightarrow EF$$ $$AC\leftrightarrow DF$$

So their ratios must satisfy

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$

A pair of congruent triangles is a special case of similar triangles: its scale factor is $1$, so corresponding sides are equal as well as corresponding angles.

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The three similarity criteria

You do not need to know every angle and every side before deciding that triangles are similar. These three criteria are shortcuts that guarantee similarity.

AA similarity: two corresponding angles

AA stands for Angle-Angle.

If two angles of one triangle are congruent to two corresponding angles of another triangle, the triangles are similar.

Why are two enough? The angles inside every triangle total $180^{\circ }$. Once two angles match, the third angle must also match.

Suppose:

$$\angle A=\angle D=58^{\circ }$$

and

$$\angle B=\angle E=37^{\circ }$$

Then the remaining angles are both:

$$180^{\circ }-(58^{\circ }+37^{\circ })=85^{\circ }$$

Therefore,

$$△ABC\sim △DEF$$

by AA similarity.

In diagram questions, the equal angles may be shown by:

• matching angle marks;
• equal given angle measures;
• vertical angles;
• angles formed by parallel lines.

You only need two corresponding pairs, though you may notice all three.

SSS similarity: three proportional side pairs

SSS stands for Side-Side-Side.

If all three pairs of corresponding sides are proportional, the triangles are similar.

Consider triangles with these side lengths:

$$AB=6,BC=8,AC=10$$ $$DE=9,EF=12,DF=15$$

Compare the corresponding sides:

$$\frac{AB}{DE}=\frac{6}{9}=\frac{2}{3}$$ $$\frac{BC}{EF}=\frac{8}{12}=\frac{2}{3}$$ $$\frac{AC}{DF}=\frac{10}{15}=\frac{2}{3}$$

All three ratios are equal. Hence,

$$△ABC\sim △DEF$$

by SSS similarity.

When the diagram does not already show corresponding angles, matching shortest side to shortest side, longest side to longest side, and middle-length side to middle-length side can help you identify the pairs. However, do not rely on how a triangle appears on the page; use its labelled lengths.

SAS similarity: two proportional sides and the included angle

SAS stands for Side-Angle-Side.

To use SAS similarity, you need:

1. two pairs of corresponding sides that are proportional;
2. the included angle between those sides to be congruent.

The included angle is the angle physically between the two sides being compared.

For example, suppose:

$$AB=5,BC=7$$ $$DE=10,EF=14$$

and

$$\angle B\cong \angle E$$

The angle $B$ lies between $AB$ and $BC$, while angle $E$ lies between $DE$ and $EF$. Now compare the sides:

$$\frac{AB}{DE}=\frac{5}{10}=\frac{1}{2}$$ $$\frac{BC}{EF}=\frac{7}{14}=\frac{1}{2}$$

The ratios match, and the included angles are congruent. Therefore,

$$△ABC\sim △DEF$$

by SAS similarity.

A frequent mistake is to use two proportional sides and an angle that is not between them. That information is called SSA, and it is not a general similarity criterion. For SAS, always trace the two sides to the angle between them.

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Choosing the correct criterion

Before calculating anything, sort the information in the question.

Information supplied	Valid criterion
Two corresponding angle pairs are congruent	AA
Three corresponding side pairs have equal ratios	SSS
Two corresponding side pairs have equal ratios, plus the included angle is congruent	SAS

A useful written routine is:

1. Identify the two triangles.
2. Mark or list the information given.
3. Check whether the information fits AA, SSS, or SAS exactly.
4. Write the similarity statement in corresponding order.
5. Use that statement to match sides before making a proportion.

The fourth step prevents many errors. For example, if you have proved

$$△PQR\sim △XYZ$$

then the correspondence is:

$$P\leftrightarrow X,Q\leftrightarrow Y,R\leftrightarrow Z$$

Thus:

$$PQ\leftrightarrow XY$$ $$QR\leftrightarrow YZ$$ $$PR\leftrightarrow XZ$$

Writing $△PQR\sim △XZY$ would describe a different correspondence, so it would give incorrect side pairs even if the triangles really are similar.

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From a similarity statement to a missing side

Once similarity has been proved, the main payoff is that corresponding sides are proportional.

Suppose:

$$△ABC\sim △DEF$$

and you know:

$$AB=8,DE=12,BC=10$$

Find $EF$.

Start with the similarity statement:

$$△ABC\sim △DEF$$

This tells you that $AB$ corresponds to $DE$, and $BC$ corresponds to $EF$. Use only those corresponding pairs:

$$\frac{AB}{DE}=\frac{BC}{EF}$$

Substitute the known values:

$$\frac{8}{12}=\frac{10}{EF}$$

Cross-multiply:

$$8(EF)=12(10)$$ $$8(EF)=120$$ $$EF=15$$

The scale factor from the first triangle to the second is:

$$\frac{12}{8}=\frac{3}{2}$$

So the check is straightforward:

$$10\left(\frac{3}{2}\right)=15$$

The answer fits the scale factor.

You could also set up the proportion in the reverse direction:

$$\frac{DE}{AB}=\frac{EF}{BC}$$

Both forms are correct. The important rule is to keep the ratio direction consistent: first-triangle side over second-triangle side throughout, or second-triangle side over first-triangle side throughout.

A clean exam-style conclusion

For a proof or a numerical question, organise your work so that the reason is visible:

$$\angle A\cong \angle D$$ $$\angle B\cong \angle E$$ $$\therefore △ABC\sim △DEF\text{ by AA similarity}$$

Then, if a length is required:

$$\frac{AB}{DE}=\frac{BC}{EF}$$

This structure makes it clear that you did not assume the side proportion before proving similarity.

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Common errors to avoid

• Using equal side lengths instead of proportional side lengths. Similar triangles can have different side lengths.
• Comparing non-corresponding sides. First establish the vertex correspondence.
• Changing ratio direction halfway through. If you begin with $\frac{AB}{DE}$, keep first-triangle sides on top.
• Using only two side ratios for SSS. SSS requires all three pairs.
• Forgetting “included” in SAS. The congruent angle must lie between the proportional side pairs.
• Naming triangles in a random order. A similarity statement is a map of corresponding vertices, not just a label.

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Key takeaways

Similarity is about same shape: corresponding angles are congruent and corresponding sides have one constant ratio.

• Use AA when two corresponding angle pairs are congruent.
• Use SSS when all three corresponding side pairs are proportional.
• Use SAS when two corresponding side pairs are proportional and their included angles are congruent.
• After proving similarity, use the vertex order in the similarity statement to write correct side proportions.

In the next lesson, you will move into coordinate geometry and use formulas to calculate the distance between points and the coordinates of a point that divides a line segment internally.