Tests for similar triangles practice questions | Year 10 Mathematics

Similarity tests for triangles

Triangles are similar when a valid test is satisfied and the correspondence is consistent. The tests used here are AA, SSS and SAS

\triangle ABC\sim\triangle DEF

Keep the correspondence Pair each side with the named corresponding side and use one direction for every ratio
Use SSS All three corresponding side ratios must be equal \(\displaystyle \frac{9}{6}=\frac{12}{8}=\frac{15}{10}\)
Use SAS Two corresponding side ratios must agree and the included angles must be equal
Use AA Two pairs of corresponding angles are enough because the third pair then follows \(\displaystyle \angle A=\angle D,\quad\angle B=\angle E\)
Use the direction Image length equals k times original length. Divide by k to work backwards
Know the limits One side ratio or one equal angle pair is insufficient. SSA is not a general test: some data allow two different shapes
Connect similarity to congruence Scale one triangle by the common ratio. Its three sides then match the other triangle, so SSS congruence explains SSS similarity
Prove before calculating Use parallel-line angle facts and the common angle to prove AA similarity. Write vertices in matching order, then equate corresponding ratios

Worked example

Triangles ABC and DEF have corresponding sides AB = 6 cm, AC = 8 cm, BC = 10 cm and DE = 9 cm, DF = 12 cm, EF = 15 cm. Decide whether the triangles are similar and state the scale factor from ABC to DEF

\triangle ABC\sim\triangle DEF\;?

\(\displaystyle \frac{DE}{AB}=\frac{9}{6}=\frac{3}{2}\)Find the first ratio from triangle ABC to triangle DEF
\(\displaystyle \frac{DF}{AC}=\frac{12}{8}=\frac{3}{2}\)The second corresponding ratio has the same value
\(\displaystyle \frac{EF}{BC}=\frac{15}{10}=\frac{3}{2}\)The third corresponding ratio also has the same value
\(\displaystyle k=\frac{3}{2}\)All three corresponding ratios agree, so the triangles are similar by SSS

Common mistakes

Using one ratio as proof

✕\(\displaystyle \frac{DE}{AB}=\frac32\)✓\(\displaystyle \frac{DE}{AB}=\frac{DF}{AC}=\frac{EF}{BC}\)

One ratio alone does not prove SSS. All three corresponding ratios must agree

Using an angle that is not included

✕\(\displaystyle \angle B=\angle E\)✓\(\displaystyle \angle A=\angle D\)

When AB corresponds to DE and AC to DF, the included vertices are A and D. Another equal angle does not complete the SAS test

Inverting the scale factor

✕\(\displaystyle k=\frac{AB}{DE}=\frac{6}{9}=\frac{2}{3}\)✓\(\displaystyle k=\frac{DE}{AB}=\frac{9}{6}=\frac{3}{2}\)

The scale factor from ABC to DEF is the second triangle length divided by the first triangle length

Difficulty levels

introCompare three side ratios and decide SSS similarity

standardUse SAS or AA, then apply a length factor when similarity holds

challengeCompare SSS data or investigate genuinely ambiguous non-included-angle data

examSelect a justified verdict from mixed SSS, SAS, AA and non-included-angle data

Questions about this topic

What does SSS mean for similar triangles?+

SSS means that all three pairs of corresponding sides have one common ratio

Why must the angle be included for SAS?+

The equal angle must be between the two side pairs being compared, otherwise the SAS test has not been met

Why is one side ratio not enough?+

Different triangle pairs can share one side ratio while their other ratios or angle conditions differ

Can a triangle be both similar and congruent?+

Yes. Congruent triangles are similar with scale factor 1. Similar triangles need not be congruent

Why do parallel lines help prove similarity?+

They give corresponding or alternate equal angles. With a second justified equal-angle pair, AA proves similarity

How do I reverse a scale factor?+

Take its reciprocal. The forward and reverse length factors multiply to 1

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 7: Geometry and similarity

In this chapter

Angle facts and parallel lines Triangle angle sum, special triangles and the exterior angle theorem Quadrilateral angle sums and special-quadrilateral properties Polygon angle sums and regular polygons Congruent triangle tests Tests for similar triangles 7F Similar triangle tests: Similar figures and scale factors 7G Applied triangle similarity: Applied similarity and scale Similar solids and scaling volume 7H Similar solids and map scale: Similar figures and scale factors 7H Similar solids and map scale: Applied similarity and scale Interpret practical networks