Triangles · Lesson 3 of 4
Criteria for Similarity of Triangles
“Basically, similar triangles are just the geometric version of a high-resolution photo and its thumbnail: exact same look, completely different file size.”
• Understand the three criteria used to prove triangle similarity. • Learn the AA (or AAA), SAS and SSS similarity criteria. • Identify which criterion fits a given question. • Solve simple similarity problems step by step. • Avoid common mistakes while matching corresponding sides and angles.
In the previous lesson, we learned that similar triangles have the same shape, even if they are different in size. Their corresponding angles are equal, and their corresponding sides are in the same ratio.
However, in most problems, we are not directly told that two triangles are similar. We have to prove it using the information given in the diagram or question. Checking all three pairs of angles and all three pairs of sides every time would take too long.
Fortunately, this is not necessary. Certain combinations of equal angles and proportional sides are enough to confirm that two triangles have the same shape. These conditions are called the criteria for similarity of triangles. In this lesson, we will learn how to recognise and use these criteria to prove that two triangles are similar.
To prove two triangles are similar, we use one of three criteria: AA, SAS or SSS.
Why Do We Need Criteria?
A triangle has only three sides and three angles. Because of this tight structure, if some parts match in the right way, the entire shape becomes fixed. That is why triangles are special: fewer conditions are enough to prove similarity.
For example, if two angles of one triangle are equal to two angles of another triangle, then the third angles will also be equal automatically. So we do not need to check all three.
1. AA (or AAA) Similarity 2. SAS Similarity 3. SSS Similarity
1. AA Similarity Criterion
AA stands for Angle-Angle. If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar.
If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
Why does this work? Because the sum of angles in a triangle is 180°. If two pairs of angles are equal, then the third pair must also be equal. Once the three angles match, the triangles have the same shape.
Problem
In triangle ABC, ∠A = 50° and ∠B = 80°. In triangle PQR, ∠P = 50° and ∠Q = 80°. Are the triangles similar?
- 1.Compare the given angles.
- 2.∠A = ∠P = 50°.
- 3.∠B = ∠Q = 80°.
- 4.Two pairs of corresponding angles are equal.
- 5.Therefore, by the AA criterion, triangle ABC is similar to triangle PQR.
Many books say AA, while some say AAA. For triangles, both ideas mean the same thing, because once two angles are equal, the third one is forced to be equal.
2. SAS Similarity Criterion
SAS stands for Side-Angle-Side. Here we compare two pairs of corresponding sides and the included angle between them.
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.
The equal angle must be the included angle between the two proportional sides. If the angle is somewhere else, you cannot use SAS directly.
Problem
In triangle ABC, AB = 6 cm, AC = 8 cm and ∠A = 40°. In triangle PQR, PQ = 9 cm, PR = 12 cm and ∠P = 40°. Show that the triangles are similar.
- 1.Compare the included angles first.
- 2.∠A = ∠P = 40°.
- 3.Now compare the sides around these angles.
- 4.AB/PQ = 6/9 = 2/3.
- 5.AC/PR = 8/12 = 2/3.
- 6.The two side ratios are equal, and the included angles are equal.
- 7.Therefore, by the SAS criterion, triangle ABC is similar to triangle PQR.
Students often compare the correct side ratios but forget to check whether the equal angle lies between those two sides. Without the included angle, SAS cannot be used.
3. SSS Similarity Criterion
SSS stands for Side-Side-Side. If all three pairs of corresponding sides are proportional, then the triangles are similar.
If the corresponding sides of two triangles are proportional, then the triangles are similar.
This criterion is especially useful when angles are not given at all, but all the side lengths are known. If the three side ratios are equal, the triangles must have the same shape.
Problem
Triangle ABC has side lengths 4 cm, 6 cm and 8 cm. Triangle DEF has side lengths 6 cm, 9 cm and 12 cm. Are the triangles similar?
- 1.Write the three ratios of corresponding sides.
- 2.4/6 = 2/3.
- 3.6/9 = 2/3.
- 4.8/12 = 2/3.
- 5.All three ratios are equal.
- 6.Therefore, by the SSS criterion, triangle ABC is similar to triangle DEF.
How to Decide Which Criterion to Use
When you face a question, do not try to remember all three criteria at once. First see what information is given. If angles are given, think about AA. If two sides and the angle between them are given, think about SAS. If all three sides are given, think about SSS.
| Given Information | Use This Criterion |
|---|---|
| Two corresponding angles are equal | AA |
| Two side pairs are proportional and the included angle is equal | SAS |
| All three corresponding side pairs are proportional | SSS |
Common Mistakes to Avoid
• Do not match sides in the wrong order. • In SAS, make sure the equal angle is the included angle. • Do not use just two proportional sides without an angle and call it SAS. • In SSS, all three side ratios must match.
To prove triangle similarity, use AA, SAS or SSS. The trick is not just knowing the names, but recognising which information the question gives you.
Quiz
If two angles of one triangle are equal to two angles of another triangle, which criterion should you use?
For SAS similarity, which angle must be equal?
If the three side ratios of two triangles are all equal, which criterion proves similarity?
In triangle ABC and triangle PQR, AB/PQ = AC/PR and ∠A = ∠P. Which criterion applies?
Practice Problems
- In △ABC and △DEF, ∠A = 45°, ∠B = 65°, ∠D = 45° and ∠E = 65°. Show that the triangles are similar and write the correct similarity statement.
- The sides of △ABC are 6 cm, 8 cm and 10 cm. The corresponding sides of △PQR are 9 cm, 12 cm and 15 cm. Determine whether the triangles are similar and name the criterion used.
- In △ABC and △DEF, AB = 8 cm, AC = 12 cm, DE = 10 cm, DF = 15 cm and ∠A = ∠D. Determine whether the triangles are similar and give a reason.
- Given that △ABC ∼ △PQR, AB = 6 cm, BC = 8 cm, AC = 10 cm and PQ = 9 cm, find the lengths of QR and PR.
- The sides of two triangles are 6 cm, 8 cm and 10 cm, and 9 cm, 12 cm and 16 cm. Determine whether the triangles are similar. Support your answer by comparing the corresponding sides.
Key Takeaways
• Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. • In the AA criterion, two pairs of corresponding angles must be equal. • In the SSS criterion, all three pairs of corresponding sides must be proportional. • In the SAS criterion, two pairs of corresponding sides must be proportional and the included angles must be equal. • The order of vertices in a similarity statement shows which vertices correspond to each other. • Similar triangles have the same shape, but they may have different sizes.