Triangle Similarity: Step-by-Step Guide
Two triangles are similar if they have the same shape but can be different sizes. This means their matching angles are equal and their matching sides are in proportion (they all grow by the same amount). There are three shortcuts: AA (two pairs of equal angles), SAS (two pairs of sides in proportion with an equal angle between them), and SSS (all three pairs of sides in proportion).
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- List what you know. Write down any equal angles and any side ratios. Look for shared angles and parallel lines, which make equal angles.\[\text{Given: }\angle A \cong \angle D,\;\angle B \cong \angle E,\;\dots\]
- Pick a similarity shortcut. AA: two pairs of equal angles. SAS: one pair of equal angles between two pairs of sides in proportion. SSS: all three pairs of sides in proportion.\[\text{AA},\;\text{SAS},\;\text{SSS}\]
- Show that the shortcut works. For AA, find two pairs of equal angles. For SAS, find the side ratios and check the angle between them. For SSS, find all three side ratios.\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}\]
- Write the similarity statement. Write 'Triangle ABC is similar to triangle DEF' with the corners in matching order.\[\triangle ABC \sim \triangle DEF\]
- Use the proportions if needed. Once the triangles are similar, all the matching sides have the same ratio (the scale factor), and all the matching angles are equal.\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k\]
Worked example
In triangles ABC and DEF, angle A = angle D = 50 degrees and angle B = angle E = 70 degrees. Prove the triangles are similar.
Step 1: The givens are angle A = angle D = 50 degrees, and angle B = angle E = 70 degrees.
Step 2: Two pairs of matching angles are equal. That is the AA shortcut.
Step 3: Two-column proof: 1) angle A = angle D, Given. 2) angle B = angle E, Given. 3) Triangle ABC is similar to triangle DEF, by AA.
Step 4: Conclusion: Triangle ABC is similar to triangle DEF by AA.
Step 5: The third angles are also equal: angle C = 180 - 50 - 70 = 60 degrees = angle F. The sides are in proportion: AB/DE = BC/EF = CA/FD.
Check: Because angles in a triangle add up to 180, two equal pairs force the third pair to be equal. So the triangles have exactly the same angles and must be similar.
Tips and common mistakes
- AA only needs two pairs of equal angles. The third pair is automatic, since the angles in a triangle add up to 180.
- The order of the corners matters. Triangle ABC is similar to triangle DEF means A goes with D, B with E, and C with F.
- Similar triangles do not have to be the same size. They need equal angles and proportional sides.
- Congruent triangles are also similar, with a scale factor of 1.
- Parallel lines cut by a transversal make equal angles. This is a good way to get AA.
- When the sides are in proportion, the scale factor is the same for all three pairs.
- In SAS similarity, the equal angle must be between the two proportional sides.
- Once triangles are similar, you can use the scale factor to find missing sides.