Similar Triangles

1Summary 重點摘要

Learn what similar triangles are, and how to use them to find unknown side lengths.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What similar triangles areSimilar triangles are two triangles of the same shape but possibly different sizes, written with the symbol ~. They have two essential properties: corresponding angles are equal, and corresponding sides are in proportion. Congruence is a special case of similarity — the case where the ratio equals 1 and the figures coincide exactly. It follows that testing every element is unnecessary: equal corresponding angles or a consistent ratio of sides is enough, which leads to the tests in the next steps.
2. Step 2: The AAA (AA) testIf the three angles of one triangle equal the three angles of another, the triangles are similar, called AAA. Since the angles of a triangle always sum to 180°, two pairs of equal angles already force the third, so the condition is usually written AA — two pairs of corresponding angles equal. In practice, the two pairs are often produced by vertically opposite angles, a common angle, or corresponding and alternate angles between parallel lines, making AA the most used similarity test in geometry problems.
3. Step 3: The other testsBesides AA there are two more tests: three pairs of sides in proportion (SSS~), where all three ratios agree; and two pairs of sides in proportion with the included angle equal (SAS~). In SAS~ the angle must be the one between the two sides — an angle opposite the sides does not qualify. When side lengths are given, order the sides of each triangle by size and compare ratios, for example 36 = 48 = 510 = 0.5; equal ratios confirm similarity.
4. Step 4: Using similarity to find unknown sidesThe greatest use of similarity is finding unknown sides: corresponding sides are proportional, so find the scale factor and multiply. For example, if a large triangle has sides 2, 4, 6 and the small triangle's longest side is 3, the ratio is 3÷6 = 0.5, so the other sides are 2×0.5 = 1 and 4×0.5 = 2. The worked example follows the same pattern: sides 3, 4, 5 with shortest side 1.5 give ratio 1.5÷3 = 0.5, hence the longest side is 5×0.5 = 2.5. The routine is fixed: match corresponding sides, find the ratio, apply the ratio.
5. Step 5: Matching sides and the direction of the ratioMatching corresponding sides correctly decides success: a corresponding side lies opposite the corresponding angle — smallest angle faces smallest side, largest faces largest — so never match by position when the figures point differently. The direction of the scale factor matters too: going from the large triangle to the small one uses small ÷ large; reversing it gives the reciprocal, such as 2 instead of 0.5, and ruins the answer. A safeguard is writing a sentence like 'small = large × ratio' and checking that the result is indeed smaller, as the question implies.
6. Step 6: Similarity and congruenceCongruence requires the same shape and the same size, with ratio exactly 1; similarity asks only for the same shape, with any positive ratio. Every pair of congruent triangles is therefore similar, but similar triangles need not be congruent. Choose words carefully: after proving similarity, conclude only that corresponding sides are proportional and corresponding angles equal — never that sides are equal; conversely, once congruence is proven, the properties of similarity follow automatically. Multiple-choice questions love this distinction, so keep the containment clear.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Similar = same shape size can be different
• Corresponding angles equal
• Corresponding sides in proportion
• AAA is similarity condition
• Used to find unknown side lengths