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AA, SAS & SSS Similarity Theorems
Triangle Similarity Theorems
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Similarity theorems let you prove that two triangles have the same shape without proving every angle and side separately. In similar triangles, corresponding angles are equal and corresponding side lengths are proportional. These ideas are used in maps, scale drawings, shadows, indirect measurement, and coordinate geometry. AA, SAS, and SSS similarity give efficient shortcuts for recognizing triangles that are scaled copies of each other.
AA similarity uses two pairs of equal angles, because the third pair must also be equal. SAS similarity uses two proportional pairs of corresponding sides and the included angle between them. SSS similarity uses all three pairs of corresponding sides in the same ratio. Once triangles are proven similar, you can write proportions to find missing side lengths, scale factors, and real-world distances.
Key Facts
- AA Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
- SAS Similarity: If two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.
- SSS Similarity: If all three pairs of corresponding sides are proportional, then the triangles are similar.
- Similarity statement order matters: If triangle ABC is similar to triangle DEF, then A corresponds to D, B corresponds to E, and C corresponds to F.
- Scale factor = corresponding side in image divided by corresponding side in original, such as k = DE/AB.
- For similar triangles, AB/DE = BC/EF = AC/DF and angle A = angle D, angle B = angle E, angle C = angle F.
Vocabulary
- Similar triangles
- Triangles that have the same shape, with equal corresponding angles and proportional corresponding side lengths.
- Corresponding parts
- Angles or sides in two figures that match because they are in the same relative position.
- Scale factor
- The number that multiplies each side length of one figure to produce the matching side length of a similar figure.
- Included angle
- The angle formed by two given sides of a triangle.
- Proportion
- An equation showing that two ratios are equal.
Common Mistakes to Avoid
- Matching sides in the wrong order: This gives incorrect ratios because corresponding sides must be paired by their positions in the similarity statement.
- Using SAS without the included angle: SAS similarity requires the equal angle to be between the two proportional side pairs, not just any angle.
- Assuming equal-looking diagrams are exact: A drawing may not be to scale, so use labels, angle marks, and given measurements instead of visual guessing.
- Mixing up scale factor direction: DE/AB and AB/DE are reciprocals, so choose the direction that matches whether you are scaling up or scaling down.
Practice Questions
- 1 Triangles ABC and DEF have angle A = angle D and angle B = angle E. If AB = 6, BC = 9, and DE = 10, find EF.
- 2 Triangles PQR and XYZ have sides PQ = 4, QR = 7, PR = 8 and XY = 12, YZ = 21, XZ = 24. Prove the triangles are similar and state the scale factor from triangle PQR to triangle XYZ.
- 3 Two triangles have side ratios 5/10 and 7/14 for two pairs of corresponding sides, and both include a 40 degree angle between those sides. Explain which similarity theorem applies and why.