2.7 – Similarity Theorems and Proportional Reasoning

Objectives

• Translate, rotate, reflect, and dilate triangle pairs to determine if they are similar.
• Discover the similarity shortcuts: AA postulate, SSS theorem, and SAS theorem.
• Apply the similarity shortcuts (AA postulate, SSS theorem, and SAS theorem) to find the missing side lengths and angle measures of triangles.
• Prove two triangles are similar by using the similarity shortcuts.

Key Terms

• AA (Angle-Angle) Similarity Postulate – A postulate stating that if two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar.
This is because of the Third Angle Theorem.
• SAS (Side-Angle-Side) Similarity Theorem – A theorem stating that if an angle of one triangle is congruent to an angle of another triangle; and, if the lengths of the sides including these angles are proportional, then the triangles are similar.
• SSS (Side-Side-Side) Similarity Theorem – A theorem stating that if the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.

Notes

Similar Triangles
• Review

Proving Similarity
• How to Prove Two Triangles Are Similar
1. Dilate one triangle to match the size of the second.
2. Translate and rotate the triangles until they overlap.
3. If the overlapped triangles match exactly, they are congruent. That means the original two triangles are similar

• If you cannot do that, you can use cross-products
1. Set up the corresponding sides as fractions that are equal to each other.
2. Cross-multiply
3. Solve for the missing side length

• Example

• Step 1: $\overline{AB}$ corresponds to $\overline {DE}$ and $\overline{AC}$ corresponds to $\overline {DF}$
• So, $\frac{3}{9}=\frac{5}{x}$
• Step 2: $3x=9\bullet5$
• Step 3: $\frac{3x}{3}=\frac{45}{3}$
• Simplify: $x = 15$
Similarity Shortcuts
• AA Similarity will never prove congruence, but it will be used to help prove similarity because…
• Angle Sum Theorem: The sum of the measures of any triangle’s angles is 180°.
• Third Angle Theorem: If two angles of a triangle are congruent to two angles of another triangle, then the third angles of the triangles are also congruent.
• The ASA postulate and AAS theorem can be reduced to the AA similarity postulate, as they each include two angles.If two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar.
• SSS for similar triangles is not the same as the SSS we used for congruent triangles.
• To show triangles are congruent, you have to show that the three pairs of corresponding sides are congruent.
• To show triangles are similar, you only have to show that the three pairs of corresponding sides are proportional.
• SAS for similar triangles is not the same as the SAS we used for congruent triangles.
• To show triangles are congruent, you have to show that two pairs of corresponding sides and their included angles are congruent.
• To show triangles are similar, you have to show that the included angle is congruent and that the two sides are proportional.

Proof