# Similarity

## Examine your intuitive notions of what makes a "good copy" and then progress toward a more formal definition of similarity. Explore similar triangles and look into some applications of similar triangles, including trigonometry.

### In This Session

Part A: Scale Drawings
Part B: Similar Triangles
Part C: Trigonometry
Homework

Similarity is one of the “big ideas” in geometry. Note that two things may be similar in colloquial English, but it is a much stronger statement to say that they are mathematically similar.

In this session, you will build on your intuitive notions of what makes a “good copy” to build a more formal definition of similarity. You will then look at applications of similar triangles, including triangle trigonometry.

For information on required and/or optional materials for this session, see Note 1.

### Learning Objectives

In this session you will learn to do the following:

• Explore geometric similarity as “reasoning about proportions”
• Study similar triangles
• Explore some basic ideas in trigonometry

### Previously Introduced

Side-Side-Side (SSS) Congruence: The side-side-side (SSS) congruence states that if the three sides of one triangle have the same lengths as the three sides of another triangle, then the two triangles are congruent.

Triangle Inequality: The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.

### New in This Session

Angle-Angle-Angle (AAA) Similarity: The angle-angle-angle (AAA) similarity test says that if two triangles have corresponding angles that are congruent, then the triangles are similar. Because the sum of the angles in a triangle must be 180°, we really only need to know that two pairs of corresponding angles are congruent to know the triangles are similar.

Cosine: If angle A is an acute angle in a right triangle, the cosine of A is the length of the side adjacent to angle A, divided by the length of the hypotenuse of the triangle. We often abbreviate this as cos A = (adjacent)/(hypotenuse).

Side-Angle-Side (SAS) Similarity: The side-angle-side (SAS) similarity test says that if two triangles have two pairs of sides that are proportional and the included angles are congruent, then the triangles are similar.

Side-Side-Side (SSS) Similarity: The side-side-side (SSS) similarity test says that if two triangles have all three pairs of sides in proportion, the triangles must be similar.

Similar: Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.

Sine: If angle A is an acute angle in a right triangle, the sine of A is the length of the side opposite to angle A divided by the length of the hypotenuse of the triangle. We often abbreviate this as sin A = (opposite)/(hypotenuse).

Tangent: If angle A is an acute angle in a right triangle, the tangent of A is the length of the side opposite to angle A divided by the length of the side adjacent to angle A. We often abbreviate this as tan A = (opposite)/(adjacent).

### Note 1

Materials Needed:

• graph paper
• blank white paper
• yardstick