Geometry Unit 11: Similar Polygons

Trigonometric Ratios

• Cosine Ratio: In a right triangle, defined as the length of the side adjacent to an acute angle divided by the hypotenuse.
• Sine Ratio: In a right triangle, the ratio of the length of the side opposite an acute angle to the hypotenuse.
• Tangent Ratio: The ratio of the length of the side opposite an acute angle to the length of the side adjacent to that angle.

Means and Proportions

• Geometric Mean: The positive real number x such that the ratio a/x equals x/b for positive real numbers a and b.
• Proportion: An equation that asserts two ratios are equal.
• Ratio: Comparison of two numbers via division, where the quotient is termed the ratio.

Similar Polygons and Triangles

• Similar Polygons: Polygons with corresponding angles equal and corresponding sides in proportion.
• Postulate 15: Triangles are similar if all three angles are equal (AAA).
• Theorem 5-1: If two angles of one triangle equal two angles of another triangle, they are similar.
• Theorem 5-2: If two sides of one triangle are proportional to two sides of another and the included angles are equal, the triangles are similar.
• Theorem 5-3: Similarity is established if three sides of one triangle are proportional to three sides of another triangle.

Properties of Similarity

• Theorem 5-4: Similarity of polygons is reflexive, symmetric, and transitive.
• Theorem 5-5: For similar polygons, the ratio of their perimeters matches the ratio of corresponding sides.

Triangle Properties and Theorems

• Theorem 5-6: A line parallel to one side of a triangle divides the other two sides proportionally.
• Theorem 5-7: An angle bisector in a triangle divides the opposite side into segments proportional to the other two sides.
• Theorem 5-8: The lengths of corresponding altitudes in similar triangles share the same ratio as corresponding sides.
• Theorem 5-9: In a right triangle, drawing the altitude to the hypotenuse forms two triangles that are similar to each other and to the original triangle.
• Theorem 5-10: The Pythagorean Theorem states that the sum of the squares of the legs equals the square of the hypotenuse in a right triangle.
• Theorem 5-11: A triangle is a right triangle if the sum of the squares of two sides equals the square of the third side.

Special Right Triangles

• Theorem 5-12: In a 30°-60°-90° triangle, the hypotenuse is double the length of the short leg; the longer leg is √3 times the short leg.
• Theorem 5-13: In a 45°-45° right triangle, the hypotenuse measures √2 times the length of each leg.