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Geometry Unit 8 Notes
Similarity
𝑎
Ratio: a comparison of two quantities in the form 𝑏 , 𝑏 ≠ 0
Proportion: An equation which states two ratios are equal.
Properties of proportions:
𝑎
𝑐
=𝑑
1) 𝑎𝑑 = 𝑏𝑐 (𝐶𝑟𝑜𝑠𝑠 𝑃𝑟𝑜𝑑𝑢𝑐𝑡)
𝑏
𝑎
𝑏
3) 𝑐 = 𝑑
(𝑏𝑐 𝑠𝑤𝑖𝑡𝑐ℎ)
𝑏
2) 𝑎 =
4)
𝑎+𝑏
𝑏
𝑑
𝑐
=
(𝑅𝑒𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦)
𝑐+𝑑
𝑑
(𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑠𝑢𝑚 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦)
Scale: The comparison of each drawn length to a given length
Scale Drawing: A drawing where each length is represented by a corresponding scale.
Similar Polygons
Similar: two polygons are similar if:
(1) their corresponding sides are proportional and (2) their corresponding angles are congruent.
Similarity symbol: ~
Approximate symbol: 
Similarity Ratio: the ratio of similarity between two similar figures
Golden Rectangle: A rectangle that can be divided into a square and a rectangle, the resulting rectangle is
similar to the original rectangle.
Golden Ratio: In a golden rectangle the ratio of the length of a rectangle to its width is  1.618
How to prove triangles are similar
AA similarity postulate: If two angles of one triangle are congruent to two angles of another triangle, then the
triangles are similar.
SSS similarity postulate: If the three sides of one triangle are proportional to the three sides of another
triangle, then the triangles are similar.
SAS Similarity Postulate: If the included angle of one triangle is congruent to the included angle of another
triangle and the including sides are proportional, then the triangles are similar.
Indirect Measurement: A way to measure the size of an object without measuring the object itself. (Mirrors)
Right Triangle Similarity
Altitude of the right angle: The altitude of a right triangle divides the right triangle into two triangles that are
both similar to each other and the original triangle.
Altitude of the right angle (2): The altitude of a right triangle is the geometric mean of the lengths of the two
segments that make up the hypotenuse.
Altitude of the right angle (3): The altitude to the hypotenuse of a right triangle separates the hypotenuse so
that the length of each leg of the triangle is the geometric mean of the length of
the adjacent hypotenuse segment and the length of the hypotenuse.
Proportional Properties of Triangles
Side-Splitter Theorem : If a line is parallel to one side of a triangle, then it divides the triangle proportionally.
Side-Splitter Corollary: If three or more parallel lines intersect two transversals, then the segments created on
the transversals are proportional.
Triangle angle-bisector theorem: if a ray bisects the vertex of a triangle, then the opposite sides are
proportional to the adjacent sides.