Y. Davis Geometry Notes Chapter 7

Ratio A comparison of two quantities in the same unit

Rate A comparison of 2 quantities in different units

Extended ratios Used to compare 3 or more quantities.

Proportion A comparison of 2 or more ratios. Extremes a and d Means b and c

Cross Product Property The product of the extremes is always equal to the product of the means. You solve proportions by cross multiplying

Similar Polygons Polygons with the same shape, but not necessarily the same size. ~ Corresponding angles are congruent. Corresponding sides are proportional.

Scale Factor The ratio of the lengths of corresponding sides of similar figures.

Theorem 7.1 Perimeters of Similar Polygons If 2 polygons are similar, then their perimeters are proportional to the scale factor of the 2 polygons.

Postulate 7.1 Angle-Angle (AA~) Similarity Postulate If 2 angles of one triangle are congruent to 2 angles of another triangle, then the 2 triangles are similar.

Theorem 7.2 Side-Side-Side (SSS~) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

Theorem 7.3 Side-Angle-Side (SAS~) Similarity Theorem If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

Theorem 7.4 Properties of Similarity Reflexive– Symmetric– Transitive–

Theorem 7.5 Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other 2 sides, then it divides the sides into segments of proportional lengths.

Theorem 7.6 Converse of Triangle Proportionality Theorem If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.

Midsegment of a Triangle A segment that connects the midpoints of 2 sides of a triangle.

Theorem 7.7 Triangle Midsegment Theorem A midsegment of a triangle is parallel to one side of the triangle, and its length is ½ the length of that side.

Corollary 7.1 Proportional Parts of Parallel Lines If 3 or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

Corollary 7.2 Congruent Parts of Parallel Lines If 3 or more parallel lines cut off segments on one transversal, then they cut off congruent segments on every transversal

Theorem 7.8 If 2 triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides.

Theorem 7.9 If 2 triangles are similar, the lengths of corresponding angle bisectors are proportional to the lengths of corresponding sides.

Theorem 7.10 If 2 triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides.

Theorem 7.11 Triangle Angle Bisector An angle bisector in a triangle separate the opposite side into 2 segments that are proportional to the lengths of the other 2 sides.

Dilation A transformation that enlarges or reduces the original figure proportionally.

Similarity Transformation Transformations that produce similar figures.

Center of Dilation The fixed point that dilations are performed with respect to.

Scale factor of dilation The ratio of a length on the image to a corresponding length on the preimage.

Enlargement A dilation with a scale factor greater than 1. (an image is larger than the preimage.)

Reduction A dilation with a scale factor between 0 & 1. (an image is smaller than the preimage.)

Scale Model (Scale Drawing) An object or drawing with lengths proportional to the object it represents.

Scale Is the ratio of a length on the model or drawing to the actual length of the object being modeled or drawn.