Geometry Chapter 7 Theorems and Postulates

Ratio

a comparison of two quantities by division

Golden Rectangle

a rectangle that can be divided into a square and a rectangle that is similar the the original rectangle

Proportion

a statement that two ratios are equal

Properties of Proportions

a/b=c/d is equivalent to a/c=b/d a/b=c/d is equivalent to d/b=c/a a/b=c/d is equivalent to b/a=d/c

Theorem 7-4: Side-Splitter Theorem

if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally

Theorem 7-5: Triangle-Bisector Theorem

if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides

Theorem 7-2:Side-Angle-Side Similarity Postulate

if an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar

Theorem 7-3: Side-Side-Side Similarity Postulate

if the corresponding sides of two triangles are proportional, then the triangles are similar

Corollary to Theorem 7-4

if three parallel lines intersect two transversal, then the segments intercepted on the transversal are proportional

Postulate 7-1:Angle-Angle Similarity Postulate

if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar

Similar Polygons

polygons having corresponding angles congruent and corresponding sides proportional

Theorem 7-3

the altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other

Corollary to Theorem 7-3

the length of the altitude to the hypotenuse of a right triangle is the geometric means of the lengths of the segments of the hypotenuse

Geometric Mean

the number x such that a/x=x/b where a,b, and x are positive numbers

Cross-Product Property

the product of the extremes is equal to the product of the means

Scale

the ratio of any length in a scale drawing to the corresponding actual length

Golden Ratio

the ratio of the length of a golden rectangle to its width. about 1.62

Similarity Ratio

the ratio of the lengths of corresponding sides