Geometry Chapter 6 Similar Triangles

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Geometry (McDougall Littell) Chapter 6 (similar triangle) review

Proportion

An equation that states that two ratios are equal

Means

In a proportion, the denominator of the first equation and the numerator of the second equation

Extremes

In a proportion, the numerator of the first equation and the denominator of the second equation

Geometric Mean

A positive number that satisfies the proportion a/x= x/b; x = sqrt(ab)

Scale Factor

The common ratio of the lengths of two corresponding sides of similar polygons

Perimeters of Similar Polygons

If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding lengths (same scale factor)

AA

Angle-Angle similarity postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar

SSS

Side-Side-Side similarity theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar

SAS

Side-Angle-Side similarity theorem: 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

Triangle Proportionality Theorem and Converse

A line is parallel to one side of a triangle IFF it intersects the other two sides proportionally

Transversal Similarity Theorem

If three parallel lines intersect two transversals, then they divide the transversals proportionally

Angle Bisector Similarity Theorem

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

Ratio

The comparison of two numbers using division

Cross Product Property

In a proportion, the product of the extremes equals the product of the means: if a/b = c/d (and b and d both not 0) then ad = bc

Mean Proportional

Another name for Geometric Mean (x = sqrt(ab))

Reciprocal Property

Given a proportion, the reciprocals of the ratios are equal: if a/b = c/d then b/a = d/c

Interchange Means

Given a proportion, if you interchange the means (or the extremes!), then you form another true proportion: if a/b = c/d then a/c = b/d

Add Denominator

Given a proportion, if you add the value of the denominator to the numerator, then you form a true proportion: if a/b = c/d then (a+b)/b = (c + d)/d

Similar Polygons

Two polygons in which the corresponding angles are congruent and corresponding side lengths are proportional; similar polygons have the same shape, but not necessarily the same size; angles are congruent and sides are in proportion (have a common scale factor)

Extended ratios

Ratios that compare more than 2 values; example: ratios of the angles of a triangle are 1:2:3 - find the measures of the angles (answer: 30, 60, 90 degrees)

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