# 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

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)

Example: