Geometry Chapter 7 Theorems and Postulates

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Questions and Answers

What does the Angle-Angle Similarity Postulate state?

If the corresponding sides of two triangles are proportional, then the triangles are similar.

If an angle of one triangle is congruent to an angle of a second triangle, then the triangles are similar.

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

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. (correct)

What is the Side-Angle-Side Similarity Postulate about?

If two angles of one triangle are congruent to two angles of another triangle.

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. (correct)

If the altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar.

If a line is parallel to one side of a triangle.

Which theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar?

Theorem 7-2

Theorem 7-1

Theorem 7-4

Theorem 7-3 (correct)

What does the Corollary to Theorem 7-3 state?

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. Signup and view all the answers

What is the 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. Signup and view all the answers

What is the definition of the Cross-Product Property?

The product of the extremes is equal to the product of the means. Signup and view all the answers

What is the Geometric Mean?

The number x such that a/x=x/b where a,b, and x are positive numbers. Signup and view all the answers

What is a Golden Rectangle?

A rectangle that can be divided into a square and a rectangle similar to the original rectangle. Signup and view all the answers

What is the Golden Ratio?

The ratio of the length of a golden rectangle to its width, approximately 1.62. Signup and view all the answers

What does a proportion imply?

A statement that two ratios are equal. Signup and view all the answers

Define ratio.

A comparison of two quantities by division. Signup and view all the answers

What is a scale in geometry?

The ratio of any length in a scale drawing to the corresponding actual length. Signup and view all the answers

What are similar polygons?

Polygons having corresponding angles congruent and corresponding sides proportional. Signup and view all the answers

What is a similarity ratio?

The ratio of the lengths of corresponding sides. Signup and view all the answers

Study Notes

Angle Similarity Postulates

Postulate 7-1: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Side Similarity Theorems

Theorem 7-2: An angle of one triangle is congruent to an angle of another triangle. If the lengths of the sides including these angles are proportional, then the triangles are similar.
Theorem 7-3: If the corresponding sides of two triangles are proportional, then the triangles are similar.

Right Triangle Properties

If an altitude is drawn to the hypotenuse of a right triangle, it divides the triangle into two smaller 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 is the geometric mean of the lengths of the segments of the hypotenuse.

Proportions and Parallel Lines

Theorem 7-4: A line parallel to one side of a triangle that intersects the other two sides divides those sides proportionally.
Corollary to Theorem 7-4: If three parallel lines intersect two transversals, the segments intercepted on the transversals are proportional.

Triangle Bisector Theorem

Theorem 7-5: If a ray bisects an angle of a triangle, it divides the opposite side into segments proportional to the other two sides.

Proportional Relationships

Cross-Product Property: The product of the extremes equals the product of the means in a proportion.
Properties of Proportions: Various equivalent relationships in proportions including:
- 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

Key Definitions

Geometric Mean: The number x such that ( a/x = x/b ), where a, b, and x are positive numbers.
Golden Rectangle: A rectangle that can be subdivided into a square and a smaller rectangle similar to the original.
Golden Ratio: The ratio of the length to width of a golden rectangle, approximately 1.62.
Proportion: A statement that two ratios are equal.
Ratio: A comparison of two quantities by division.
Scale: The ratio of any length in a scale drawing to the corresponding actual length.
Similar Polygons: Polygons with congruent corresponding angles and proportional corresponding sides.
Similarity Ratio: The ratio of the lengths of corresponding sides in similar polygons.

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Description

Test your understanding of the theorems and postulates in Geometry Chapter 7. This quiz covers crucial concepts such as the Angle-Angle Similarity and Side-Angle-Side Similarity Postulates. Perfect for students looking to reinforce their knowledge of triangle similarity!