Questions and Answers
What does the Angle-Angle Similarity Postulate state?
What is the Side-Angle-Side Similarity Postulate about?
Which theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar?
What does the Corollary to Theorem 7-3 state?
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What is the Side-Splitter Theorem?
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What is the definition of the Cross-Product Property?
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What is the Geometric Mean?
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What is a Golden Rectangle?
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What is the Golden Ratio?
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What does a proportion imply?
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Define ratio.
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What is a scale in geometry?
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What are similar polygons?
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What is a similarity ratio?
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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.
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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!