Podcast
Questions and Answers
What is a ratio?
The ratio of one number to another is the quotient when the first number is divided by the second.
What is a proportion?
Stating two ratios are equal.
What is the Cross Product in the context of proportions?
Product of your extremes = product of your means.
What does the Exchange Property state?
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What is the Reciprocal Property?
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Explain the Add One Property.
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When are two polygons considered similar?
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What is a Scale Factor?
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All regular polygons are similar.
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Two equilateral triangles are sometimes similar.
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A right triangle and an obtuse triangle can be similar.
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Two equilateral pentagons are similar.
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What is the AA Similarity Postulate?
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Define the SAS Similarity Theorem.
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What is the SSS Similarity Theorem?
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What does the Triangle Proportionality Theorem state?
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Explain the Transversal Theorem.
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What is the Ray Theorem?
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Study Notes
Key Geometry Concepts
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Ratio: Quotient when one number is divided by another; used to compare sizes of two quantities.
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Proportion: An expression indicating that two ratios are equal, crucial for solving many geometric problems.
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Cross Product: Method for proving proportions; states that the product of the extremes equals the product of the means.
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Exchange Property: A property indicating that swapping the means in a proportion does not change its validity.
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Reciprocal Property: Demonstrates the relationship between two ratios; if a/b equals c/d, then b/a equals d/c.
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Add One Property: Allows addition of the same number to both denominators of two ratios while maintaining the equality of the proportions.
Polygons and Similarity
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Similar Polygons: Polygons whose corresponding angles are congruent and corresponding sides are in proportion.
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Scale Factor: The ratio of the lengths of two corresponding sides in similar figures, determining their relative size.
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Regular Polygons: Defined as both equiangular and equilateral; all regular polygons are similar, regardless of size.
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Equilateral Triangles: Always similar due to having identical angles.
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Isosceles Triangles: Sometimes similar, depending on the angles they contain.
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Right Triangle vs. Obtuse Triangle: These two triangle types are never similar due to differing angle measures.
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Equilateral Pentagons: Two equilateral pentagons are not considered similar despite equal angles.
Theorems and Postulates
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AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, the triangles are confirmed to be similar.
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SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle, and the sides including these angles are proportional, then the triangles are similar.
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SSS Similarity Theorem: States that if the sides of two triangles are proportional, then the triangles are similar.
Proportionality Theorems
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Triangle Proportionality Theorem: If a line parallel to one triangle’s side intersects the other two sides, it divides those sides proportionally.
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Transversal Theorem: States that if three parallel lines intersect two transversal lines, they divide those lines proportionally.
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Ray Theorem: A ray that bisects an angle within a triangle creates segments on the opposite side that are proportional to the other two sides’ lengths.
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Description
Enhance your understanding of geometry concepts with these Chapter 7 flashcards. Each card provides key definitions including ratio, proportion, and more. Perfect for mastering the intricate relationships found in geometric studies.