Questions and Answers
What is a ratio?
What is an extended ratio?
What does 3:4:5 represent?
It can be written as 3x:4x:5x and solved as an algebraic equation.
A ______ is two fractions set equal to each other.
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What should you compare when looking at two triangles?
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What are the means in a proportion?
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What are the extremes in a proportion?
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Cross products in a proportion are equal.
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What does the cross product property state?
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What do similar polygons have in common?
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In similar polygons, corresponding angles are congruent.
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What is the scale factor in similar polygons?
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The ratio of the perimeters of similar polygons equals the ratio of the sides.
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Angle-angle similarity means triangles are similar when two angles are congruent.
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What similarity relationships work for triangles?
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What is a midsegment of a triangle?
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The triangle midsegment theorem states that a midsegment is parallel to one side of the triangle.
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What does the triangle proportionality theorem describe?
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What is the property of proportional parts of parallel lines?
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What do congruent parts of parallel lines imply?
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Study Notes
Ratios and Proportions
- Ratio: A comparison of two quantities through division.
- Extended Ratio: Compares three or more quantities.
- 3:4:5: Can be represented as 3x:4x:5x; used in triangular problems, set equal to 180 or another specified number.
- Proportion: Two fractions that are set equal to each other.
- Cross Products: The products of the extremes and means in a proportion must be equal.
- Cross Product Property: In a proportion, the product of the extremes equals the product of the means.
Triangle Similarity
- Similar Polygons: Same shape with proportional side lengths; sizes may differ.
- Corresponding Angles: In similar polygons, these angles are congruent.
- Scale Factor: The ratio of lengths of corresponding sides in similar polygons.
- Perimeters: The ratio of perimeters equals the ratio of the corresponding sides.
Triangle Properties
- Angle Angle Similarity: Two triangles are similar if two angles are congruent.
- Side-Side-Side (SSS) Similarity: Triangles are similar if the corresponding side lengths are proportional.
- Side-Angle-Side (SAS) Similarity: If one angle is congruent and the sides enclosing the angle are proportional, the triangles are similar.
- Triangle Proportionality Theorem: Discussed in Chapter 7, Section 4.
- Converse of Triangle Proportionality Theorem: Also detailed in Section 7-4.
Triangle Segments
- Midsegment of a Triangle: A segment connecting the midpoints of two sides of a triangle.
- Triangle Midsegment Theorem: A midsegment is parallel to one side and half the length of that side.
Parallel Lines and Proportionality
- Proportional Parts of Parallel Lines: If three or more parallel lines intersect two transversals, they cut off segments proportional to one another.
- Congruent Parts of Parallel Lines: If three or more parallel lines cut off congruent segments on one transversal, they will also cut off congruent segments on all transversals.
Special Segments in Similar Triangles
- Altitude Ratios: The ratio of altitudes in similar triangles is the same as the ratio of their sides.
- Angle Bisectors and Medians: Ratios of these segments also reflect the ratio of the sides in similar triangles.
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Description
Explore essential concepts from Geometry Chapter 7 with these flashcards. Each card provides definitions for key terms like ratio, extended ratio, and proportions, aiding in your understanding of geometric relationships. Perfect for quick review and study.
