Podcast
Questions and Answers
What is a ratio?
What is a ratio?
What are equivalent ratios?
What are equivalent ratios?
Two ratios that have the same simplified form.
Define proportion.
Define proportion.
Two equal ratios.
What are extremes in a proportion?
What are extremes in a proportion?
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What are means in a proportion?
What are means in a proportion?
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What does the cross product property of proportions state?
What does the cross product property of proportions state?
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If a/b=c/d, b≠0 and d≠0, then ______________.
If a/b=c/d, b≠0 and d≠0, then ______________.
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What is the geometric mean?
What is the geometric mean?
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What is expressed by x^2?
What is expressed by x^2?
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What is the definition of similar polygons?
What is the definition of similar polygons?
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The symbol ~ means __________.
The symbol ~ means __________.
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If two polygons are similar, the ratio of the lengths of two corresponding sides is called ____________.
If two polygons are similar, the ratio of the lengths of two corresponding sides is called ____________.
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If two polygons are similar, then the ratios of their ____________ are equal to the ratio of their corresponding ___________.
If two polygons are similar, then the ratios of their ____________ are equal to the ratio of their corresponding ___________.
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What does the AA similarity postulate state?
What does the AA similarity postulate state?
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What does the SSS similarity theorem state?
What does the SSS similarity theorem state?
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What does the SAS similarity theorem state?
What does the SAS similarity theorem state?
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What does the reflexive property of similarity state?
What does the reflexive property of similarity state?
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What does the symmetric property of similarity state?
What does the symmetric property of similarity state?
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What does the transitive property of similarity state?
What does the transitive property of similarity state?
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What does the triangle proportionality theorem state?
What does the triangle proportionality theorem state?
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What does the triangle proportionality converse state?
What does the triangle proportionality converse state?
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Define the midsegment of a triangle.
Define the midsegment of a triangle.
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What does the midsegment theorem state?
What does the midsegment theorem state?
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If three _____________ __________ intersect two _____________, then they divide the transversals proportionally.
If three _____________ __________ intersect two _____________, then they divide the transversals proportionally.
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If _____________ _______ parallel lines cut off _____________ segments on one transversal, then they cut off _____________ segments on every transversal.
If _____________ _______ parallel lines cut off _____________ segments on one transversal, then they cut off _____________ segments on every transversal.
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What is altitude in a triangle?
What is altitude in a triangle?
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If two triangles are similar, the lengths of _____________ ____________ are proportional to the lengths of corresponding sides.
If two triangles are similar, the lengths of _____________ ____________ are proportional to the lengths of corresponding sides.
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If two triangles are similar, the lengths of corresponding __________ _____________ are proportional to the lengths of corresponding sides.
If two triangles are similar, the lengths of corresponding __________ _____________ are proportional to the lengths of corresponding sides.
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If two triangles are similar, the lengths of corresponding ____________ are proportional to the lengths of corresponding sides.
If two triangles are similar, the lengths of corresponding ____________ are proportional to the lengths of corresponding sides.
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If a ray bisects an angle of a triangle, then it divides the __________ __________ into segments whose lengths are _____________ to the lengths of the other two sides of the triangle.
If a ray bisects an angle of a triangle, then it divides the __________ __________ into segments whose lengths are _____________ to the lengths of the other two sides of the triangle.
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Study Notes
Ratios and Proportions
- A ratio compares two quantities of the same unit, expressed as a/b or a : b, where b ≠ 0.
- Equivalent ratios maintain the same simplified form.
- A proportion consists of two equal ratios.
- In a proportion, the extremes are the first and last terms (a and d), while the means are the middle terms (b and c).
- The cross product property states that in a proportion, the product of the extremes equals the product of the means: ad = bc.
Geometric Mean
- The geometric mean between two positive numbers a and b is a positive number x such that a/x = x/b.
- Algebraically, x^2 equals the product of the two numbers: x^2 = ab, and x represents the square root of ab.
Triangle Similarity
- Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional.
- The symbol ~ denotes similarity.
- The scale factor is the ratio of the lengths of two corresponding sides in similar polygons.
- Ratios of perimeters of similar polygons are equal to the ratios of their corresponding sides.
Similarity Postulates and Theorems
- AA similarity postulate: Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.
- SSS similarity theorem: If corresponding sides of two triangles are proportional, they are similar.
- SAS similarity theorem: If an angle of one triangle is congruent to an angle of another, and the lengths of included sides are proportional, the triangles are similar.
Properties of Similarity
- Reflexive property: Triangle ABC is similar to itself (ABC ~ ABC).
- Symmetric property: If triangle ABC is similar to triangle DEF, then DEF is similar to ABC (ABC ~ DEF implies DEF ~ ABC).
- Transitive property: If triangle ABC ~ triangle DEF and triangle DEF ~ triangle XYZ, then ABC ~ XYZ.
Proportionality in Triangles
- Triangle proportionality theorem: A line parallel to one side of a triangle intersects the other sides, dividing them proportionally.
- Triangle proportionality converse: If a line divides two sides proportionally, it is parallel to the third side.
- The midsegment connects midpoints of two sides and is parallel to the third side while being half its length.
Proportional Segments
- If three parallel lines intersect two transversals, they divide the transversals proportionally.
- If three or more parallel lines cut off congruent segments on one transversal, they create congruent segments on all transversals.
- The altitude of a triangle creates perpendicular lines from a vertex to the opposite side.
Corresponding Lengths in Similar Triangles
- The lengths of corresponding altitudes in similar triangles are proportional to the lengths of corresponding sides.
- The lengths of angle bisectors in similar triangles correspond proportionally to side lengths.
- The lengths of medians in similar triangles follow the same proportionality as the sides.
Angle Bisectors and Segments
- An angle bisector in a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of key geometry concepts with these flashcards from Chapter 7. Learn essential terms such as ratio, equivalent ratios, and proportion to strengthen your mathematical skills.
