## Questions and Answers

What is a ratio?

What are equivalent ratios?

Two ratios that have the same simplified form.

Define proportion.

Two equal ratios.

What are extremes in a proportion?

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What are means in a proportion?

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What does the cross product property of proportions state?

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If a/b=c/d, b≠0 and d≠0, then ______________.

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What is the geometric mean?

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What is expressed by x^2?

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What is the definition of similar polygons?

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The symbol ~ means __________.

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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 ___________.

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What does the AA similarity postulate state?

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What does the SSS similarity theorem state?

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What does the SAS similarity theorem state?

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What does the reflexive property of similarity state?

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What does the symmetric property of similarity state?

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What does the transitive property of similarity state?

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What does the triangle proportionality theorem state?

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What does the triangle proportionality converse state?

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Define the midsegment of a triangle.

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What does the midsegment theorem state?

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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.

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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.

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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.

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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.

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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.

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## 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.