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# Geometry Chapter 7 Flashcards

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@TenaciousFeynman9892

### What is a ratio?

• a + b
• a/b or a : b where b ≠ 0 (correct)
• a * b
• a - b
• ### What are equivalent ratios?

Two ratios that have the same simplified form.

### Define proportion.

Two equal ratios.

### What are extremes in a proportion?

<p>a and d.</p> Signup and view all the answers

### What are means in a proportion?

<p>b and c.</p> Signup and view all the answers

### What does the cross product property of proportions state?

<p>The product of the extremes equals the product of the means.</p> Signup and view all the answers

### What is the geometric mean?

<p>It is the positive number x that satisfies a/x = x/b.</p> Signup and view all the answers

### What is expressed by x^2?

<p>ab.</p> Signup and view all the answers

### What is the definition of similar polygons?

<p>Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional.</p> Signup and view all the answers

### The symbol ~ means __________.

<p>similar.</p> Signup and view all the answers

### If two polygons are similar, the ratio of the lengths of two corresponding sides is called ____________.

<p>scale factor.</p> Signup and view all the answers

### If two polygons are similar, then the ratios of their ____________ are equal to the ratio of their corresponding ___________.

<p>perimeters, sides.</p> Signup and view all the answers

### What does the AA similarity postulate state?

<p>If 2 angles of one triangle are congruent to 2 angles of another triangle, then the two triangles are similar.</p> Signup and view all the answers

### What does the SSS similarity theorem state?

<p>If corresponding sides of two triangles are proportional, then the two triangles are similar.</p> Signup and view all the answers

### What does the SAS similarity theorem state?

<p>If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.</p> Signup and view all the answers

### What does the reflexive property of similarity state?

<p>Triangle ABC ~ triangle ABC.</p> Signup and view all the answers

### What does the symmetric property of similarity state?

<p>If triangle ABC ~ triangle DEF, then triangle DEF ~ triangle ABC.</p> Signup and view all the answers

### What does the transitive property of similarity state?

<p>If triangle ABC ~ triangle DEF, and triangle DEF ~ triangle XYZ, then triangle ABC ~ triangle XYZ.</p> Signup and view all the answers

### What does the triangle proportionality theorem state?

<p>If a line parallel to one side of a triangle intersects the other sides, then it divides the two sides proportionally.</p> Signup and view all the answers

### What does the triangle proportionality converse state?

<p>If a line divides two sides of a triangle proportionally, then it is parallel to the third one.</p> Signup and view all the answers

### Define the midsegment of a triangle.

<p>The line that connects the midpoints of any two sides.</p> Signup and view all the answers

### What does the midsegment theorem state?

<p>The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.</p> Signup and view all the answers

### If three _____________ __________ intersect two _____________, then they divide the transversals proportionally.

<p>parallel lines, transversals.</p> Signup and view all the answers

### If _____________ _______ parallel lines cut off _____________ segments on one transversal, then they cut off _____________ segments on every transversal.

<p>three or more, congruent, congruent.</p> Signup and view all the answers

### What is altitude in a triangle?

<p>Vertex to side creates perpendicular lines.</p> Signup and view all the answers

### If two triangles are similar, the lengths of _____________ ____________ are proportional to the lengths of corresponding sides.

<p>corresponding altitudes.</p> Signup and view all the answers

### If two triangles are similar, the lengths of corresponding __________ _____________ are proportional to the lengths of corresponding sides.

<p>angle bisectors.</p> Signup and view all the answers

### If two triangles are similar, the lengths of corresponding ____________ are proportional to the lengths of corresponding sides.

<p>medians.</p> Signup and view all the answers

### 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.

<p>opposite side, proportional.</p> Signup and view all the answers

## 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.

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