Geometry Chapter 7 Flashcards

Questions and Answers

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?

a and d. Signup and view all the answers

What are means in a proportion?

b and c. Signup and view all the answers

What does the cross product property of proportions state?

The product of the extremes equals the product of the means. Signup and view all the answers

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

ad=bc Signup and view all the answers

What is the geometric mean?

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

What is expressed by x^2?

ab. Signup and view all the answers

What is the definition of similar polygons?

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

The symbol ~ means __________.

similar. Signup and view all the answers

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

scale factor. Signup and view all the answers

If two polygons are similar, then the ratios of their __ are equal to the ratio of their corresponding _.

perimeters, sides. Signup and view all the answers

What does the AA similarity postulate state?

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

What does the SSS similarity theorem state?

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

What does the SAS similarity theorem state?

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. Signup and view all the answers

What does the reflexive property of similarity state?

Triangle ABC ~ triangle ABC. Signup and view all the answers

What does the symmetric property of similarity state?

If triangle ABC ~ triangle DEF, then triangle DEF ~ triangle ABC. Signup and view all the answers

What does the transitive property of similarity state?

If triangle ABC ~ triangle DEF, and triangle DEF ~ triangle XYZ, then triangle ABC ~ triangle XYZ. Signup and view all the answers

What does the triangle proportionality theorem state?

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

What does the triangle proportionality converse state?

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

Define the midsegment of a triangle.

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

What does the midsegment theorem state?

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. Signup and view all the answers

If three _ intersect two ___, then they divide the transversals proportionally.

parallel lines, transversals. Signup and view all the answers

If _________ _ parallel lines cut off _ segments on one transversal, then they cut off ___ segments on every transversal.

three or more, congruent, congruent. Signup and view all the answers

What is altitude in a triangle?

Vertex to side creates perpendicular lines. Signup and view all the answers

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

corresponding altitudes. Signup and view all the answers

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

angle bisectors. Signup and view all the answers

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

medians. 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.

opposite side, proportional. 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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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.