Ratios and Proportions in Similar Polygons

Questions and Answers

What is the ratio?

first & last

equation stating that two ratios are equal

middle

comparison of two quantities using division (correct)

What is a proportion?

An equation stating that two ratios are equal.

What are cross products?

Can be found by multiplying the means and extremes.

What are extremes in a ratio?

First & last. Signup and view all the answers

What are means in a ratio?

Middle. Signup and view all the answers

What are similar polygons?

Polygons that have the same shape but may be different in size. Signup and view all the answers

What is the Similar Polygons Rule?

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

What is a scale factor?

The numerical ratio you get when comparing the lengths of corresponding sides of similar figures. Signup and view all the answers

What does proportionate mean?

All reduce to the same ratio. Signup and view all the answers

To show that 2 triangles are similar, show corresponding angles are __________.

congruent Signup and view all the answers

To show that 2 triangles are similar, show corresponding sides are __________.

proportionate Signup and view all the answers

How to find distance?

Distance = rate x time. Signup and view all the answers

What are shortcuts for similar triangles?

AA Similarity Postulate, SSS Similarity Postulate, SAS Postulate. Signup and view all the answers

What does the AA similarity postulate state?

If two of three corresponding angles of two triangles are congruent, then the triangles are similar. Signup and view all the answers

What does the SSS similarity theorem state?

If all three corresponding sides in two triangles are proportionate, then the triangles are similar. Signup and view all the answers

What does the SAS similarity theorem state?

If two corresponding sides of two triangles are proportionate and the included corresponding angles are congruent, then the two triangles are similar. Signup and view all the answers

What is The Triangle Proportionality Theorem?

If a line parallel to the base of a triangle is drawn inside the triangle, then it cuts the legs of the triangles into proportionate parts. Signup and view all the answers

What is the converse of the triangle proportionality theorem?

If a line cuts the legs of a triangle into proportionate parts then it is parallel to the base of a triangle. Signup and view all the answers

What is a midsegment?

A segment with midpoints that are the midpoints of two sides of the triangle. Signup and view all the answers

What is The Triangle Midsegment Theorem?

The midsegment of a triangle is parallel to the base of the triangle and is half the length of the base. Signup and view all the answers

What is The Proportional Perimeter Theorem?

If two triangles are similar then their perimeters are proportional to the measures of the corresponding sides. Signup and view all the answers

What is the special segments of similar triangles theorem?

Altitudes, bisectors, and medians are also proportional to corresponding sides within similar triangles. Signup and view all the answers

Study Notes

Ratios and Proportions

Ratio: A comparison of two quantities expressed through division.
Proportion: An equation that shows two ratios are equivalent.
Cross Products: Calculated by multiplying the means and extremes in a proportion.
Extremes: The first and last terms in a proportion.
Means: The middle terms in a proportion.

Similar Polygons

Similar Polygons: Polygons with the same shape, but potentially different sizes.
Similar Polygons Rule: Two polygons are similar if their corresponding angles are congruent and corresponding sides are proportional.
Scale Factor: The ratio comparing the lengths of corresponding sides in similar figures.

Triangle Similarity Criteria

Similarity Criteria for Triangles:
- Show that corresponding angles are congruent.
- Show that corresponding sides are proportional.
- Write similarity statements.
AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS Similarity Theorem: If all three pairs of corresponding sides are proportional, then the triangles are similar.
SAS Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, the triangles are similar.

Triangle Proportionality Theorems

Triangle Proportionality Theorem: A line drawn parallel to the base of a triangle divides the legs into proportional segments.
Converse of the Triangle Proportionality Theorem: If a line intersects the legs of a triangle in proportion, it is parallel to the triangle's base.
Midsegment: A line connecting the midpoints of two sides of a triangle.
Triangle Midsegment Theorem: The midsegment is parallel to the triangle's base and is half its length.

Perimeter and Special Segments

Proportional Perimeter Theorem: Similar triangles have perimeters that are proportional to the lengths of their corresponding sides.
Special Segments of Similar Triangles Theorem: Altitudes, bisectors, and medians in similar triangles are proportional to their corresponding sides.

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Description

This quiz explores the concepts of ratios, proportions, and their applications in similar polygons and triangles. Understand the definitions and rules that govern similarity, as well as how to identify and apply the AA similarity postulate. Test your knowledge with engaging questions on these important geometric concepts.