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Similar Triangles and Polygons

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More Lessons for High School Geometry
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A series of free, online High School Geometry Video Lessons.
Videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

  • proportional segments between parallel lines or triangle proportionality theorem
  • corresponding parts of similar triangles
  • angle bisectors and opposite sides ratio or triangle angle bisector theorem
  • similarity and perimeter ratio
  • similarity and area ratio
  • similarity and volume ratio

Proportional Segments Between Parallel Lines

When a line is drawn parallel to one side in a triangle, two similar triangles are formed because corresponding angles yield the AA similarity shortcut. Because the triangles are similar, the segments formed by the parallel line are proportional segments. When finding one of the bases of the triangles, be careful in setting up the proportion since the ratio is equal to the small triangle's side to the large triangle's.
How to prove two triangles are similar using a line parallel to a base?

This video states and proves the triangle proportionality theorem. Using the Properties of the Triangle Proportionality Theorem to Solve for Unknown Values

Corresponding Parts of Similar Triangles

If two triangles are similar, then the ratio of corresponding sides is equal to the ratio of the angle bisectors, altitudes, and medians of the two triangles. To find a missing angle bisector, altitude, or median, use the ratio of corresponding sides.
Similar Triangle shortcuts, altitudes, medians, perimeter comparison

Angle Bisectors and Opposite Side Ratios

When an angle bisector is drawn in a triangle, the ratio of the opposite sides forming the bisected angle is equal to the ratio of the segments formed by bisector intersecting the opposite side. This ratio applies to all types of triangles and for an angle bisector drawn from any angle.
This video states and proves the triangle angle bisector theorem.

Using the Properties of the Triangle Angle Bisector Theorem to Determine Unknown Values Using the Triangle-Angle-Bisector Theorem to solve a problem.

Similarity and Perimeter Ratios

If two triangles are similar, then their corresponding sides and perimeters are proportional Proof -- similar triangles perimeters share same proportion as corresponding sides.

Similarity and Area Ratios

If two triangles are similar, then their corresponding sides are proportional. Since sides are a length and lengths are one dimensional, the side ratio will not predict the ratio of the areas. To find the area ratios, raise the side length ratio to the second power. This applies because area is a square or two-dimensional property.
How to convert between length and area ratios of similar polygons A lesson on finding the ratio of areas of similar figures

Similarity and Volume Ratios

If two solids are similar, then their corresponding sides are all proportional. The ratio of their surface areas is the side ratio squared and note that the ratios of the areas does not give the actual surface areas. The volume ratio for the two solids is the side length ratio raised to the third power. Again, this is not the solids' volume, only the ratio of the volumes.
How to identify if two solids are similar and how to convert 1D ratios to surface area and volume ratios. Similarity ratios with volume and lengths.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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