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Similar Triangles and Polygons

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

In these lessons, we will learn

  • how to solve proportions
  • the properties of similar polygons
  • how to determine similar triangles
  • similar triangles in circles and right triangles (Altitude-on-Hypotenuse Theorems)
  • how to use proportions to find an unknown length or distance in similar figures (indirect measurement)

Solving Proportions

Solving proportions is a crucial skill when studying similar polygons. The ratio of corresponding side lengths between similar polygons are equal and two equivalent ratios are a proportion. For solving proportions problems, we set up the proportions and solve for the missing side length - it will be a variable, or a variable expression.
How to solve a simple proportion.
This video demonstrates how to use cross-multiplication to solve simple proportion problems.
3 examples on solving proportions using cross-multiplication.

Properties of Similar Polygons

Two polygons are similar if their corresponding angles are congruent and the corresponding sides have a constant ratio (in other words, if they are proportional). Typically, problems with similar polygons ask for missing sides. To solve for a missing length, find two corresponding sides whose lengths are known. After we do this, we set the ratio equal to the ratio of the missing length and its corresponding side and solve for the variable.
This video defines similar polygons and shows how to use the properties of similar triangles to solve for unknown values.
Learn about similar polygons.


Triangle Similarity

There are four triangle congruence shortcuts: SSS, SAS, ASA, and AAS. We have triangle similarity if
(1) two pairs of angles are congruent (AA)
(2) two pairs of sides are proportional and the included angles are congruent (SAS),
or (3) if three pairs of sides are proportional (SSS).
Notice that AAA, AAS, and ASA are not listed -- to include them would be redundant since they all have two congruent angles.
SSS, SAS and AA Triangle Similarity Tutorial
How to determine if two triangles are similar using a shortcut.
Working with similar triangles, determining similar triangles.

Similar Triangles in Circles and Right Triangles

Two triangles in a circle are similar if two pairs of angles have the same intercepted arc. Sharing an intercepted arc means the inscribed angles are congruent. Since these angles are congruent, the triangles are similar by the AA shortcut. If an altitude is drawn from the right angle in a right triangle, three similar triangles are formed, also because of the AA shortcut.
How to determine if two triangles in a circle are similar and how to prove that three similar triangles exist in a right triangle with an altitude.
A lesson on the altitude on hypotenuse theorems.

A review of problems using the altitude on hypotenuse theorems.
Altitude-on-Hypotenuse Theorems.

Indirect Measurement

Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures. Two common ways to achieve indirect measurement involve (1) using a mirror on the ground and (2) using shadow lengths and find an object's height.
Method 1 measures the person's height and the distances between the person, mirror, and object. Method 2 measures shadows and the person's height
How to use similar triangles to measure the height of objects.
Indirect measurement is a brilliant method for measuring distances that would otherwise be difficult or impossible to measure - using the geometrical properties of similar triangles.
This video explains how to use the properties of similar triangles to determine the height of a tree.


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other 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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