*Contributor: Marlene Vogel. Lesson ID: 11792*

180 degrees! That sounds awfully hot, doesn't it? But we're not talking about temperature; we're talking about Theorem 2-1! Don't be obtuse; finish this lesson to learn all the angles about triangles!

categories

subject

Math

learning style

Kinesthetic, Visual

personality style

Lion, Beaver

Grade Level

High School (9-12)

Lesson Type

Dig Deeper

Triangles may seem simple and downright dull, but they are important to many fields, including architecture, engineering, GPS, politics, and art (see the above picture). Are you ready to "tri" your hand at identifying the different types of triangles?

It is important to begin this lesson by reviewing or learning the following vocabulary words:

- An
*acute triangle*consists of interior angles that are less than 90°, which are called acute angles. *Adjacent*means next to or neighboring.- If a general theorem has a special case worth noting, it is called a
*corollary*. - An
*equiangular triangle*consists of three congruent angles. - An
*equilateral triangle*consists of three congruent sides. - The angle between one side of a polygon and the extension of an adjacent side is called the
*exterior angle of a polygon*. - An angle on the interior of a plane figure is called an
*interior angle*. - An
*isosceles triangle*contains two sides that are the same length. - An
*obtuse triangle*contains an interior angle that measures more than 90° and less than 180°, which is called an obtuse angle. - If all of the sides in a closed plane figure are line segments. it is called a
*polygon*. - Any interior angle that is not next to an exterior angle is called a
*remote interior angle*. - A
*right triangle*contains a 90° interior angle, which is called a right angle. - A scalene triangle consists of three sides that are all different lengths.
- A theorem is any idea that can be proven true using the rules of logic.

As you are probably aware, geometry is full of theorems. As you begin your understanding of triangles and the angles associated with them, you will need to be familiar with two more theorems and a corollary to a theorem. This lesson focuses on introducing you to the different types of triangles and the theorems used to find the measurements of the angles of the triangles.

Below are illustrations of the different types of triangles:

For a visual representation of an equiangular triangle, refer to the picture of the equilateral triangle. It is shaped exactly the same.

Now, it is time to become familiar with the theorems that will help you understand how we find the measurements of the angles in triangles.

**Theorem 2-1: Triangle Angle-Sum Theorem**

The sum of the measures of the angles of a triangle is 180. This theorem tells you the sum of any and every triangle's three angles will always be 180°.

**Theorem 2-2: Exterior Angle Theorem**

This theorem relates to the exterior angle of a triangle. The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Below is an illustration of a triangle with an exterior angle:

To find the exterior angle, identify the two remote interior angles. In the illustration above, the exterior angle is Angle C. The two remote interior angles to Angle C are Angles A and B. This theorem states that if you add the measurements of angles A and B in the above illustration, their sum will be the same as the measurement of angle C. See example below:

**Corollary for Theorem 2-2 **(the corollary for the above theorem)

The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

This corollary, which is just an additional explanation for Theorem 2-2, states that the measure of the exterior angle is always going to be greater than the measure of either of the remote interior angles. As you can see in the illustration above, the measurement of the exterior angle C is 130°, which is greater than the measurement of remote interior angle A, which is 60°, and the measurement of remote interior angle B, which is 70°.

Once you have absorbed the above information, continue on to the *Got It?* section for opportunities to practice your new geometry skills!

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