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Math for Kids23 chapters | 325 lessons

Instructor:
*Danielle Wilson*

Danielle is a certified elementary, middle school math, and special education teacher. She has a master's degree in elementary education and special education.

In this lesson, you'll learn about the triangle sum theorem proof, as well as the special trait all triangles share that can help you find missing angles in geometry problems.

A triangle is a three-sided polygon with three angles. The sides of the triangle can be all the same length, or they can be all different lengths.

Equilateral triangles have all the same lengths and angles. Isosceles triangles have two lengths and angles that are the same and one length and angle that are different. A triangle with three different side lengths and angles is called a scalene triangle. A right triangle has one angle that is 90°.

All triangles have one thing in common: all of their interior angles equal 180°. We call this common attribute the **triangle sum theorem**. **Interior angles** are the angles that you find inside of a polygon. If you add up the measure of each of the angles, you get the interior angle sum. The triangle sum theorem is very helpful in the study of geometry.

Any two triangles can be put together to make a quadrilateral. A quadrilateral is a four-sided polygon. A rectangle is a quadrilateral made up of four angles that each equal 90°. If we add up all of the angles of a rectangle, the total interior angle sum is 360°.

If we cut this rectangle in half, moving from one **vertex** (or corner) to the next, it divides into two triangles. Since one of these triangles is half of the rectangle, the total interior angle sum of a triangle is half of 360°, or 180°. Mathematicians found this to be true of every type of quadrilateral, including trapezoids. Therefore all triangles must equal 180°.

The triangle sum theorem is helpful for finding the missing angle in a triangle. When two angles are given, you can figure out the missing sum by doing the opposite operation. The sum of the angles will always equal 180°. Subtract the given angles from 180° to find the missing angle.

For example, the scalene triangle below has one angle that measures 46° and another angle that measures 52°. Subtract 46° and 52° from 180° to get the missing angle of 82°.

The triangle sum theorem makes it easy to find the interior angle sum of other polygons too. Say that you have a pentagon and want to find the interior angle sum for that shape. If you split the shape into triangles by drawing a line from one vertex to the other ones, you will make three triangles. You know each triangle equals 180°, so 3 times 180° equals 540°.

A hexagon has 6 sides, and it can be split into 4 triangles. The total interior angle sum of a hexagon will be 4 times 180°, or 720°. The number of triangles in a polygon is always 2 less than the number of sides. A 10-sided polygon will have 8 triangles, so 8 times 180° equals 1,440°.

The **triangle sum theorem** states that the **interior angles** of all triangles are equal to 180°. If you cut a triangle in half at its **vertex**, you'll find two triangles. The triangle sum theorem makes it easy for to find the missing angle of a triangle or the total interior angle sum of a polygon.

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31 in chapter 2 of the course:

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Math for Kids23 chapters | 325 lessons

- Surface Area Lesson for Kids
- Types of Angles: Lesson for Kids
- Types of Triangles: Lesson for Kids
- Geometric Shapes: Lesson for Kids
- Isosceles Triangle Lesson for Kids
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- Finding the Area of an Irregular Hexagon 3:38
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