The triangle sum theorem and exterior angle theorem

standard 8.G.5

Triangle sum theorem

• Triangle sum theorem- the sum of the interior angles of any triangle is equal to 180 degrees
• a+b+c=180
• Interior angles of triangle adds up to 180 degrees
• A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle . If c is the length of the longest side, then, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate.

Difference

• In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. The sum of the internal angle and the external angle on the same vertex is 180°.
• Interior angles are the ones inside the shape or inside the angle, for example on a square the interior angles are each 90 degrees
• Exterior angles are the angles that are on the outside of the shape

Exterior angle theorem

• The exterior angle theorem when you extend the side of a triangle is equal to the sum of its non-adjacent angles. our non-adjacent angles are those that do not touch the angle we are working with. So, when we extend the side of an angle, creating a straight line that goes beyond the triangle, an exterior, or outside, angle that equals the sum of the two angles inside of the triangle that it does not touch.

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Example of an interior triangle

Example:

Find the value of x in the following triangle.

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Solution:

x + 24° + 32° = 180° (sum of angles is 180°)

x + 56° = 180°

x = 180° – 56° = 124°

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Example of exterior

Find the values of x and y in the following triangle.

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Solution:

x + 50° = 92° (sum of opposite interior angles = exterior angle)

x = 92° – 50° = 42°

y + 92° = 180° (interior angle + adjacent exterior angle = 180°.)

y = 180° – 92° = 88°

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