Unit 2 MM1G3 a Sum of the Interior and Exterior angles in Polygons!

Interior Angles of a Polygon

Exterior Angles of a Polygon

Interior/Exterior Angle Relationship

Since the sum of the measures of the interior angles of a triangle is 180 o, we can use this fact to help us find the sum of the measures of the interior angles of any convex n-gon. Example 1: Find the sum of the measures of the interior angles of quadrilateral ABCD below. Begin by drawing diagonal AC. As you can see, this diagonal divides quadrilateral ABCD into two triangles. Therefore, the sum of the measures of the interior angles is 180 o x 2 triangles = 360 o. B A CD A B CD

Example 2: Find the sum of the measures of the interior angles of a pentagon. ED C B A Draw diagonals AC and AD. These diagonals divide the pentagon into three triangles. Therefore, the sum of the measures of the interior angles of the pentagon is 180 o x 3 triangles = 540 o

The same method can be applied to convex polygons with many sides. PolygonNumber of sides Number of triangles Sum of the interior angles Triangle311 x 180 o = 180 o Quadrilateral422 x 180 o = 360 o Pentagon533 x 180 o = 540 o Hexagon644 x 180 o = 720 o Heptagon755 x 180 o = 900 o Octagon866 x 180 o = 1080 o n-gonnn – 2(n – 2) x 180 o Therefore, the sum of the measures of the interior angles of any convex polygon can be found by using (n – 2) x 180 o where n is the number of sides.

Example 3: Find the sum of the measures of the interior angles of a decagon. Solution: A decagon has 10 sides. Using the formula (n – 2) x 180 o, we can find the sum. (10 – 2) x 180 o = 8 x 180 o = 1440 o Therefore, the sum of the measures of the interior angles of a decagon is 1440 o.

Example 4: Find the value of x in the following figure. 114° x° 135° 102° 85° 115°

Solution: Since the figure has six sides, the sum of the measures of the interior angles is 720 o. (n – 2) · 180° = (6 – 2) · 180° = 720° Solving for x: 135° + 102° + 85° + 115° + 114° + x° = 720° 551° + x° = 720° x° = 169° 114° x° 135°102° 85° 115°

If we know the sum of the measures of the interior angles of a polygon, we can work backwards to find how many sides it has. Example 5:The sum of the measures of the interior angles of a convex polygon is 720º. How many sides does it have? Begin with the formula Now, solve for n (the number of sides). Therefore, the polygon has 6 sides.

A regular polygon is both equilateral and equiangular. The measure of one interior angle can be found by dividing the sum of the measures of the interior angles by the number of sides. Regular polygon Number of sides Number of triangles Sum of the interior angles Measure of one interior angle Triangle311 x 180 o = 180 o 180 o / 3 = 60 o Quadrilateral422 x 180 o = 360 o 360 o / 4 = 90 o Pentagon533 x 180 o = 540 o 540 o / 5 = 108 o Hexagon644 x 180 o = 720 o 720 o / 6 = 120 o Heptagon755 x 180 o = 900 o 900 o / 7 = o Octagon866 x 180 o = 1080 o 1080 o / 8 = 135 o n-gonnn – 2(n – 2) x 180 o

Example 6: Find the measure of one interior angle in a regular octagon. Solution: An octagon has 8 sides. The sum of the measures of the interior angles is (8 – 2) x 180 o = 1080 o To find the measure of one interior angle, divide this sum by the number of angles. Therefore, each interior angle in a regular octagon has a measure of 135 o.

If we know the measure of one interior angle of a regular convex polygon, then we can work backwards to find how many sides it has. Example 7:The measure of one interior angle of a regular convex polygon is 144º. How many sides does it have? Begin with the formula for the measure of one interior angle. Now solve for n (the number of sides) Therefore, the polygon has 10 sides.

Summary The sum of the measures of the interior angles of a convex n-gon = (n – 2) x 180 o. The measure of one interior angle of a regular convex n- gon =

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Exterior Angles of a Polygon Examples

An exterior angle of a polygon is formed by extending one side of the polygon. An exterior angle and its adjacent interior angle are supplementary. Interior angle exterior angle 12

Example 1: Find the sum of the measures of the exterior angles of the triangle below.

We can add the equations together. Since we know the sum of the interior angles of the triangle is 180°, we can substitute and solve. So, the sum of the measures of the exterior angles of the triangle is 360°.

Example 2: Find the sum of the measures of the exterior angles of a quadrilateral

Again, we know that each exterior angle and its adjacent interior angle are supplementary

Notice that the sum of the measures of the exterior angles of the quadrilateral is also 360º.

Further exploration shows us that the sum of the measures of the exterior angles of any convex polygon is always 360º. Example 3: In a pentagon, there are 5 exterior/interior angle pairs. Each pair is supplementary. 5 x 180° = 900° We know the sum of the measures of the interior angles is (5 – 2) 180° or 540°. 900° - 540° = 360° (the sum of the measures of the exterior angles. Interior angle Interior angle Interior angle Interior angle Interior angle exterior angle exterior angle exterior angle exterior angle exterior angle

We can also easily find the measure of one exterior angle of a regular convex polygon. Example 4: Find the measure of one exterior angle of a regular pentagon. Solution: Since the sum of the measures of the exterior angles of ANY convex polygon is 360º, then we simply divide by the number of sides. Therefore, the measure of each exterior angle of a regular pentagon is 72º exterior angle exterior angle exterior angle exterior angle exterior angle

In a regular polygon, we can use the formula where n is the number of sides to find the measure of each exterior angle. Example 4: Find the measure of one exterior angle of a regular convex 15-gon. Solution: The sum of the measures of the exterior angles of a regular 15-gon is 360°. Therefore, the measure of each exterior angle of a regular convex 15-gon is 24°.

Example 5: Each exterior angle of a certain regular convex polygon measures 20º. How many sides does the polygon have? Solution: We can work the formula backwards to find the number of sides. Therefore, the regular polygon has 18 sides.

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