FIND ANGLE MEASURES IN POLYGONS

Ch 8.1

Vocab

Consecutive vertices: vertices that are next to each other or part of the same side.

Diagonal

A diagonal is a segment that connects 2 non-consecutive edges.

Polygon Interior Angles Theorem Theorem 8.1

What is the sum of the interior angles of… A triangle:

A quadrilateral:

A pentagon:

A hexagon:

A heptagon:

180)2( n180)23( 180

180)24( 360

180)25( 540

180)26( 720

180)27( 900

Corollary to Interior Angle Theorem

If I have the Sum of the interior angles how do I figure out how many sides there are?

anglesofsumn 180)2(1800issumthethatGiven

1800180)2( n

180

1800

180

180)2(

n

102n12n

If I have the Sum of the interior angles how do I figure out how many sides there are?

anglesofsumn 180)2(1980issumthethatGiven

1980180)2( n

180

1980

180

180)2(

n

112n13n

How would you find x?

What kind of figure is this?What is the sum of the interior angles?

36059121108 x

360288 x72x

What kind of figure is this?

What is the sum of the interior angles of a pentagon?(5 – 2)(180) = 540

110+92+100+84 + x = 540386 + x = 540

x = 154

What kind of figure is it?How do you find x?

What is the sum of the interior angles of a heptagon?

(7 – 2)(180) = 900

145+112+99+133+156+2x+x = 900645 + 3x = 900

3x = 255x = 85

Finding exterior angles

How do you find x?

67 + 96 + 59 + 86 + x = 360

308 + x = 360

x = 52

Find x

X = 71

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Word Problems

120 ̊

What is the measure of an exterior angle in a regular 10-gon?360 ÷ 10 = 36 ̊

The measures of the exterior angles of a convex pentagon are 54, 72, 2x, 3x and x. What is the measure of the largest angle? The sum of the measure of the exterior

angles equals 360, so we’ll add the angles and find x.

54 + 72 + 2x + 3x + x = 360 126 + 6x = 360 6x = 234 X = 39 3x = 3(39) = 117

Find an interior and exterior angle for a regular 11-gon 360 ÷ 11 = 32.73 ̊

X + 32.73 = 180

X = 147.27

How do you find how many sides a figure has, when given one angle measure?

How do you find the sum of the interior angles?

(n – 2)180 = interior angle sum If we have a regular figure all the angles

are the same, so we would divide the interior angle sum by the number of sides, n, to find the measure of one angle.

measureangleonen

n

180)2(

An interior angle of a regular polygon has a measure of 150 . ̊How many sides does the figure have?

measureangleonen

n

180)2(

150180)2(

n

nMultiply both sides by n!

nn

nn

150

180)2(

nn 150180)2(

nn 150360180 Distribute the 180!

n30360 n12

Subtract the 180n!

An interior angle of a regular polygon has a measure of 120 . ̊How many sides does the figure have?

measureangleonen

n

180)2(

120180)2(

n

nMultiply both sides by n!

nn

nn

120

180)2(

nn 120180)2( nn 120360180

Distribute the 180!

n60360 n6Subtract the 180n!

Find the measure of and interior and exterior angle of a regular hexagon.

Find the measure of and interior and exterior angle of a regular 20-gon.

A regular polygon has an angle measure of 108 ̊ how many sides does the polygon have?