ANGLE MEASURES OF POLYGONS

Angle Measures of Polygons In this presentation, you will learn: The Interior Angles of a Quadrilateral Theorem. The Interior Angle Sum Theorem. The Exterior Angle Sum Theorem. How to determine the measures of interior angles of regular polygons. How to determine the measures of exterior angles of polygons.

Theorem: Interior Angles of a Quadrilateral The sum of the measures of the interior angles of a convex quadrilateral is 360 o.

Example 1: Find m<F, m<G, and m<H. xoxo xoxo 55 o HG EF

Example 1: Find m<F, m<G, and m<H. xoxo xoxo 55 o HG EF

Example 1: Find m<F, m<G, and m<H. xoxo xoxo 55 o HG EF m<E + m<F + m<G + m<H = 360 o

Example 1: Find m<F, m<G, and m<H. xoxo xoxo 55 o HG EF m<E + m<F + m<G + m<H = 360 o 55 o + x o + 55 o + x o = 360 o

Example 1: Find m<F, m<G, and m<H. xoxo xoxo 55 o HG EF m<E + m<F + m<G + m<H = 360 o 55 o + x o + 55 o + x o = 360 o (2x) o o = 360 o (2x) o = 250 o x = 125

Example 1: Find m<F, m<G, and m<H. xoxo xoxo 55 o HG EF m<E + m<F + m<G + m<H = 360 o 55 o + x o + 55 o + x o = 360 o (2x) o o = 360 o (2x) o = 250 o x = 125 m<F = 125 o, m<G = 55 o, and m<H = 125 o.

Example 2: Find m<K, m<L, and m<M. Is quadrilateral JKLM regular? xoxo xoxo 80 o ML JK (x – 20) o

Example 2: Find m<K, m<L, and m<M. Is quadrilateral JKLM regular? m<J + m<K + m<L + m<M = 360 o xoxo xoxo 80 o ML JK (x – 20) o

Example 2: Find m<K, m<L, and m<M. Is quadrilateral JKLM regular? m<J + m<K + m<L + m<M = 360 o 80 o + x o + (x – 2) o + x o = 360 o xoxo xoxo 80 o ML JK (x – 20) o

Example 2: Find m<K, m<L, and m<M. Is quadrilateral JKLM regular? m<J + m<K + m<L + m<M = 360 o 80 o + x o + (x – 2) o + x o = 360 o 78 o + (3x) o = 360 o (3x) o = 282 o x = 94 xoxo xoxo 80 o ML JK (x – 20) o

Example 2: Find m<K, m<L, and m<M. Is quadrilateral JKLM regular? m<J + m<K + m<L + m<M = 360 o 80 o + x o + (x – 2) o + x o = 360 o 78 o + (3x) o = 360 o (3x) o = 282 o x = 94 m<J = 80 o, m<K = 94 o, m<L = 74 o, m<M = 94 o No. JKLM is not regular. xoxo xoxo 80 o ML JK (x – 20) o

Diagonal A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

Diagonal A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

Quadrilatera l 4 sides 2 triangles Sum = 360 o

Quadrilatera l 4 sides 2 triangles Sum = 360 o Pentagon 5 sides 3 triangles Sum = 540 o

Quadrilatera l 4 sides 2 triangles Sum = 360 o Pentagon 5 sides 3 triangles Sum = 540 o Hexagon 6 sides 4 triangles Sum = 720 o

Quadrilatera l 4 sides 2 triangles Sum = 360 o Pentagon 5 sides 3 triangles Sum = 540 o Hexagon 6 sides 4 triangles Sum = 720 o Sum = 180 o (n – 2)

Theorem: Polygon Interior Angle Theorem The sum of the measures of the interior angles of a convex n-gon is Sum = 180 o (n – 2).

Example 3: What is the sum of the measures of the interior angles of a convex decagon?

Example 3: What is the sum of the measures of the interior angles of a convex decagon? Sum = 180 o (n – 2)

Example 3: What is the sum of the measures of the interior angles of a convex decagon? Sum = 180 o (n – 2) Sum = 180 o (10 – 2)

Example 3: What is the sum of the measures of the interior angles of a convex decagon? Sum = 180 o (n – 2) Sum = 180 o (10 – 2) Sum = 180 o (8) Sum = 1440 o

Example 4: The sum of the measures of the interior angles of a convex polygon is 2700 o. How many sides does the polygon have?

Example 4: The sum of the measures of the interior angles of a convex polygon is 2700 o. How many sides does the polygon have? Sum = 180 o (n – 2)

Example 4: The sum of the measures of the interior angles of a convex polygon is 2700 o. How many sides does the polygon have? Sum = 180 o (n – 2) 2700 o = 180 o (n – 2)

Example 4: The sum of the measures of the interior angles of a convex polygon is 2700 o. How many sides does the polygon have? Sum = 180 o (n – 2) 2700 o = 180 o (n – 2) 2700 o = 180 o n – 360 o 3060 o = 180 o n 17 = n

Example 4: The sum of the measures of the interior angles of a convex polygon is 2700 o. How many sides does the polygon have? Sum = 180 o (n – 2) 2700 o = 180 o (n – 2) 2700 o = 180 o n – 360 o 3060 o = 180 o n 17 = n The polygon has 17 sides.

Regular Polygons & Interior Angles

Example 5: Find the measure of each interior angle of a regular dodecagon.

Theorem: Polygon Exterior Angle The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°

Theorem: Polygon Exterior Angle The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°

Example 6: Find the value of y. (2y) o yoyo yoyo

Example 6: Find the value of y. (2y) o yoyo yoyo Sum = 360 o

Example 6: Find the value of y. (2y) o yoyo yoyo Sum = 360 o (2y) o + y o + (2y) o + y o = 360 o

Example 6: Find the value of y. (2y) o yoyo yoyo Sum = 360 o (2y) o + y o + (2y) o + y o = 360 o (6y) o = 360 o y = 60

Example 7: The measure of each exterior angle of a regular polygon is 40 o. How many sides does the polygon have?

Angle Measures of Polygons Summary