Warm-Up #28 Monday 5/2 Solve for x 2. Find AB

Warm-Up #29 Tuesday, 5/3 1. Prove why they are similar and then solve for x. 2. Solve for x and y

Interior and Exterior Angles of Polygons Homework Interior and Exterior Angles of Polygons

Naming Polygons closed figure A polygon is a _____________ in a plane formed by segments, called sides. closed figure angles A polygon is named by the number of its _____ or ______. sides tri A triangle is a polygon with three sides. The prefix ___ means three.

Prefix Number of Sides Name of Polygon Naming Polygons Prefixes are also used to name other polygons. Prefix Number of Sides Name of Polygon tri- 3 triangle quadri- 4 quadrilateral penta- 5 pentagon hexa- 6 hexagon hepta- 7 heptagon octa- 8 octagon nona- 9 nonagon deca- 10 decagon

Terms Naming Polygons A vertex is the point of intersection of two sides. Consecutive vertices are the two endpoints of any side. U T S Q R P A segment whose endpoints are nonconsecutive vertices is a diagonal. Sides that share a vertex are called consecutive sides.

An equilateral polygon has all _____ congruent. sides An equiangular polygon has all ______ congruent. angles A regular polygon is both ___________ and ___________. equilateral equiangular equilateral but not equiangular equiangular but not equilateral regular, both equilateral and equiangular Investigation: As the number of sides of a series of regular polygons increases, what do you notice about the shape of the polygons?

Diagonals and Angle Measure Number of Diagonals from One Vertex Make a table like the one below. 1) Draw a convex quadrilateral. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Draw a convex pentagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Draw a convex hexagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Draw a convex heptagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900 n-gon n n - 3 n - 2 (n – 2)180 Theorem If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

Theorem of interior angles If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

Ex. 1: Finding measures of Interior Angles of Polygons Find the value of x in the diagram shown: 142 88 136 105 136 x

SOLUTION: 88 136 142 105 x The sum of the measures of the interior angles of any hexagon is (6 – 2) ● 180 = 4 ● 180 = 720. Add the measure of each of the interior angles of the hexagon.

SOLUTION: 136 + 136 + 88 + 142 + 105 +x = 720. 607 + x = 720 The sum is 720 Simplify. Subtract 607 from each side. The measure of the sixth interior angle of the hexagon is 113.

EXTERIOR ANGLE THEOREMS

EXTERIOR ANGLE THEOREMS

Ex. 2: Finding the Number of Sides of a Polygon The measure of each interior angle is 140. How many sides does the polygon have?

Solution: = 140 Corollary to Thm. 11.1 (n – 2) ●180= 140n Multiply each side by n. 180n – 360 = 140n Distributive Property Addition/subtraction props. 40n = 360 n = 9 Divide each side by 40.

Ex. 3: Finding the Measure of an Exterior Angle

Ex. 3: Finding the Measure of an Exterior Angle

Ex. 3: Finding the Measure of an Exterior Angle

Using Angle Measures in Real Life Ex Using Angle Measures in Real Life Ex. 4: Finding Angle measures of a polygon

Using Angle Measures in Real Life Ex Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon

Using Angle Measures in Real Life Ex Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon

Using Angle Measures in Real Life Ex Using Angle Measures in Real Life Ex. 5: Using Angle Measures of a Regular Polygon Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of: 135°? 145°?

Using Angle Measures in Real Life Ex Using Angle Measures in Real Life Ex. : Finding Angle measures of a polygon