120 ⁰ 15x+5 ⁰ 22x+4 ⁰

 “If two lines are parallel and a transversal is perpendicular to one line, then it is perpendicular to the other.  Reason: Corresponding angles are congruent

 The exterior angle of a triangle equals the sum of the 2 remote interior angles.  a=m+h  Why??? m h a

Prove: m+h+g=180 g+a=180 g=180-a m+h+(180-a)=180 m+h-a=0 m+h=a m h ag Triangle angle sum theorem Defn. of supplementary Subtraction property Substitution Subtraction property Addition property m+h=a

O B K J C RT P A L U Fill in a missing angle in the picture.

p z q StatementsReasons Construct segment PA so that it is parallel to segment QZ

3.5 The Polygon Angle-Sum Theorem

“a closed plane figure with at least three sides that are segments. Sides intersect only at their endpoints and no adjacent sides are collinear.”

 Name like naming planes (go in order clockwise or counterclockwise)  Vertices are the letters at the points  Sides are segments that form the polygon K H MG B D

Convex “has no diagonal with points outside the polygon” Concave “has at least one diagonal with points outside the polygon”

convex Convex Concave Convex convex Concave

3 sides: 4 sides: 5 sides: 6 sides: 7 sides: 8 sides: 9 sides: 10 sides: 11 sides: 12 sides: Triangle Quadrilateral Pentagon Hexagon Heptagon Decagon Nonagon Octagon Dodecagon Undecagon

The angles “inside” a polygon. There is a special rule to find the sum of the interior angle measures. Can you figure it out? Get with a partner Pg. 159 Activity (top) Do all 8 sides (skip the quadrilateral portion) Diagonals cannot overlap or cross each other; connect only vertices

PolygonNumber of SidesNumber of Triangles Formed Sum of interior angle measures

“The sum of the measures of the interior angles of an n- gon is (n-2)180.” Ex.) Sum of angles in a triangle. Tri=3 sides (3-2)180=180 Ex.) Sum of the angles in a quadrilateral (4 sides). (4-2)180=360 Ex.) The sum of the interior angles in a 23-gon…

According to the theorem, the interior angles should sum to 720 degrees. Why? 180(n-2) n=number of sides 6 triangles, so 6(180) degrees…but we want 4(180). What’s going on??

Polygon Exterior Angle-Sum Theorem “The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.”

What do you know about exterior angles?

Prove that the sum of the exterior angles of an n- gon is always 360. In an n-sided polygon, there are n vertices. Thus, we can construct n lines from each vertice. The sum of the measures of these is 180n because of n lines each 180 degrees in measure. The sum of the interior angles is 180(n-2) by the interior angle sum theorem. To calculate the sum of the exterior angles, we subtract the interior sum from the total measure of all angles. Thus we have 180n-(180(n-2)). StatementsReasons

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