WARM UP: SOLVE FOR X

120⁰

15x+5⁰

22x+4⁰

PERPENDICULAR TRANSVERSAL THEOREM “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

TRIANGLE EXTERIOR ANGLE THEOREM The exterior angle of a triangle equals

the sum of the 2 remote interior angles.

a=m+h Why???

m

h

a

PROOF:

Prove:

m+h+g=180g+a=180g=180-am+h+(180-a)=180m+h-a=0m+h=a

m

h

ag

Triangle angle sum theorem

Defn. of supplementary

Subtraction property

Substitution

Subtraction property

Addition property

m+h=a

O

BK

J

C

R TP

AL

U

4842

62

Fill in a missing angle in the picture.

TRIANGLE INTERIOR ANGLE SUM THEOREM (PROOF BOOK). PROVE THAT THE INTERIOR ANGLES IN A TRIANGLE HAVE A MEASURE SUM OF 180.

p

zq

Statements

Reasons

Construct segment PA so that it is parallel to segment QZ

LEQ: HOW DO WE CLASSIFY POLYGONS AND FIND THEIR ANGLE MEASURE SUMS?

3.5 The Polygon Angle-Sum Theorem

WHAT IS A POLYGON?

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

NAMING POLYGONS

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

TWO MAIN TYPES OF POLYGONS

Convex

“has no diagonal with points outside the polygon”

Concave

“has at least one diagonal with points outside the polygon”

CLASSIFY WHICH ARE CONCAVE AND WHICH ARE CONVEX

convex

ConvexConcave Convex

convexConcave

Concave

CLASSIFYING BY SIDES

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

HWK: FINISH RIDDLE WKST (BACK) AND COPY TRIANGLE EXTERIOR ANGLE THM & VERTICAL ANGLES THM INTO PROOF BOOK

INTERIOR ANGLES

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 partnerPg. 159 Activity (top) Do all 8 sides (skip the quadrilateral portion)Diagonals cannot overlap or cross each

other; connect only vertices

Polygon Number of Sides

Number of Triangles Formed

Sum of interior angle measures

POLYGON INTERIOR ANGLE-SUM THEOREM

“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…

SO WHY DOES IT WORK??

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.”

PROOF OF EXTERIOR ANGLE-SUM

What do you know about exterior angles?

IN PROOF BOOK: UNDER POLYGON EXTERIOR ANGLE SUM THM:

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)).

Statements Reasons

CLASS/HOMEWORK :

p. 161-162: 1-25, 47-49, 56