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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
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”
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 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.”
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