Transcript Perpendicular transversal Theorem

WARM UP: SOLVE FOR X
15x+5⁰
120⁰
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???

h
m
a
PROOF:
Prove: m+h=a
h
Triangle angle sum
theorem
m+h+g=180
Defn. of supplementary
g+a=180
Subtraction property
g=180-a
m+h+(180-a)=180 Substitution
Subtraction property
m+h-a=0
Addition property
m+h=a
m
g
a
Fill in a missing angle in the picture.
R
P
T
L
U
A
48
42
O
J
62
C
K
B
TRIANGLE INTERIOR ANGLE SUM THEOREM
(PROOF BOOK). PROVE THAT THE INTERIOR
ANGLES IN A TRIANGLE HAVE A MEASURE SUM
Construct segment PA so that
OF 180.
it is parallel to segment QZ
Statements
p
q
z
Reasons
3.5 The Polygon Angle-Sum Theorem
LEQ: HOW DO WE CLASSIFY POLYGONS
AND FIND THEIR ANGLE MEASURE SUMS?
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
D
H
K
B
G
M
TWO MAIN TYPES OF POLYGONS
Convex
Concave
“has no diagonal with
points outside the
polygon”
“has at least one diagonal
with points outside the
polygon”
CLASSIFY WHICH ARE CONCAVE AND WHICH
ARE CONVEX
convex
Convex
convex
Concave
Concave
Concave
Convex
CLASSIFYING BY SIDES
3 sides:
4 sides:
5 sides:
Triangle
Quadrilateral
Pentagon
8 sides:
Octagon
9 sides:
Nonagon
10 sides: Decagon
6 sides:
Hexagon
11 sides:
Undecagon
7 sides:
Heptagon
12 sides:
Dodecagon
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 partner
Pg. 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 ngon 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??
180(n-2) n=number of sides
6 triangles, so 6(180)
degrees…but we want 4(180).
What’s going on??
According to the theorem, the interior angles should
sum to 720 degrees. Why?
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 ngon 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