Transcript Slide 1

Section 6.1
 Identify and classify polygons.
 Find angle measures of quadrilaterals.
 Polygon
 Side of a polygon
 Vertex of a polygon
 Diagonal of a polygon
 6.1 Quadrilateral Interior Angles Thoerem
 Polygon – a closed figure whose sides are formed by a
finite number of coplanar segments called sides.
 the sides that have a common endpoint are
noncollinear
 each side intersects exactly two other sides, but only at
their endpoints.
 The vertex of each angle is a vertex of the polygon. A
polygon is named by using the letters of its vertices,
written in consecutive order.
 State whether the figure is
a polygon. If it is not,
explain why.
 Not D – has a side that isn’t
a segment – it’s an arc.
 Not E– because two of the
sides intersect only one
other side.
 Not F because some of its
sides intersect more than
two sides.
A
C
B
F
E
D
Figures A, B, and C are polygons.
Is the figure a polygon? Explain your reasoning.
a.
b.
c.
d.
SOLUTION
a. Yes. The figure is a polygon formed by four straight sides.
b. No. The figure is not a polygon because it has a side that is not a
segment.
c. No. The figure is not a polygon because two of the sides intersect
only one other side.
d. Yes. The figure is a polygon formed by six straight sides.
 If the lines of any segment of the polygon are
drawn and any of the lines contain points that lie
in the interior of the polygon, then it is concave.
 Otherwise, it is convex (no points of the lines are
in the interior).
 Identify the polygon and
state whether it is convex
or concave.

Polygons are classified by the number of
sides they have. A polygon with n number
of sides is an n-gon.
Polygon
Number of Sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Hendecagon
11
Dodecagon
12
3 sides
HEXAGON
TRIANGLE
4 sides
QUADRILATERAL
5 sides
PENTAGON
6 sides
7 sides
HEPTAGON
8 sides
OCTAGON
Decide whether the figure is a polygon. If so, tell what type. If not,
explain why.
a.
b.
c.
d.
SOLUTION
a. The figure is a polygon with four sides, so it is a quadrilateral.
b. The figure is not a polygon because it has some sides that are not
segments.
c. The figure is a polygon with five sides, so it is a pentagon.
d. The figure is not a polygon because some of the sides intersect
more than two other sides.
Decide whether the figure is a polygon. If so, tell what type. If not,
explain why.
1.
2.
3.
4.
ANSWER
No; two of the sides
intersect only one other side.
ANSWER
yes; pentagon
ANSWER
yes; quadrilateral
ANSWER
No; one side is not a segment.
 A segment that connects any two nonconsecutive
vertices is a diagonal.
 Like triangles,
quadrilaterals have both
interior and exterior angles.
If you draw a diagonal in a
quadrilateral, you divide it
into two triangles, each of
which has interior angles
with measures that add up
to 180°. So you can
conclude that the sum of
the measures of the interior
angles of a quadrilateral is
2(180°), or 360°.
B
C
A
B
D
C
A
A
C
D
 The sum of the measures
of the interior angles of a
quadrilateral is 360°.
2
3
1
m1 + m2 + m3 + m4 = 360°
4
Find the measure of S.
SOLUTION
Use the fact that the sum of the measures of the interior angles of a
quadrilateral is 360°.
mP + mQ + mR + mS = 360°
70° + 80° + 70° + mS = 360°
220° + mS = 360°
mS = 140°
ANSWER
The measure of S is 140°.
Quadrilateral Interior
Angles Theorem
Substitute angle measures.
Simplify.
Subtract 220° from each
side.
Find the measure of A.
1.
ANSWER
90°
ANSWER
65°
ANSWER
72°
2.
3.
P
80°
 Find mQ and mR.
70°
 Find the value of x. Use
the sum of the measures
of the interior angles to
write an equation
involving x. Then, solve
the equation. Substitute
to find the value of R.
x°
2x°
Q
x°+ 2x° + 70° + 80° = 360°
R
S
P
80°
70°
x°
x°+ 2x° + 70° + 80° = 360°
3x + 150 = 360
3x = 210
x = 70
Q
S
2x°
R
Sum of the measures of int. s of
a quadrilateral is 360°
Combine like terms
Subtract 150 from each side.
Divide each side by 3.
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°
 Pg. 306 – 308 #1 – 27 odd, 31 - 35