drill 1)if two angles of a triangle have a sum of 85 degrees find the third angle. 2) the three...
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DrillDrill1) If two angles of a triangle have a
sum of 85 degrees find the third angle.
2) The three angles of a triangle are 2x, 3x, and 2x + 40 find each angle.
2.2 PolygonsPolygons
Polygon Is a closed figure with at least three
sides, so that each segment intersects exactly two segments at their endpoints.
Polygon TerminologyPolygon Terminology
Sides Vertex
Interior
A
B
C D
E
FDiagonal
ConsecutiveVertices
A polygon can also be classified as convex or concave.
If all of the diagonalslie in the interior of the figure, then the
polygon is ______.convex
If any part of a diagonal liesoutside of the figure, then thepolygon is _______.concave
Naming PolygonsNaming Polygons
Types of Polygons# of Sides Name/Draw
3
4
5
6
7
8
9
10
12
QUADRILATERAL
PENTAGON
HEXAGON
OCTAGON
HEPTAGON
NONAGON
DECAGON
DODECAGON
TRIANGLE
Diagonals and Angle Measure Diagonals and Angle Measure
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2 2(180) = 360
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
Make a table like the one below.
Diagonals and Angle Measure Diagonals and Angle Measure
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2 2(180) = 360
1) Draw a convex pentagon.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
pentagon 5 2 3 3(180) = 540
Diagonals and Angle Measure Diagonals and Angle Measure
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2 2(180) = 360
1) Draw a convex hexagon.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
pentagon 5 2 3 3(180) = 540
hexagon 6 3 4 4(180) = 720
Diagonals and Angle Measure Diagonals and Angle Measure
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2 2(180) = 360
1) Draw a convex heptagon.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
pentagon 5 2 3 3(180) = 540
hexagon 6 3 4 4(180) = 720
heptagon 7 4 5 5(180) = 900
Diagonals and Angle Measure Diagonals and Angle Measure
Convex
Polygon
Number
of Sides
Number of Diagonals from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral 4 1 2 2(180) = 360
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
pentagon 5 2 3 3(180) = 540
hexagon 6 3 4 4(180) = 720
heptagon 7 4 5 5(180) = 900
n-gon n n - 3 n - 2 (n – 2)180
Theorem 10-1If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.
Diagonals and Angle Measure Diagonals and Angle Measure
57°48°
74°
55°54°
72°
In §7.2 we identified exterior angles of triangles.
Likewise, you can extend the sides of anyconvex polygon to form exterior angles.
The figure suggests a method for finding thesum of the measures of the exterior angles of a convex polygon.
When you extend n sides of a polygon,
n linear pairs of angles are formed.
The sum of the angle measures in each linear pair is 180.
sum of measure of exterior angles
sum of measures of linear pairs
sum of measures of interior angles=
=
–
–n•180 180(n – 2)= –180n 180n + 360
= 360sum of measure of exterior angles
Polygon Interior Angle-Sum TheoremPolygon Interior Angle-Sum Theorem
The sum of the measures of the interior angles of an n-gon is (n-2)180.
Polygon Exterior Angle-Sum TheoremPolygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.
HomeworkHomework
Pages 79 – 80Pages 79 – 80
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