1. (over Lesson 10-1)
Slide 1 of 1
1-1a

2. Diagonals and Angle Measure
What You'll Learn
You will learn to find measures of interior and exterior angles
of polygons.
Vocabulary
Nothing New!

3. Diagonals and Angle Measure Number of Diagonals from One Vertex
Make a table like the one below.
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
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

4. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex pentagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
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
pentagon 5 2 3
3(180) = 540

5. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex hexagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
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
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720

6. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Draw a convex heptagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
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
pentagon 5 2 3
3(180) = 540
hexagon 6 3 4
4(180) = 720
heptagon 7 4 5
5(180) = 900

7. Diagonals and Angle Measure Number of Diagonals from One Vertex
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
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
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 10-1
If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

8. Diagonals and Angle Measure
In §7.2 we identified exterior angles of triangles.
Likewise, you can extend the sides of any
convex polygon to form exterior angles.
48°
57°
74°
The figure suggests a method for finding the
sum of the measures of the exterior angles of a convex polygon.
72°
55°
54°
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
sum of measure of
exterior angles =
360

9. Diagonals and Angle Measure
Theorem 10-2
In any convex polygon, the sum of the measures of the
exterior angles, (one at each vertex), is 360.
Java Applet

10. Diagonals and Angle Measure
End of Section 10.2