6-1 Properties and Attributes of Polygons
Bellringer:
1. Tell whether the
figure is a polygon. If
it is a polygon, name
it by the number of its
sides.
not a polygon
2. Name the polygon by
the number of its
sides. Then tell
whether the polygon is
regular or irregular,
concave or convex.
Nonagon, irregular, concave
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
To find the sum of the interior angle measures of a
convex polygon, draw all possible diagonals from
one vertex of the polygon. This creates a set of
triangles. The sum of the angle measures of all the
triangles equals the sum of the angle measures of
the polygon.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Remember!
By the Triangle Sum Theorem, the sum of the
interior angle measures of a triangle is 180°.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
In each convex polygon, the number of triangles
formed is two less than the number of sides n. So
the sum of the angle measures of all these triangles
is (n — 2)180°.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Example 3A: Finding Interior Angle Measures and
Sums in Polygons
Find the sum of the interior angle measures of a
convex heptagon.
(n – 2)180°
Polygon ∠ Sum Thm.
(7 – 2)180°
A heptagon has 7 sides,
so substitute 7 for n.
900°
Holt McDougal Geometry
Simplify.
6-1 Properties and Attributes of Polygons
Example 3B: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each interior angle of a
regular 16-gon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon ∠ Sum Thm.
Substitute 16 for n
(16 – 2)180° = 2520°
and simplify.
Step 2 Find the measure of one interior angle.
The int. ∠s are ≅, so divide by 16.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Example 3C: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each
interior angle of pentagon
ABCDE.
(5 – 2)180° = 540°Polygon ∠ Sum Thm.
Polygon ∠
m∠A + m∠B + m∠C + m∠D + m∠E = 540° Sum Thm.
35c + 18c + 32c + 32c + 18c = 540
135c = 540
c=4
Substitute.
Combine like terms.
Divide both sides by 135.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Example 3C Continued
m∠A = 35(4) = 140°
m∠B = m∠E = 18(4) = 72°
m∠C = m∠D = 32(4) = 128°
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Check It Out! Example 3a
Find the sum of the interior angle measures of
a convex 15-gon.
(n – 2)180°
Polygon ∠ Sum Thm.
(15 – 2)180° A 15-gon has 15 sides, so
substitute 15 for n.
2340°
Holt McDougal Geometry
Simplify.
6-1 Properties and Attributes of Polygons
Check It Out! Example 3b
Find the measure of each interior angle of a
regular decagon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon ∠ Sum Thm.
(10 – 2)180° = 1440°
Substitute 10 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. ∠s are ≅, so divide by 10.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
In the polygons below, an exterior angle has been
measured at each vertex. Notice that in each case,
the sum of the exterior angle measures is 360°.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Remember!
An exterior angle is formed by one side of a
polygon and the extension of a consecutive side.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Example 4A: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each exterior angle of a
regular 20-gon.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Example 4B: Finding Interior Angle Measures and
Sums in Polygons
Find the value of b in polygon
FGHJKL.
Holt McDougal Geometry