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Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Warm Up
1. A ? is a three-sided polygon.
2. A ? is a four-sided polygon.
Evaluate each expression for n = 6.
3. (n 4) 12
4. (n 3) 90
Solve for a.
5. 12a + 4a + 9a = 100
trianglequadrilateral
24
270
4
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Classify polygons based on their sides and angles.
Find and use the measures of interior and exterior angles of polygons.
Objectives
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
side of a polygonvertex of a polygondiagonalregular polygonconcaveconvex
Vocabulary
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
You can name a polygon by the number of its sides. The table shows the names of some common polygons.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
Remember!
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 1A: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, hexagon
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 1B: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, heptagon
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 1C: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
not a polygon
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 1a
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 1b
Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
polygon, nonagon
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 1c
Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
not a polygon
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 2A: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 2B: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 2C: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 2a
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 2b
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
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.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180.
Remember!
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
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 trianglesis (n 2)180.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
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
(7 2)180
900
Polygon Sum Thm.
A heptagon has 7 sides, so substitute 7 for n.
Simplify.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
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.
Step 2 Find the measure of one interior angle.
(n 2)180
(16 2)180 = 2520
Polygon Sum Thm.
Substitute 16 for n and simplify.
The int. s are , so divide by 16.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
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.
mA + mB + mC + mD + mE = 540Polygon
Sum Thm.
35c + 18c + 32c + 32c + 18c = 540 Substitute.
135c = 540 Combine like terms.
c = 4 Divide both sides by 135.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 3C Continued
mA = 35(4) = 140
mB = mE = 18(4) = 72
mC = mD = 32(4) = 128
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 3a
Find the sum of the interior angle measures of a convex 15-gon.
(n 2)180
(15 2)180
2340
Polygon Sum Thm.
A 15-gon has 15 sides, so substitute 15 for n.
Simplify.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Find the measure of each interior angle of a regular decagon.
Step 1 Find the sum of the interior angle measures.
Step 2 Find the measure of one interior angle.
Check It Out! Example 3b
(n 2)180
(10 2)180 = 1440
Polygon Sum Thm.
Substitute 10 for n and simplify.
The int. s are , so divide by 10.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
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.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
An exterior angle is formed by one side of a polygon and the extension of a consecutive side.
Remember!
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 4A: Finding Interior Angle Measures and Sums in Polygons
Find the measure of each exterior angle of a regular 20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360.
A regular 20-gon has 20 ext. s, so divide the sum by 20.
The measure of each exterior angle of a regular 20-gon is 18.
Polygon Sum Thm.
measure of one ext.
=
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 4B: Finding Interior Angle Measures and Sums in Polygons
Find the value of b in polygon FGHJKL.
15b + 18b + 33b + 16b + 10b + 28b = 360
Polygon Ext. Sum Thm.
120b = 360 Combine like terms.
b = 3 Divide both sides by 120.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Find the measure of each exterior angle of a regular dodecagon.
Check It Out! Example 4a
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360.
A regular dodecagon has 12 ext. s, so divide the sum by 12.
The measure of each exterior angle of a regular dodecagon is 30.
Polygon Sum Thm.
measure of one ext.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 4b
Find the value of r in polygon JKLM.
4r + 7r + 5r + 8r = 360 Polygon Ext. Sum Thm.
24r = 360 Combine like terms.
r = 15 Divide both sides by 24.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Example 5: Art Application
Ann is making paper stars for party decorations. What is the measure of 1?
1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360.
A regular pentagon has 5 ext. , so divide the sum by 5.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
Check It Out! Example 5
What if? Suppose the shutter were formed by 8 blades instead of 10 blades. What would the measure of each exterior angle be?
CBD is an exterior angle of a regular octagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360.
A regular octagon has 8
ext. , so divide the sum by 8.
Angle Relationships in TrianglesProperties and Attributes of Polygons
Holt Geometry
1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex.
2. Find the sum of the interior angle measures of a convex 11-gon.
Lesson Review
nonagon; irregular; concave
1620
3. Find the measure of each interior angle of a regular 18-gon.
4. Find the measure of each exterior angle of a regular 15-gon.
160
24
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