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Holt McDougal Geometry 6-1 Properties and Attributes of Polygons 6-1 Properties and Attributes of Polygons Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry
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6-1. Properties and Attributes of Polygons. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Geometry. 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 - PowerPoint PPT Presentation
Transcript of 6-1
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons6-1 Properties and Attributes of Polygons
Holt Geometry
Warm UpLesson PresentationLesson Quiz
Holt McDougal Geometry
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsWarm Up
1. A ? is a three-sided polygon.2. A ? is a four-sided polygon.
Evaluate each expression for n = 6.3. (n – 4) 124. (n – 3) 90
Solve for a.5. 12a + 4a + 9a = 100
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Classify polygons based on their sides and angles.Find and use the measures of interior and exterior angles of polygons.
Objectives
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
side of a polygonvertex of a polygondiagonalregular polygonconcaveconvex
Vocabulary
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
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.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
You can name a polygon by the number of its sides. The table shows the names of some common polygons.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
Remember!
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 1A: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, hexagon
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 1B: Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, heptagon
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 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
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck 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
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck 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
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck 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
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
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.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 2A: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 2B: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 2C: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck It Out! Example 2a
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck It Out! Example 2b
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsTo 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
By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.
Remember!
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 trianglesis (n — 2)180°.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 3A: Finding Interior Angle Measures and Sums in
PolygonsFind 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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 3B: Finding Interior Angle Measures and Sums in
PolygonsFind 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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 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 = 540° Polygon Sum Thm.
35c + 18c + 32c + 32c + 18c = 540 Substitute.135c = 540 Combine like terms.
c = 4 Divide both sides by 135.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 3C Continued
mA = 35(4°) = 140°mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck 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.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
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.
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
An exterior angle is formed by one side of a polygon and the extension of a consecutive side.
Remember!
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 4A: Finding Interior Angle Measures and Sums in
PolygonsFind 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. =
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 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.
Holt McDougal Geometry
6-1 Properties and Attributes of Polygons
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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck 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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsExample 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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsCheck 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.
Holt McDougal Geometry
6-1 Properties and Attributes of PolygonsLesson Quiz
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.
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.
