geometry chapter 6
DESCRIPTION
Geometry Chapter 6. Quadrilaterals Lesson 1: Angles of Polygons. 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. 12 a + 4 a + 9 a = 100. triangle. - PowerPoint PPT PresentationTRANSCRIPT
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Geometry Geometry Chapter 6Chapter 6
QuadrilateralsLesson 1:
Angles of Polygons
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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) 124. (n – 3) 90
Solve for a.5. 12a + 4a + 9a = 100
trianglequadrilateral
24270
4
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Classify polygons based on their sides and angles.Find and use the measures of interior and exterior angles of polygons.
Your Math Goal Today…
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side of a polygonvertex of a polygondiagonalregular polygonconcaveconvex
Vocabulary
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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.
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You can name a polygon by the number of its sides. The table shows the names of some common polygons.
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A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
Remember!
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Example 1A: Identifying Polygons
If it is a polygon, name it by the number of sides.
polygon, hexagon
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Example 1B: Identifying Polygons
If it is a polygon, name it by the number of sides.
polygon, heptagon
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Example 1C: Identifying Polygons
If it is a polygon, name it by the number of sides.
not a polygon
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In Your Notes
If it is a polygon, name it by the number of its sides.
not a polygon
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In Your Notes
If it is a polygon, name it by the number of its sides.
polygon, nonagon
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In Your Notes
If it is a polygon, name it by the number of its sides.
not a polygon
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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.
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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.
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Example 2A: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex
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Example 2B: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave
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Example 2C: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex
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In Your Notes
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex
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In Your Notes
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave
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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.
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By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.
Remember!
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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°.
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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.
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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.
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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 = 540° Polygon Sum Thm.
35c + 18c + 32c + 32c + 18c = 540 Substitute.135c = 540 Combine like terms.
c = 4 Divide both sides by 135.
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Example 3C Continued
mA = 35(4°) = 140°mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
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In Your Notes
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.
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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.
In Your Notes
(n – 2)180°
(10 – 2)180° = 1440°
Polygon Sum Thm.Substitute 10 for n and simplify.
The int. s are , so divide by 10.
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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°.
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An exterior angle is formed by one side of a polygon and the extension of a consecutive side.
Remember!
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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. =
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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.
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Find the measure of each exterior angle of a regular dodecagon.
In Your Notes
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.
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In Your Notes
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.
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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.
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In Your Notes
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.