Math 132 Day 4 (2/8/18) CCBC Dundalk

All the sides are congruent in an equilateral polygon 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.

http://www. brainpop. com/math/geometryandmeasurement/polygons/preview http://www.brainpop.com/math/geometryandmeasurement/polygons/preview.weml "Triangle" uses the Latin "angle" (angulus) rather than the Greek "gon" which means the same thing, so it's just the Latin equivalent of the Greek "trigon." "Quadrilateral" is even more distinctive, since it not only comes from Latin but means "four sides" rather than "four angles"; and in fact we DO use the word "quadrangle" sometimes (and also "trilateral")

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.

Complete How Many Degrees Inside? By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember!

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°.

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° Simplify.

Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. Polygon  Sum Thm. (5 – 2)180° = 540° Polygon  Sum Thm. mA + mB + mC + mD + mE = 540° 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.

Example 3C Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

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° Simplify.

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 and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

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. Substitute 10 for n and simplify. (10 – 2)180° = 1440° Step 2 Find the measure of one interior angle. The int. s are , so divide by 10.

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°.

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°. Polygon  Sum Thm. A regular 20-gon has 20  ext. s, so divide the sum by 20. measure of one ext.  = The measure of each exterior angle of a regular 20-gon is 18°.

Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext.  Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° 120b = 360 Combine like terms. b = 3 Divide both sides by 120.

Check It Out! Example 4a Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular dodecagon has 12  ext. s, so divide the sum by 12. measure of one ext. The measure of each exterior angle of a regular dodecagon is 30°.

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.

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.

Congruent Segments and Angles Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Regular Polygons All sides are congruent and all angles are congruent. A regular polygon is equilateral and equiangular. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Triangles and Quadrilaterals Right triangle a triangle containing one right angle Acute triangle a triangle in which all the angles are acute Obtuse triangle a triangle containing one obtuse angle Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Triangles and Quadrilaterals Scalene triangle a triangle with no congruent sides Isosceles triangle a triangle with at least two congruent sides Equilateral triangle a triangle with three congruent sides Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Triangles and Quadrilaterals Trapezoid a quadrilateral with one pair of parallel sides Kite a quadrilateral with two pairs of adjacent sides congruent and opposite sides not congruent. (NO parallel sides) Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Triangles and Quadrilaterals Isosceles trapezoid a trapezoid with congruent non-parallel sides & base angles Parallelogram a quadrilateral in which each pair of opposite sides is parallel Rectangle a parallelogram with a right angle. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Triangles and Quadrilaterals Rhombus a parallelogram with two adjacent sides congruent Square a rectangle with two adjacent sides congruent Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Hierarchy Among Polygons Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Vertical Angles Vertical angles created by intersecting lines are a pair of angles whose sides are two pairs of opposite rays. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles. Vertical angles are congruent. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Supplementary Angles The sum of the measures of two supplementary angles is 180°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Complementary Angles The sum of the measures of two complementary angles is 90°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. The Sum of the Measures of the Interior Angles of a Convex Polygon with n sides The sum of the measures of the interior angles of any convex polygon with n sides is (n – 2)180°. The measure of a single interior angle of a regular n-gon is Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

The Sum of the Measures of the Exterior Angles of a Convex n-gon The sum of the measures of the exterior angles of a convex n-gon is 360°. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-12 a. Find the measure of each interior angle of a regular decagon. The sum of the measures of the angles of a decagon is (10 − 2) · 180° = 1440°. The measure of each interior angle is Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 11-12 (continued) b. Find the number of sides of a regular polygon each of whose interior angles has measure 175°. Since each interior angle has measure 175°, each exterior angle has measure 180° − 175° = 5°. The sum of the exterior angles of a convex polygon is 360°, so there are exterior angles. Thus, there are 72 sides. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4A: Finding Arc Length Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. FG Use formula for area of sector. Substitute 8 for r and 134 for m.  5.96 cm  18.71 cm Simplify.

an arc with measure 62 in a circle with radius 2 m Example 4B: Finding Arc Length Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. an arc with measure 62 in a circle with radius 2 m Use formula for area of sector. Substitute 2 for r and 62 for m.  0.69 m  2.16 m Simplify.

Use formula for area of sector. Find each arc length. Give your answer in terms of  and rounded to the nearest hundredth. GH Use formula for area of sector. Substitute 6 for r and 40 for m. =  m  4.19 m Simplify.

an arc with measure 135° in a circle with radius 4 cm Find each arc length. Give your answer in terms of  and rounded to the nearest hundredth. an arc with measure 135° in a circle with radius 4 cm Use formula for area of sector. Substitute 4 for r and 135 for m. = 3 cm  9.42 cm Simplify.

Find each measure. Give answers in terms of  and rounded to the nearest hundredth. 1. length of NP 2.5  in.  7.85 in.

2. The gear of a grandfather clock has a radius of 3 in 2. The gear of a grandfather clock has a radius of 3 in. To the nearest tenth of an inch, what distance does the gear cover when it rotates through an angle of 88°?  4.6 in.