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Transcript
1.5: Describe Angle Pair Relationships 1.6: Classify Polygons Objectives: 1. To use special angle relationships to find angle measures 2. To define, name, and classify polygons Vocabulary (make sure you know these) Complementary Supplementary Linear Pair Vertical Angles Polygon Diagonal (n.) Convex Concave Equilateral Equiangular Regular Define these in your notebook C Comes Before S… m1 m2 90 m3 m4 90 m5 m6 180 m7 m8 180 Example 1a 1. Given that <1 is a complement of <2 and m<1 = 68°, find m<2. 220 2. Given that <3 is a supplement of <4 and m<3 = 56°, find m<4. 1240 Example 1b 1. What is the sum of complementary angles in radians? Π 2 2. What is the sum of supplementary angles in radians? Π 3. What is complement for the angle that measures π/3? Π6 4. What is the supplement for the angle that measures 3π/4? Π 4 Example 2 Let <A and <B be complementary angles and let m<A = (2x2 + 35)° and m<B = (x + 10)°. What is (are) the value(s) of x? What are the measures of the angles? Set up the equation and solve X = 4.5 or -5 m<A = 14.5 or 85 m<B = 75.5 or 5 Check to make sure the sum is 90 Linear Pairs of Angles Linear Pairs of Angles • Two adjacent angles form a linear pair if their noncommon sides are opposite rays. • The angles in a linear pair are supplementary. Vertical Angles Vertical Angles • Two nonadjacent angles are vertical angles if their sides form two pairs of opposite rays. • Vertical angles are formed by two intersecting lines. Check them out HERE Example 3 Identify all of the linear pairs of angles and all of the vertical angles in the figure. Example 4: SAT y z In the figure 5 and 4 , what is the value x x of x? x=18, y=90 and z=72 HOW did I do that? x y z 3-D Rendering 3-D rendering in digital graphics is based upon polygons. 3-D Rendering The higher the polygon count, the smoother the surface. – Tomb Raider (1996) 3-D Rendering The higher the polygon count, the smoother the surface. – Tomb Raider Underworld (2008) What Makes a Polygon? So, what makes a polygon a polygon? Polygons A closed plane figure is a polygon if it is formed by 3 or more line segments (sides), joined endpoint to endpoint (vertices) with each side intersecting exactly two others. Parts of a Polygon Consecutive Angles Consecutive Vertices Consecutive Sides What’s the name of this polygon? Example 5 Why are the following not polygons? Names of Polygons (memorize these) Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon • Polygons come in many flavors. • They are classified by the number of sides they have. • A polygon with more than 12 sides is commonly called an n-gon, where n is the number of sides. Names of Polygons Sides Name Sides Name Triangle 13 Tridecagon 4 Quadrilateral* 14 Tetradecagon 5 Pentagon 15 Pentadecagon 6 Hexagon 16 Hexadecagon 7 Heptagon 17 Heptadecaton 8 Octagon 18 Octadecagon 9 Nonagon 19 Nonadecagon 10 Decagon 20 Icosagon 11 Undecagon 100 Hectagon 12 Dodecagon *Also called a Tetragon 3 1,000,000 Hecatommyriagon Example 6 Name each polygon. hexagon C B quadrilateral T W A F D C G U Example 7 When you buy a 42” television, how or where is that 42 inches measured? Diagonal Diagonal A diagonal is a line segment that joins two nonconsecutive vertices of a polygon. Example 8 How many diagonals are there in an octagon? (Do you really want to draw that? Heck no! In your notebook make a table and find a pattern!) Convex & Concave Polygons Convex & Concave Polygons Convex polygons have all their diagonals in the interior of the polygon. Concave polygons have at least one diagonal on the exterior of the polygon. Example 9 Tell whether the figure is a polygon and whether it is convex or concave. Equilateral Polygon An equilateral polygon is a polygon in which all of its sides are congruent. Equiangular Polygon An equiangular polygon is a polygon in which all its interior angles are congruent. Regular Polygon A regular polygon is a polygon that is equilateral and equiangular. Example 10 Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. Example 11: SAT In the figure, RS = ST and the coordinates of S are (k, 3). What is the value of k? y S T x -3 R O (1, 0) Example 12 Given that the figure is regular, find the values of x and y. x=12, y=8