* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Cloudfront.net
Survey
Document related concepts
Steinitz's theorem wikipedia , lookup
History of geometry wikipedia , lookup
Multilateration wikipedia , lookup
Shape of the universe wikipedia , lookup
Noether's theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
Transcript
4.1 Apply Triangle Sum Properties Objectives: 1. To classify triangles by sides and angles 2. To find the measures of the interior and exterior angles of a triangle 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. 3-D Rendering 3-D rendering in digital graphics is based upon polygons. Vocabulary In your notebook, define Classified by Sides each of these types of Isosceles triangles without your Scalene book. Draw a picture Equilateral for each word and leave a bit of space Classified by Angles for additions and Acute Right revisions. Obtuse Equiangular Types of Triangles Triangles in Architecture Triangles in Architecture Interior vs. Exterior Angles All the angles inside the triangle are called interior angles. If you extend the sides of the triangle, then the angles that form a linear pair with the interior angles are called exterior angles. Investigation 1 Click on the button to investigate the Triangle Sum Theorem. Triangle Sum Theorem The sum of the measures --?--. Write and complete this statement in your notebook Click the arrow buttons to watch the Flash animation of the Triangle Sum Theorem. Example 1 In ABC below, find the measures of 1 and 2. 1 C A 60 2 <1 = 80 <2 = 100 40 B Example 2 Prove the Triangle Sum Theorem. F Given: ABC Prove: ma + mb + mc = 180° C 1 c G 2 HINT: Uses parallel lines theorem and linear angle sum a A TRY THIS in your notebook! b B Non-Euclidean Triangles Does the Triangle Sum Conjecture hold true in either hyperbolic or elliptic geometry? The answer may reveal secrets of the universe. C B A Hyperbolic Elliptic Big Unanswered Question So what shape is the universe? Is it basically flat, or is it curved? Big Unanswered Question So what shape is the universe? Is it basically flat, or is it curved? Example 3 In the right triangle below, what is the value of x + y? 45 y x Triangle Sum Theorem Corollary The acute angles of a right triangle are complementary. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of its two remote interior angles. remote interior Example 4 Find m<JKL. m<JKL = 35 Example 5 Find the values of a, b, and c. 75 c 40 <a = 65 <b = 60 <c = 55 120 b a Exercise 6: SAT What is the value of c? 80 c a a 160 b b Example 7 Rewrite the Triangle Sum Theorem and its Corollary in terms of radians. Do this in your notebook Example 8 Find the measure of each angle in radians. <A & <C = 2π 7 <B = 3π 7