* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Introduction to Geometry Angles
Anatomical terms of location wikipedia , lookup
History of geometry wikipedia , lookup
Projective plane wikipedia , lookup
Penrose tiling wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Tessellation wikipedia , lookup
Rotation formalisms in three dimensions wikipedia , lookup
Plane of rotation wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Technical drawing wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Rational trigonometry wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Integer triangle wikipedia , lookup
Math 952 (Spring ’09) 10.1 "Angles and Triangles" Skills Objectives: * Recognize the terms point, line, and plane. * Know the de…nition of angles and learn how to classify angles by their measure as: acute, right, obtuse, or straight. * Know the meaning of the terms complementary angles, and supplementary angles. * Learn how to classify a triangle by its sides and by its angles. * Know that for any triangle: - The sum of the measures of its angles is 180 - The sum of the lengths of any two sides must be greater than the length of the third side. Preliminaries: Plane Geometry is the study of the properties of …gures in a plane. The most basic ideas in plane geometry are point, line, and plane. These simple terms are then used to de…ne higher-level ideas such as rays, angles, triangles, circles, and so on. Introduction to Geometry De…nitions: Term 1.Point "Point", "Line", and "Plane" Representation Discussion A dot represents a point. Points are labeled with capital letters. 2. Line A line has no beginning or end. Lines are labeled with small letters or two points on the line. 3. Plane Flat surfaces, such as table tops or walls, represent planes. Planes are labeled with capital letters. Angles De…nitions: Term 1. ray 2. angle 3. vertex 4. side "Ray", "Angle", "Vertex", and "Sides" Representation Discussion A ray consists of a point (called the endpoint) and all the points on one side of that point. An angle consists of two rays (called the sides) with a common endpoint. The common endpoint is called the vertex. Page: 1 Notes by Bibiana Lopez Prealgebra by Franklin Wright Three Ways of Labeling Angles: M ethod 1. Three Capital letters 10.1 Illustration : 2. Using Numbers 3. One Capital Letter Classifying Angles by Their Measures: Name Measure Illustration Acute Obtuse Right straight Example 1: (classifying angles by their measures) Use the …gure below to determine whether each of the following angles is acute, right, obtuse, or straight: b) 6 XOZ c) 6 W OY a) 6 1 De…nition: 1. Two angles are complementary if the sum of their measures is 2. Two angles are supplementary if the sum of their measures is 3. Two angles are equal if they have the same measure. Question: What is the supplement of a right angle? Question: What is the complement of an acute angle? Page: 2 Notes by Bibiana Lopez Prealgebra by Franklin Wright 10.1 Example 2: (Naming complementary and supplementary angles) In the …gure shown on the right: a. Name two pairs of supplementary angles. b. Name two pairs of complementary angles. Example 3: (Finding complementary and supplementary angles) If 6 3 and 6 4 are supplementary and m6 3 = 45 , what is m6 4? De…nitions: "vertical angles", "adjacent angles" Term Representation 1. vertical angles Discussion If two lines intersect, then two pairs of vertical angles are formed. Vertical angles are also called opposite angles. NOTE: Vertical angles have the same measure. 2. adjacent angles Two angles are adjacent if they have a common side. Example 4: (Using properties of vertical angles) In the …gure shown below, l, m, and n are straight lines with m6 1 = 20 and m6 6 = 90 a. Find the measures of the other four angles. b. Which angle is supplementary to 6 6? Page: 3 Notes by Bibiana Lopez Prealgebra by Franklin Wright 10.1 Triangles De…nition: Term "triangle" Representation 1. triangle Discussion A triangle consists of three line segments which join three points. The line segments are called the the sides of the triangle, and the points are the vertices. Properties of Triangles: - The sum of the measures of its angles is 180 - The sum of the lengths of any two sides must be greater than the length of the third side. Triangles Classi…ed by Sides Type Property Illustration Scalene Isosceles Equilateral Triangles Classi…ed by Angles Type Property Illustration Acute Obtuse Right Equiangular Page: 4 Notes by Bibiana Lopez Prealgebra by Franklin Wright 10.1 Example 5: (Naming triangles) Name the following triangles in two ways: a. b. Example 6: (Determining the existence of a triangle) Suppose that the lengths of the sides of a triangle are 13; 6; 19. Is this possible? Example 7: In 4ST U below, m6 T = 50 and m6 U = 40 : a. What is m6 S? b. What kind of triangle is 4ST U ? c. Which side is opposite m6 T ? d. Which sides include m6 T ? e. Is 4ST U a right triangle? Why or why not? Page: 5 Notes by Bibiana Lopez