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Transcript
Section 9.1 Plane Figures Mathematical Systems Undefined Terms – point, line, plane Defined Terms – collinear, half plane, line segment, ray, angle Axioms – statements we assume are true and do not try to prove Theorems – Statements that can be proven with axioms, defined terms, undefined terms and deductive reasoning Defined Terms Half-Plane – A line in a plane will partition the plane into two half planes. Line Segment -- two points on a line and all the points between them. The line segment with endpoints A and B is denoted by AB . The midpoint C bisects AB A C B Defined Terms Ray– A point on a line partitions a line into two half-lines. A ray is a point on a line and all the points in one of the half-lines. Angle – An angle is formed by two rays or line segments that have a common endpoint called the vertex. Angles Right angle – 90o Acute angle – Less than 90o Obtuse angle –More than 90o and less than 180o Straight angle – 180o Reflex angle – More than 180o 8) Place the corner of a sheet of paper on the angles of the polygons to check for right angles a. Which angles, if any, are acute? b. Which angles, if any, are right angles? c. Which angles if any, are reflex angles? More on Angles Complementary – If the sum of two angles is 90o, the angles complementary. Supplementary – If the sum of two angles is 180o, the angles are supplementary. Adjacent angles – two angles having the same vertex and sharing a common side. More on Angles Vertical Angles –The nonadjacent angles formed by two intersecting lines are called vertical angles. Vertical angles are congruent. 10) Use the angles in the following figure to identify the pairs of angles. a. Three pairs of adjacent supplementary angles b. Three pairs of vertical angles c. Two pairs of adjacent complementary angles Perpendicular and Parallel Lines Perpendicular lines – intersect to form right angles Parallel Lines – do not intersect. Transversal – two parallel lines can be intersected by a third line called a transversal. Alternate Interior Angles If two lines are intersected by a transversal, the lines are parallel if and only if the alternate interior angles created by the transversal have the same measure. 12) If l and m are parallel lines, explain why the angles in each pair in parts a through d have the same measure. a. b. c. d. e. Angle 2 and Angle 8 Angle 2 and Angle 4 Angle 4 and Angle 8 Angle 1 and Angle 7 Explain why Angle 3 and Angle 8 are supplementary angles. 14) If s and t are parallel lines and the measure of Angle f is 32o and the measure of Angle a is 40o, determine the measure of each of the following angles. a.Angle b b.Angle c c.Angle d d.Angle e e.Angle g Curves Simple Curve – starts and stops without intersecting itself. Simple Closed Curve – starts and stops at the same point. Closed Curve – like a simple closed curve except it intersects itself. Classify each of the curves as simple, simple closed, or none of these. 1 2 3 4 Concave vs. Convex Concave - A set is concave if it contains two points such that the line segment joining them does not completely lie in the set. Concave sets are also called nonconvex Convex – not concave Classify each region as concave or convex. 1 2 Circles Tangent Chord Diameter Circle – Each point on a circle is the same distance from a fixed point called the center. Chord – A line segment whose endpoints are on the circle Radius Diameter – a chord that passes through the center Radius – a line segment from a point on the circle to its center Tangent – a line that intersects a circle in exactly one point 16) Use twelve hour clock to answer the following. a. How many degrees will the minute hand of the clock move through when the time changes from 8 o’clock to 8:25? b. How many hours will have passed when the hour hand has moved through 120o? c. What is the measure of the obtuse angle formed by the hour hand and the minute hand if the time is 2:30? Polygons Polygon – a simple closed curve that is the union of line segments called sides. The endpoints of these line segments are called vertices. Two sides of a polygon are adjacent sides if they share a common vertex, and two vertices are adjacent vertices if they share a common side. Regular Polygons Triangles Right Triangle (contains 1 right angle) Equilateral Triangle (all 3 sides of equal lengths) Scalene Triangle (all 3 sides of different lengths) Isosceles Triangle (at least two sides of equal length) Quadrilaterals Rectangle (pairs of opposite sides parallel and of equal length and all right angles) Square Parallelogram (all sides of equal length and all right angles) (pairs of opposite sides parallel and of equal length) Rhombus Trapezoid (opposites sides of parallel and all sides of equal length) (exactly 1 pair of opposite sides parallel) 24) Determine whether the following statements are true or false. For each false statement, show a counterexample. a. The two diagonals of a parallelogram have the same length. b. Any two angles in a parallelogram that share a common side are supplementary. c. The two diagonals of a rectangle have the same length.