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Section 9.1- Basic Notions

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c Kendra Kilmer May 19, 2009 Section 9.1- Basic Notions The fundamental building blocks of geometry are points, lines, and planes. There is no formal definition of these terms but we can present an intuitive notion of each: Linear Notions: • Collinear points are points on the same line. (Any two points are collinear but not every three points have to be collinear.) • Point B is between points A and C on line l if B 6= A, B 6= C and B is on the part of the line connecting A and C. • A line segment, or segment, is a subset of a line that contains two points of the line and all points between those two points. • A ray is a subset of a line that contains the endpoint and all points on the line on one side of the point. 1 c Kendra Kilmer May 19, 2009 Planar Notions: • Points in the same plane are coplanar. • Noncoplanar points cannot be placed in single plane. • Lines in the same plane are coplanar lines. • Skew lines are lines that do not intersect, and there is no plane that contains them. • Intersecting lines are two coplanar lines with exactly one point in common. • Concurrent lines are lines that contain the same point. • Two distinct coplanar lines m and n that have no points in common are parallel lines. Example 1: Let’s think about the following questions: a) How many different lines can be drawn through two points? b) Can skew lines be parallel? 2 c Kendra Kilmer May 19, 2009 Properties of Points, Lines, and Planes: 1. There is exactly one line that contains any two distinct points. 2. If two points lie in a plane, then the line containing the points lies in the plane. 3. If two distinct planes intersect, then their intersection is a line. 4. There is exactly one plane that contains any three distinct noncollinear points. 5. A line and a point not on the line determine a plane. 6. Two parallel lines determine a plane. 7. Two intersecting lines determine a plane. Example 2: Euclid assumed the first four of the preceding properties as axioms (statements pre- sumed true). Let’s show that properties 5-7 follow logically from the first four properties. Example 3: Given 15 points, no 3 of which are collinear, how many different lines can be drawn through the 15 points? 3 c Kendra Kilmer May 19, 2009 Other Planar Notions: • Two distinct planes either intersect in a line or are parallel (have no points in common). • A line and a plane can be related in one of three ways: – If a line and a plane have no points in common, the line is parallel to the plane. – If two points of a line are in the plane, then the entire line containing the points is contained in the plane. – If a line intersects a plane but is not contained in the plane, it intersects the plane at only one point. • If a line is contained in a plane, it separates the plane into two half-planes. The plane is the union of three disjoint sets: the two half-planes and the line that separates them. Angles & Angle Measurement: • When two rays share an endpoint, an angle is formed. The rays of an angle are the sides of the angle, and the common endpoint is the vertex of the angle. • Adjacent angles share a common vertex and a common side and do not have overlapping interiors. • An angle is measured according to the amount of ”opening” between its sides. The degree is commonly used to measure angles. A complete rotation about a point has a measure of 360 degrees, written 360◦. 1 Thus, one degree is of a complete rotation. We can use a protractor to measure an angle. 360 • A degree is subdivided into 60 equal parts, minutes, and each minute is further divided into 60 equal parts, seconds. Example 4: Convert 18:35◦ to degrees, minutes, and seconds. Example 5: Convert 48◦1804500 to decimal degrees. Note: The degree is not the only unit used to measure angles. Other units of measurement include the radian and the grad. 4 c Kendra Kilmer May 19, 2009 Example 6: Use the figure below to find the measure of 6 ABC if m(6 1) = 52◦340 and m(6 2) = 46◦510. Types of Angles: • An acute angle measures less than 90◦. • A right angle measures exactly 90◦. • An obtuse angle measures greater than 90◦ and less than 180◦. • A straight angle measures exactly 180◦. Perpendicular: • When two lines intersect so that the angles formed are right angles, we say the lines are perpendicular lines and we write m ? n. • A line perpendicular to a plane is a line that is perpendicular to every line in the plane through its intersection with the plane. Example 7: Let’s think about the following questions: a) Is it possible for a line intersecting a plane to be perpendicular to exactly one line in the plane through its intersection with the plane? b) Is it possible for a line intersecting a plane to be perpendicular to two distinct lines in a plane going through its point of intersection with the plane, and yet not be perpendicular to the plane? c) Can a line be perpendicular to infinitely many lines? Theorem 9-1: A line perpendicular to two distinct lines in the plane through its intersection with the plane is perpendicular to the plane. Section 9.1 Homework Problems: 1, 2, 4, 5, 7, 8, 9, 10, 12, 19, 20, 26 5 c Kendra Kilmer May 19, 2009 Section 9.2 - Polygons Definitions: • A path that is drawn without picking up your pencil and without retracing any part of the path except single points is known as a curve. • A simple curve does not cross itself except the starting and stopping points may be the same. • A closed curve can be drawn starting and stopping at the same point. • Polygons are simple and closed and have sides that are line segments. • A point where two sides of a polygon meet is a vertex. • A curve is convex if it is simple, closed, and if a segment connecting any two points in the interior of the curve is wholly contained in the interior of the curve. (i.e. there are no indentations) • A concave curve is simple, closed, and not convex. • A polygonal region includes the polygon and its interior. Example 1: Classify each of the following figures: Polygons are classifed according to the number of sides or vertices they have: Number of Sides Name of Polygon 3 4 5 6 7 8 9 10 n 6 c Kendra Kilmer May 19, 2009 More About Polygons: • Any two sides of a polygon having a common vertex determine an interior angle. • An exterior angle of a convex polygon is determined by a side of the polygon and the extension of a contiguous side of the polygon. • Any line segment connecting nonconsecutive vertices of a polygon is a diagonal of the polygon. Congruent Segments and Angles: • Two segments are congruent if a tracing of one line segment can be fitted exactly on top of the other line segment. • Two angles are congruent if they have the same measure. • The symbol =∼ is used to represent congruent. • Polygons in which all the interior angles are congruent and all the sides are congruent are regular polygons. 7 c Kendra Kilmer May 19, 2009 Triangle Classifications: • A triangle containing one right angle is a right triangle. • A triangle in which all the angles are acute is an acute triangle. • A triangle containing one obtuse angle is an obtuse triangle. • A triangle with no congruent sides is a scalene triangle. • A triangle with at least two congruent sides is an isosceles triangle. • A triangle with three congruent sides is an equilateral triangle. Quadrilateral Classifications: • A trapezoid is a quadrilateral with at least one pair of parallel sides. (Note: Some elementary texts define a trapezoid as a quadrilateral with exactly one pair of parallel sides.) • A kite is a quadrilateral with two adjacent sides congruent and the other two sides also congruent. • An isosceles trapezoid is a trapezoid with exactly one pair of congruent sides. (Equivalently, an isosce- les trapezoid is a trapezoid with two congruent base angles.) • A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. 8 c Kendra Kilmer May 19, 2009 • A rectangle is a parallelogram with a right angle. (Equivalently, a rectangle is a quadrilateral with four right angles.) • A rhombus is a parallelogram with two adjacent sides congruent. (Equivalently, a rhombus is a quadri- lateral with all sides congruent.) sides congruent.) • A square is a rectangle with two adjacent sides congruent. (Equivalently, a square is a quadrilateral with four right angles and four congruent sides.) Example 2: Determine whether each of the following statements is true or false. • An equilateral triangle is isosceles. • A square is a regular quadrilateral. • If one angle of a rhombus is a right angle, then all of the angles of the rhombus are right angles. • A square is a rhombus with a right angle. • All the angles of a rectangle are right angles. • If a kite has a right angle, then it must be a square. Section 9.2 Homework Problems: 1, 3-7, 12, 13, 19, 20 9 c Kendra Kilmer May 19, 2009 Section 9.3 - More About Angles Definitions: • Vertical Angles are pairs of angles opposite each other at the intersection of two lines.
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