Basics of Geometry Chapter 1

Points, Lines and Planes I can name points, lines, and planes.

Points, Lines and Planes Vocabulary (page 4 in Student Journal) undefined terms: words that do not have formal definitions, but have an agreed upon meaning point: indicates a location and has no dimension, which is represented by a dot and named with a capital letter

Points, Lines, and Planes line: a straight path that extends in 2 opposite directions without end and has 1 dimension, which is named using 2 points on the line or with a single lowercase letter A line contains infinitely many points.

Points, Lines and Planes plane: a flat surface that extends without end and has 2 dimensions, which is named by at least 3 points in the plane that do not all lie on the same line or by a capital letter collinear points: points that lie on the same line coplanar points: points that lie in the same plane

Points, Lines and Planes defined terms: terms that can be described using known words segment: part of a line that consists of two endpoints and all the points between them, which is named by its endpoints ray: part of a line that consists of one endpoint and all the points on the line on one side of the endpoint, which is named by the endpoint and one other point on the ray

Points, Lines and Planes opposite rays: two rays that share the same endpoint and form a line intersection: the set of points 2 or more geometric figures have in common

Points, Lines, and Planes Examples (space on pages 4 and 5 in Student Journal) a) Give 2 other names for line DE. b) Give 2 other names for plane C. c) Name 3 points that are collinear. d) Name 4 points that are coplanar.

Points, Lines, and Planes Solutions line DA, line AE b) plane DAF, plane AEF D, A, E D, A, E, F

Points, Lines, and Planes e) Give another name for segment TR. f) Name a pair of opposite rays.

Points, Lines, and Planes Solutions e) segment RT f) ray PT and ray PR

Points, Lines, and Planes g) Sketch 2 intersecting lines a and b that lie in plane W. h) Sketch 2 planes R and S that intersect in line AB.

Points, Lines, and Planes Solutions g) h)

Measuring and Constructing Segments I can measure and construct line segments.

Measuring and Constructing Segments Vocabulary (page 9 in Student Journal) postulate (axiom): a rule that is accepted without proof coordinate: the real number that corresponds to a point distance: the absolute value of the difference of the coordinates of 2 points

Measuring and Constructing Segments construction: a geometric drawing using a compass and a straightedge congruent segments: line segments that have the same length

Measuring and Constructing Segments Core Concepts (pages 9 and 10 in Student Journal) Ruler Postulate (Postulate 1.1) Every point on a line can be paired with a real number, which makes a one-to-one correspondence between the points on the line and the real numbers.

Measuring and Constructing Segments Segment Addition Postulate (Postulate 1.2) If 3 points A, B and C are collinear and B is between A and C, then AB + BC = AC.

Measuring and Constructing Segments Example (space on page 10 in Student Journal) Use the Segment Addition Postulate to find the distances for JK and KL if JK = 4x + 6, KL = 7x + 15, and JL = 120.

Measuring and Constructing Segments Solution a) JK = 42, KL = 78

Using Midpoint and Distance Formulas I can find the midpoint and length of a line segment in the coordinate plane.

Using Midpoint and Distance Formulas Vocabulary (page 14 in Student Journal) midpoint: a point that divides a segment into 2 congruent segments segment bisector: a point, line, ray or segment that intersects a segment at its midpoint

Using Midpoint and Distance Formulas Core Concepts (pages 14 and 15 in Student Journal) The Midpoint Formula ((x1 + x2)/2 , (y1 + y2)/2) The Distance Formula AB = sqrt((x2 – x1)2 + (y2 – y1)2)

Using Midpoint and Distance Formulas Examples (space on page 14 in Student Journal) Use the definition of midpoint and other postulates to find the following distance. a) TU = 8x + 11, UV = 12x - 1, find TU, UV, and TV.

Using Midpoint and Distance Formulas Solution a) TU = 35, UV = 35, TV = 70

Using Midpoint and Distance Formulas b) The endpoints of segment AB are A (-8,7) and B (5,1). Find the coordinates of midpoint M. c) The midpoint of segment PQ is M (2, -3). One endpoint P is P (4,1). Find the coordinates of endpoint Q.

Using Midpoint and Distance Formulas Solutions b) (-1.5,4) c) (o,-7)

Using Midpoint and Distance Formulas d) Find the distance between the points A (13,2) and B (7,10). e) You ride your bike 5 miles east and then 2 miles north to a friend’s house. Determine the distance between your house and your friend’s house.

Using Midpoint and Distance Formulas Solutions d) 10 units e) 5.39 miles

Perimeter and Area in the Coordinate Plane I can find the perimeter and area of a polygon in a coordinate plane.

Perimeter and Area in the Coordinate Plane Vocabulary (page 19 in Student Journal) polygon: a closed plane figure formed by 3 or more line segments vertex: a point where the sides of a polygon meet

Perimeter and Area in the Coordinate Plane convex: no line that contains a side of the polygon contains a point in the interior of the polygon concave: a polygon that is not convex

Perimeter and Area in the Coordinate Plane Classifying Polygons Sides Name 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon n N-gon

Perimeter and Area in the Coordinate Plane Example (space on page 19 in Student Journal) a) Find the perimeter and area of triangle ABC with vertices A (1,3), B (3,-3), and C (-2,-3).

Perimeter and Area in the Coordinate Plane Solution perimeter = 18.03 units area = 15 square units

Measuring and Constructing Angles I can measure and classify an angle.

Measuring and Constructing Angles Vocabulary (page 24 in Student Journal) angle: formed by 2 rays with the same endpoint, which can be named by its vertex, a point on each ray and the vertex, or by a number vertex of an angle: the common endpoint of the 2 rays sides of an angle: the rays

Measuring and Constructing Angles measure of an angle: the absolute value of the difference of the real numbers paired with 2 rays congruent angles: angles with the same measure angle bisector: a ray, line, or segment, that divides an angle into 2 congruent angles with its endpoint at the vertex of the angle

Measuring and Constructing Angles Core Concepts (page 25 in Student Journal) Protractor Postulate (Postulate 1.3) Every ray can be paired one to one with a real number from 0 to 180 Angle Addition Postulate (Postulate 1.4) If point B is in the interior of angle AOC, then measure of angle AOB + measure of angle BOC = measure of angle AOC

Measuring and Constructing Angles Examples (space on page 25 in Student Journal) a)

Measuring and Constructing Angles Solution a) angle 3, angle U, angle VUT or angle TUV

Measuring and Constructing Angles b) Use the Angle Addition Postulate to find the following angle measures if angle DEF is a straight angle, measure of angle DEC = 11x - 12, measure of angle CEF = 2x + 10. c) Ray VB bisects angle AVC and the measure of angel AVC is 158 degrees. Find the measure of angle BVC.

Measuring and Constructing Angles Solutions b) measure of angle DEF = 180 degrees, measure of angle DEC = 142 degrees, measure of angle CEF = 38 degrees c) 79 degrees

Describing Pairs of Angles I can describe angle pair relationships and use the descriptions to find angle measures.

Describing Pairs of Angles Vocabulary (page 29 in Student Journal) complementary angles: 2 angles whose measures have a sum of 90 degrees supplementary angles: 2 angles whose measures have a sum of 180 degrees adjacent angles: 2 coplanar angles with a common side, a common vertex and no common interior points

Exploring Angle Pairs linear pair: a pair of adjacent angles whose noncommon sides are opposite rays The angles of a linear pair form a straight line. vertical angles: 2 angles whose sides are opposite rays

Exploring Angle Pairs Example (space on pages 29 and 30 in Student Journal) a) Angle ADB and angle BDC are a linear pair. The measure of angle ADB = 3x + 14, measure of angle BDC = 5x – 2. Find the measure of angle ADB and angle BDC.

Exploring Angle Pairs Solution a) measure of angle ADB = 77 degrees, measure of angle BDC = 103 degrees