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Transcript
Unit 1 Learning Outcomes 1: Describe and Identify the three undefined terms Learning Outcomes 2: Understand Angle Relationships Part 1 Definitions: Points, Lines and Planes Undefined Terms Points, Line and Plane are all considered to be undefined terms. – This is because they can only be explained using examples and descriptions. – They can however be used to define other geometric terms and properties A < B > Point – A location, has no shape or size – Label: Line – A line is made up of infinite points and has no thickness or width, it will continue infinitely.There is exactly one line through two points. – Label: Line Segment – Part of a line – Label: Ray – A one sided line that starts at a specific point and will continue on forever in one direction. – Label: F < E A B > Collinear – Points that lie on the same line are said to be collinear – Example: Non-collinear – Points that are not on the same line are said to be non-collinear (must be three points … why?) – Example: Plane – A flat surface made up of points, it has no depth and extends infinitely in all directions. There is exactly one plane through any three non-collinear points Coplanar – Points that lie on the same plane are said to be coplanar Non-Coplanar – Points that do not lie on the same plane are said to be non-coplanar Intersect The intersection of two things is the place they overlap when they cross. – When two lines intersect they create a point. – When two planes intersect they create a line. Space Space is boundless, three-dimensional set of all points. Space can contain lines and planes. Practice Use the figure to give examples of the following: 1. 2. 3. 4. Name two points. Name two lines. Name two segments. Name two rays. 5. 6. 7. 8. 9. Name a line that does not contain point T. Name a ray with point R as the endpoint. Name a segment with points T and Q as its endpoints. Name three collinear points. Name three non-collinear points. QuickTime™ and a decompressor are needed to see this picture. Part 2 Distance, Midpoint and Segments Distance Between Two Points Distance on a number line • PQ = B A or A B Distance on coordinate plane – The distance d between two points with coordinates x1, y1 and x2 , y2 is given by d x 2 x1 y 2 2 y1 2 Examples Example 1: – Find the distance between (1,5) and (-2,1) Examples 2: – Find the distance between Point F and Point B < E B -6 -1 > Congruent When two segments have the same measure they are said to be congruent Symbol: Example: A B > < < > D C AB CD Between Point B is between point A and C if and only if A, B and C are collinear and AB BC AC < A B C > Midpoint Midpoint – Halfway between the endpoints of the segment. If X is the MP of AB then AX XB < A X B > Finding The Midpoint Number Line – The coordinates of the midpoint of a segment whose endpoints have coordinates a and b is ab 2 Coordinate Plane – The coordinates of midpoint of a segment whose endpoints have coordinates x1, y1 and x2 , y2 are x1 x2 , y1 y2 2 2 Examples The coordinates on a number line of J and K are -12 and 16, respectively. Find the coordinate of the midpoint of Find the coordinate of the midpoint of for G(8,-6) and H(-14,12). Segment Bisector A segment bisector is a segment, line or plane that intersects a segment at its midpoint. Segment Addition Postulate – if B is between A and C, then AB + BC = AC – If AB + BC = AC, then B is between A and C Part 3 Angles Angle An angle is formed by two non-collinear rays that have a common endpoint. The rays are called sides of the angle, the common endpoint is the vertex. Kinds of angles Right Angle Acute Angle Obtuse Angle Straight Angle / Opposite Rays Congruent Angles Just like segments that have the same measure are congruent, so are angles that have the same measure. Angle Bisector A ray that divides an angle into two congruent angles is called an angle bisector. Angle Addition Postulate – If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS – If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS Measuring Angles How to use a protractor. – 1.) Line up the base line with one ray of your angle. – 2.) Follow the base line out to zero, if you are at 180 switch the protractor around. – 3.) Trace to protractor up until you reach the second ray of your angle. – 4) The number your finger rests on is your angle measure. Part 4 Angle Relationships Pairs of Angles Adjacent Angles - are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points Vertical Angles-are two non-adjacent angles formed by two intersecting lines Linear Pair - is a pair of adjacent angles who are also supplementary Angle Relationships Complementary Angles - Two angles whose measures have a sum of 90 Supplementary Angles - are two angles whose measures have a sum of 180 Part 5 Angle Theorems Theorem 2.3 Supplement Theorem – If two angles form a linear pair, then they are supplementary angles Theorem 2.4 Complement Theorem – If the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles. Theorem 2.6 Angles supplementary to the same angle or to congruent angles are congruent Theorem 2.7 Angles complementary to the same angle or to congruent angles are congruent Theorem 2.8 Vertical Angles Theorem – If two angles are vertical, then they are congruent Part 6 Perpendicular Lines and their theorems Perpendicular Lines Lines that form right angles are perpendicular – Perpendicular lines intersect to form 4 right angles – Perpendicular lines form congruent adjacent angles – Segments and rays can be perpendicular to lines or to other line segments or rays – The right angle symbol in a figure indicates that the lines are perpendicular. Theorems Theorem 2.9 - Perpendicular lines intersect to form four right angles Theorem 2.10 - All right angles are congruent Theorem 2.11 - Perpendicular lines form congruent adjacent angles More Theorems Theorem 2.12 - If two angles are congruent and supplementary, the each angle is a right angle Theorem 2.13 - If two congruent angles form a linear pair, then they are right angles. Unit 1 The End!