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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 Adjacent 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 Complimentary Angles Complementary Angles - Two angles whose measures have a sum of 90 Complement Theorem – If the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles. Angles complementary to the same angle or to congruent angles are congruent Supplementary Angles Supplementary Angles - are two angles whose measures have a sum of 180 Angles supplementary to the same angle or to congruent angles are congruent Vertical Angles Vertical Angles-are two non-adjacent angles formed by two intersecting lines Vertical Angles Theorem – If two angles are vertical, then they are congruent Linear Pair Linear Pair - is a pair of adjacent angles who are also supplementary Supplement Theorem – If two angles form a linear pair, then they are supplementary angles Part 5 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!