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## Essentials of Geometry

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**Essentials of Geometry**Geometry Chapter 1**This Slideshow was developed to accompany the textbook**• Larson Geometry • By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L. • 2011 Holt McDougal • Some examples and diagrams are taken from the textbook. • Slides created by • Richard Wright, Andrews Academy rwright@andrews.edu**1.1 Identify Points, Lines, and Planes**What is it? What is it like? Undefined Term Dot Point A • No Dimension How is it named? Point A A**1.1 Identify Points, Lines, and Planes**What is it? What is it like? Undefined Term No Thickness Goes forever m A B Line • • Straight Through any two points there is exactly one line. How is it named? 1 Dimension m**1.1 Identify Points, Lines, and Planes**What is it? What is it like? Undefined Term No Thickness Goes forever in both directions Plane • • • A B C Through any three noncollinear points, there is exactly one plane. 2 Dimensions How is it named? Plane Plane ABC**1.1 Identify Points, Lines, and Planes**• Give two other names for • Give another name for plane • Name three collinear points • Name four coplanar points ℓ A D m B E C**1.1 Identify Points, Lines, and Planes**What is it? What is it like? Part of a Line Part of a line between two endpoints Segment • • A B Does NOT go on forever How is it named? Named by endpoints**1.1 Identify Points, Lines, and Planes**What is it? What is it like? Part of a Line Part of a line starting at one endpoint and continuing forever Ray • • A B Goes forever in one direction only How is it named? Named by endpoint followed by one other point If two rays have the same endpoint and go in opposite directions, they are called opposite rays. A B C**1.1 Identify Points, Lines, and Planes**• Give another name for • Name all rays with endpoint Q • Which of these rays are opposite rays? P T Q R S The intersection of two lines is a point.**1.1 Identify Points, Lines, and Planes**• The intersection of two planes is a line.**1.1 Identify Points, Lines, and Planes**• Sketch a plane and two intersecting lines that intersect the plane at separate points. • Sketch a plane and two intersecting lines that do not intersect the plane. • Sketch a plane and two intersecting lines that lie in the plane.**1.1 Identify Points, Lines, and Planes**• 5 #1, 4-38 even, 44-58 even = 27 total**1.1 Grading and Quiz**• 1.1 Answers • 1.1 Homework Quiz**1.2 Use Segment and Congruence**• Postulate – Rule that is accepted without proof • Theorem – Rule that is proven -2 -1 0 1 2 3 4 Ruler Postulate A B Any line can be turned into a number line**1.2 Use Segment and Congruence**What is it? What is it like? Length measurement Difference of the coordinates of the points Distance AB = | x2 – x1 | -2 -1 0 1 2 3 4 How is it named? Find AB AB BA A B**1.2 Use Segment and Congruence**What is it? What is it like? Point Placement On the segment with the other points as the endpoints Between Does not have to be the midpoint What are examples? A B C A B C**1.2 Use Segment and Congruence**• Find CD AB BC A B C AC If AB + BC = AC, then B is between A and C 42 Segment Addition Postulate D C 17 E If B is between A and C, then AB + BC = AC**1.2 Use Segment and Congruence**Graph X(-2, -5) and Y(-2, 3). • Find XY.**1.2 Use Segment and Congruence**What is it? What is it like? Comparing shapes Segments with same length Congruent Segment means segments are exactly alike What are examples? = means lengths are equal A B C D**1.2 Use Segment and Congruence**• 12 #4-36 even, 37-45 all = 26 total**1.2 Grading and Quiz**• 1.2 Answers • 1.2 Homework Quiz**1.3 Use Midpoint and Distance Formulas**What is it? What is it like? Part of a Segment Very middle of the segment Point that divides the segment into two congruent segments. Midpoint B A M What are some examples? Segment Bisector is something that intersects a segment at its midpoint. AM = MB M is the midpoint of**1.3 Use Midpoint and Distance Formulas**• bisects at Q. If PQ = 22.6, find PN. • Point S is the midpoint of . Find ST. N M Q O P R S T 5x - 2 3x + 8**1.3 Use Midpoint and Distance Formulas**• Find the midpoint of G(7, -2) and H(-5, -6) Midpoint Formula**1.3 Use Midpoint and Distance Formulas**• The midpoint of is M(5, 8). One endpoint is A(2, -3). Find the coordinates of endpoint B.**1.3 Use Midpoint and Distance Formulas**• What is PQ if P(2, 5) and Q(-4, 8)? • 19 #2, 4, 6, 10-20 even, 24, 26, 28, 32, 36, 38, 42, 44, 48, 54-64 all = 29 total • Extra Credit 22#2, 8 Distance Formula**1.3 Grading and Quiz**• 1.3 Answers • 1.3 Homework Quiz**1.4 Measure and Classify Angles**What is it? What is it like? Shape Two rays with common endpoint (vertex) Angle Formed when two lines cross How is it named? BAC CAB A B sides A C vertex**1.4 Measure and Classify Angles**What is it? What is it like? Angle measurement Difference coordinates of each ray on a protractor Measure Protractor Postulate A protractor can be used to measure angles How is it named? mA = |x2–x1| mBAC mA**1.4 Measure and Classify Angles**• Classifying Angles • Acute • Less than 90° • Right • 90° • Obtuse • More than 90° • Straight • 180° It’s such a cute angle!**1.4 Measure and Classify Angles**• Find the measure of each angle and classify. • DEC • DEA • CEB • DEB B C A E D**1.4 Measure and Classify Angles**• Name all the angles in the diagram. • Which angle is a right angle?**1.4 Measure and Classify Angles**• If mRST = 72°, find mRSP and mPST R 2x – 9 P 3x + 6 Angle Addition Postulate S T If P is in the interior of RST, then mRST = mRSP + mPST**1.4 Measure and Classify Angles**What is it? What is it like? Comparing shapes Angles with same measure Congruent Angles means angles are exactly alike What are examples? = means measures are equal ABC DEF**1.4 Measure and Classify Angles**• Identify all pairs of congruent angles in the diagram. • In the diagram, mPQR = 130, mQRS = 84, and mTSR = 121 . Find the other angle measures in the diagram.**1.4 Measure and Classify Angles**Angle Bisector is a ray that divides an angle into two angles that are congruent. • bisects PMQ, and mPMQ = 122°. Find mPMN. • 28 #4-26 even, 30, 34-42 even, 48, 50, 52, 56, 60, 64-72 even = 28 total N Q P M**1.4 Grading and Quiz**• 1.4 Answers • 1.4 Homework Quiz**1.5 Describe Angle Pair Relationships**What is it? What is it like? Angles that share a ray and vertex Angles Pairs Adjacent Angles Are next to each other What are examples? Are not inside each other**1.5 Describe Angle Pair Relationships**• Complementary and Supplementary Angles do not have to be adjacent Complementary Angles Supplementary Angles Two angles whose sum is 90° Two angles whose sum is 180°**1.5 Describe Angle Pair Relationships**• In the figure, name a pair of • complementary angles, • supplementary angles, • adjacent angles. • Are KGH and LKG adjacent angles? • Are FGK and FGH adjacent angles? Explain.**1.5 Describe Angle Pair Relationships**• Given that 1 is a complement of 2 and m2 = 8o, find m1. • Given that 3 is a supplement of 4 and m3 = 117o, find m4.**1.5 Describe Angle Pair Relationships**• LMN and PQR are complementary angles. Find the measures of the angles if mLMN = (4x – 2)o and mPQR = (9x + 1)o.**1.5 Describe Angle Pair Relationships**What is it? What is it like? Angles that make a line Angles Pairs Linear Pair Linear Pair What are examples? Adjacent angles**1.5 Describe Angle Pair Relationships**What is it? What is it like? Angles formed when 2 line cross Angles Pairs Vertical Angles Are opposite sides of the intersection What are examples? Are not necessarily above each other Vertical Angles are congruent.**1.5 Describe Angle Pair Relationships**• Do any of the numbered angles in the diagram below form a linear pair? • Which angles are vertical angles?**1.5 Describe Angle Pair Relationships**• Two angles form a linear pair. The measure of one angle is 3 times the measure of the other. Find the measure of each angle.**1.5 Describe Angle Pair Relationships**• Things you can assume in diagrams. • Points are coplanar • Intersections • Lines are straight • Betweenness • Things you cannot assume in diagrams • Congruence unless stated • Right angles unless stated**1.5 Describe Angle Pair Relationships**• 38 #4-28 even, 32-44 even, 54, 58, 60, 62 = 24 total • Extra Credit 41 #2, 6**1.5 Grading and Quiz**• 1.5 Answers • 1.5 Homework Quiz**What happens when you leave the door of the birdcage open.**1.6 Classify Polygons What is it? What is it like? Sides are straight lines Shape Polygon Sides only intersect at ends What are examples? Closed shape No curves