# Lesson 17. Introduction to Geometry. Objectives

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1 Student Name: Date: Contact Person Name: Phone Number: Lesson 17 Introduction to Geometry Objectives Understand the definitions of points, lines, rays, line segments Classify angles and certain relationships between lines

2 uthors: Jason March, Tim Wilson, Editor: Graphics: Linda Shanks Tim Wilson Jason March Eva McKendry National PSS Center OCES Geneseo Migrant Center 27 Lackawanna venue Mount Morris, NY (585) (585) (fax) wwwmigrantnet/pass Developed by the National PSS Center under the leadership of the National PSS Coordinating Committee with funding from the Region 20 Education Service Center, San ntonio, Texas, as part of the Mathematics chievement = Success (MS) Migrant Education Program Consortium Incentive project In addition, program support from the Opportunities for Success for Out-of-School Youth (OSY) Migrant Education Program Consortium Incentive project under the leadership of the Kansas Migrant Education Program

3 The word geometry comes from Greek It literally means Earth-measure It is the branch of math that studies the interesting relationships caused by the size and shape of objects To study geometry, we need to first know some basic terms point is s a location in space It is shown with a dot line goes straight through two points in opposite directions and never ends o The arrows show that the line has no end It goes on forever in both directions The notation we use to represent this line is i or ray goes through two points, has one endpoint, and extends without end in one direction o The ray above shows an endpoint at It continues forever in one direction through the point The notation for the ray is Imagine we have a line, say, drawn below and we cut the line at points and This is called a line segment Math On the Move Lesson 17 1

4 line segment has two points Each point is an endpoint X Y o The segment above has no arrows Points and are its endpoints The notation for a line segment is XY or YX Example Classify the following figures, then write them using the proper notation P Solution H K I J N M From left to right, we have point P,, line segment HI, ray JK, and line MN Classify each figure by writing point, line, ray, or line segment next to it Then T hen,, name each using the proper notation 1 H G 2 Q Math On the Move 2

5 3 T W 4 S I When you are in a car, you have to stop at red lights at every intersection With roads, an intersection is when the paths of two roads cross The same is true with geometry The point where a line, a segment, or a ray crosses another line, segment, or ray, is called an intersection intersection In this picture of a pool table, the dotted line represents the path the red ball must travel in order to drop in the top right pocket The path of the ball forms an angle Math On the Move Lesson 17 3

6 n angle is formed by two lines, rays,, or segments that t meet at a common point The common point is called the vertex o R This angle is formed by two rays with vertex Q There are three ways to name the angle: PQR RQP Q Q P The symbols all mean angle We use this arc to help show the angle we are focusing on,, and When naming angles, make sure the vertex is the middle letter Example Determine the vertex, and name the angle in three ways E F D Solution Even though there is no point drawn where the two rays meet, we still call E the vertex of the angle This means that E is the t middle letter when we name the angle We can name the angle DEF, FED, or E Math On the Move 4

7 Look at this example If we wanted to name the angles below, we cannot name them more than one angle with M as its vertex It is not clear if M is referring to JML M There is JMK, KML, or J K M L Each angle has a measure The unit of measure for angles is called degrees Degrees ( ) are a unit of measure for angles They are different from degrees Fahrenheit and degrees Celsius, because those measure temperature Take a look around you ngles are everywhere! To make the chair you re sitting on, a furniture maker had to measure many angles The angle measurements told him how to fit the chair together Just as angles are in furniture and buildings, they occur everywhere in nature, too! The golden angle, about 1375, is found in many plants It is the angle between the seeds of a sunflower ngles come in all sizes Scientists and mathematicians classify or organize them based on their measures Math On the Move Lesson 17 5

8 n angle whose measure is exactly 90 is a right angle We use the little box to show the angle is a right angle n angle whose measure is less than 90 is an acute angle The measure of this angle is around 20 n angle whose measure is between 90 and 180 (but not equal to 90 and 180 ) is an obtuse angle The measure of this angle is around 150 n angle whose measure is exactly 180 is a straight angle Example Given figure PQRS, Q R P Classify the angles as acute,, right, or obtuse S benchmark is used to compare things and classify them! Math On the Move 6

9 Solution y looking at the figure, you can tell that S is a right angle Not only does it have the right angle symbol, it also looks as if it is 90 ecause of this, a right angle can be used as a benchmark to tell whether angles are greater than, or less than 90 Compared to than 90 S, we can see that P and R are acute, because their measures are less Q is obtuse because its measure is greater than 90 Given the following figure, F G E C D Classify the following angles as acute, right, obtuse, or straight 5 : 9 FGE : 6 C : 10 E : 7 CDE : 11 CD : 8 FE : In geometry, right angles, and straight angles are very important Math On the Move Lesson 17 7

10 Complementary angles are a pair of angles whose measures have a sum of 90 D and DC are complementary, since D C m D + m DC = = 90 Supplementary angles are a pair of angles that, together, form a straight line Supplementary angles measures have a sum of 180 D D DC are supplementary, since C m D + m DC = = 180 Here is a great way to remember complementary and supplementary Look at the first letters of complementary and supplementary C S The c for complementary can be made into a 9 for 90 The s for supplementary can be made into an 8 for 180 C S nother way to remember it is that 90 comes before 180, just as c comes before s in the alphabet Math On the Move 8

11 Determine whether the angles are complementary, supplementary, or neither Show the work you used to make your decision 12 D and DC 45 D 45 C K 35 13) JMK and KML J 145 M 14) JML and KML L There is one more type of angle to discuss When two lines intersect, the angles opposite each other are called vertical angles In the diagram, 1 and 3 are vertical angles So are 2 and 4 Math On the Move Lesson 17 9

12 Let s take a closer look at vertical angles Lines and intersect to form 1, 2, 3, We notice that 1 and 2 are supplementary Since they are supplementary, the sum of their measures is 180 That means that m 1+ m 2 = 180 We can use this fact to say that the measure of angle 1 is, m 1 = 180 m 2 We also notice that 3 and 2 are supplementary ecause they are supplementary, the sum of their measures is 180 as well Once again, this means that m 3 + m 2 = 180 We can use this to say that the measure of angle 3 is, m 3 = 180 m 2 Look at this We just showed that the measures for angles 1 and 3 are the same m 1 = 180 m m 3 = 180 m ecause the measures of angles 1 and 3 are both equal to the same thing, we can say that 2 2 m 1 = m 3 Math On the Move 10

13 This method will always work If you found f it confusing, that s okay Just remember this fact Vertical angles always have equal measure! The following figure f shows the intersections of 3 line segments to form angles 1 through 12 Identify all vertical angles, then state which angles are equal in measure Vertical angles: 16) ngles equal in measure: Math On the Move Lesson 17 11

14 Sometimes, the relationship between two lines has a special name Two lines are perpendicular,, if they intersect to form four right angles m l Line l is perpendicular to line m We write this as, l m Notice that we only need to write one right angle symbol This is due to our supplementary angle property See if you can understand how this works yourself Lines, segments,, or rays are parallel if,, when extended forever, they never touch p q Lines p and q are parallel To show this, we write p q 1 Decide whether the following lines, line segments,, or rays are perpendicular, parallel, or neither 17 Lines a and b a b Math On the Move 12

15 18 Lines l and m m n l 19 Line segments and CD D C These definitions are used all the time in geometry, so be sure you are familiar with them Review 1 Highlight the following definitions: a point b line c ray d line segment e intersection f angle g vertex h right angle i acute angle j obtuse angle k straight angle Math On the Move Lesson 17 13

16 l complementary angles m supplementary angles n vertical angles o perpendicular p parallel 2 Write one question you would like to ask your mentor, or one new thing you learned in this lesson Practice Problems Math On the Move Lesson 17 Directions: Write your answers in your math journal Label this exercise Math On the Move Lesson 17, Set and Set Set 1 re the following angles acute, obtuse, right, or straight? a) b) c) d) Math On the Move 14

17 2 Identify the perpendicular and parallel lines Then state which angles are vertical angles k n E l C D F G m K H I J L Set 1 Look around your house, or outside Find 2 examples of each of the following angles: acute angle right angle obtuse angle straight angles Now look for the following lines: intersecting lines or line segments perpendicular lines parallel lines 1 line segment GH or HG 2 point Q 3 line TW or WT 4 ray SI 5 acute 6 obtuse 7 right 8 obtuse Math On the Move Lesson 17 15

18 9 straight 10 obtuse 11 obtuse 12 complementary m D = 45 ; m DC = = supplementary m JMK = 135 ; m KML = = neither 15 1 and 3, 2 and 4, 5 and 7, 6 and 8, 9 and 11, 10 and ngles with the same measure are all the vertical angle pairs 17 Parallel 18 Perpendicular 19 Neither End of Lesson 17 Math On the Move 16

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