Geometry Chapter 2 Segments and Angles

Transcription

1 Geometry Chapter 2 egments and Angles Key Concepts: By the end of this lesson, you should be able to understand and define:! The measure of a line segment! A coordinate! The distance between two points! Betweenness! Midpoints! Congruent segments! Angles! Bisectors! Congruent Angles Now that you have learned the basics of geometry, it is time to discover what the concepts you ve learned so far can do. We are still learning the fundamental knowledge to grasp this subject, but we are well on our way to really understanding this new type of math! Let's see what Lesson 2 has to offer. ection 2-1 In this section, we are learning about line segments. As you are learning, segments have a lot of useful properties. Take a look at these theorems and postulates and then we will try to work out some problems. Postulate 8 - The Ruler Postulate a. To every distinct pair of points there corresponds exactly one positive number. This number is the distance between the two points. b. To every real number there corresponds a point on the number line. To every point on the number line there corresponds a real number. Theorem 2-1 ingular Midpoint Theorem A segment has exactly one midpoint.

2 Theorem 2-2 Midpoint Theorem If M is the midpoint of AB, then AM =.AB Use this diagram to answer all of the questions for section 2-1. A B C D E What is the distance between 6 and 1? The distance is the number of spaces between the 6 mark on the number line and the 1 mark on the number line. If you count the number of spaces, you ll find that the distance between these two numbers is 7. You can also get this answer by taking the absolute value of the difference of the two numbers. 6 1 = 7 Find BD. This question is asking you to find the length of the line segment BD. o first we have to figure out where these points are on the number line. Point B is at -1 and point D is at 6. o we can either count the spaces between points B and D or take the absolute value of the difference. 1 6 = 7

3 Find a pair of congruent segments on the number line. Congruent segments are segments of the same length. AB and BC are three units long. Therefore, they are congruent! What is the midpoint of BE? By definition, the length of the midpoint of BE is 2 1 BE. First, we should find the value of BE. The coordinate of B is 1 and the coordinate of E is 7, so BE=8. o the length of the midpoint would be 4 (because 4 is half of 8). Then we have to add 4 to the coordinate of B. o the midpoint is 3 (because 1+4=3). ection 2-2 In the last section we learned about segments. Now we are going to take that one step further and learn about angles. Let s see what we can learn! And remember, if at any point you are not sure of what something means, you can check the Key Concepts. Here we go! Postulate 9 The Protractor Postulate a. To every angle there corresponds exactly one real number between and 18. This number is the measure of the angle. b. Let C be the endpoint of opposite rays CA and CB such that CA is paired with and CB is paired with 18. If P is a point in the plane where P is not on AB and n is a real number such that < n < 18, then there is exactly one ray CE, with E on the same side of AB as P, such that CE is paired with n and m ACE= n.

4 Theorem 2-3 ingular Bisector Theorem An angle has exactly one bisector. Theorem 2-4 Angle Bisector Theorem If AY is the bisector of XAZ, then m XAY = 1 2 m XAZ. Use this diagram to answer the following 2 questions. E U Find the measure of HO. H O HO is an angle that goes from to 8 degrees. o the measure of this angle is 8 degrees U H O Find the measure of EHU E EHU goes from 3 degrees to 9 degrees. o the measure of the angle is the difference between 3 and 9. o this angle is 6 degrees 9 H 8 3 U O

5 If Q bisects TQR and m = 4 Find m. TQR QR degrees, T ince Q bisects TQR QR = TQR = 2 QR. o m TQR, it cuts the degrees of the angle in half. ince m 4, then m m is 8 degrees. Q R We are now finished with Lesson 2! You should now be able to answer all of the problems for your submission. In the next lesson we are going to look at different types of angles. It should be really interesting and lots of fun. GLOARY: ANGLE An Angle is the figure formed by two non-collinear rays having the same endpoint. BETWEENNE OF POINT FOR A CIRCLE If B is a point on arc AC, then the sum of the measure of arc AB and the measure of arc BC is equal to the measure of arc AC. A B C

6 BIECTOR OF A LINE EGMENT The bisector of a line segment divides the segment into two equal halves. The bisector forms two equal (congruent) halves. bisector CONGRUENT ANGLE Congruent angles are angles whose measures are equal. 6 6 CONGRUENT EGMENT Congruent segments are segments of the same length. COORDINATE On a line the number that is matched with a point. DITANCE The distance, AB, between two points A and B with coordinates a and b respectively can be written as: AB = a b or b a MEAURE OF A LINE EGMENT The measure of a line segment is the length of the segment from endpoint to endpoint. The measure of segment T is 8. 8 T

7 MIDPOINT The midpoint of a line segment divides the segment into two equal halves. The midpoint is equidistant from the endpoints. midpoint