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GEOMETRY Lesson 3 NYS COMMON CORE MATHEMATICS CURRICULUM Name:_________________________________ M1 Date:________________ Lesson 3: Bisect an Angle Opening Exercise In the following figure, circles have been constructed so that the endpoints of the diameter of each circle coincide with the endpoints of each segment of the equilateral triangle. a. What is special about points π·, πΈ, and πΉ? Explain how this can be confirmed with the use of a compass. Μ Μ Μ Μ , and πΉπ· Μ Μ Μ Μ . What kind of triangle must β³ DEF b. Draw Μ Μ Μ Μ π·πΈ , πΈπΉ be? c. What is special about the four triangles within β³ ABC? d. How many times greater is the area of β³ ABC than the area of β³ CDE? Discussion Define the terms angle, interior of an angle, and angle bisector. Angle: An angle is Draw examples of angles based on the definition: Interior: The interior of angle β π΅π΄πΆ is the set of points in the intersection of the half-plane of β‘π΄πΆ that contains π΅ and the half-plane of β‘π΄π΅ that contains C. The interior is easy to identify because it is always the βsmallerβ region of the two regions defined by the angle (the region that is convex). The other region is called the exterior of the angle. Draw an example of the definition of the interior of an angle: Angle Bisector: If πΆ is in the interior of β π΄ππ΅, Draw an example of the definition of an angle bisector: When we say πβ π¨πΆπͺ = πβ πͺπΆπ©, we mean that the angle measures are equal. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.12 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Geometry Assumptions In this lesson, we move from working with line segments to working with anglesβ specifically with bisecting angles. Before we do this, we need to clarify our assumptions about measuring angles. These assumptions are based upon what we know about a protractor that measures up to 180Λ angles: 1. To every angle β π΄ππ΅ there corresponds a real number πβ π΄ππ΅ called the degree or measure of the angle so that π° < πβ π¨πΆπ© < πππ°. This number, of course, can be thought of as the angle measurement (in degrees) of the interior part of the angle, which is what we read off of a protractor when measuring an angle. In particular, we have also seen that we can use protractors to βadd anglesβ: 2. If πΆ is a point in the interior of β π΄ππ΅, then πβ π¨πΆπͺ + πβ πͺπΆπ© = πβ π¨πΆπ© . Draw an example: Two angles β π΅π΄πΆ and β πΆπ΄π· form a linear pair if π΄π΅ and π΄π· are opposite rays on a line, and π΄πΆ is any other ray. In earlier grades, we abbreviated this situation and the fact that the angles on a line add up to 180Λ as, ββ π on a line.β Now we state it formally as one of our assumptions: 3. If two angles β π΅π΄πΆ and β πΆπ΄π· form a linear pair, then they are supplementary, i.e., πβ π©π¨πͺ + πβ πͺπ¨π« = πππ°. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Example 1: Investigate How to Bisect an Angle (Teacher will demo first, observe, do not write yet!) Letβs list the steps needed to bisect an angle: __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ You will need a compass and a straightedge. Letβs bisect an angle together: a. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.14 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Now you try: b. c. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.15 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. d. e. f. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.16 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Name:______________________________________ Date:__________________ Geo M1L3 HW Bisect Angles Period:________________ Consider how the use of circles aids the construction of an angle bisector. Be sure to label the construction as it progresses and to include the labels in your steps. Experiment with the angles below to determine the correct steps for the construction. 1. What steps did you take to bisect an angle? List the steps below: Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2. 3. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.18 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Name:_____________________________ Date:___________________ Opening Review Exercise: Define an angle:____________________________________________________________________________ Draw an angle & bisect an angle: Example 2: Investigate How to Copy an Angle (Teacher will demo first, observe, do not write yet!) Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.19 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. You will need a compass and a straightedge. You and your partner will be provided with a list of steps (in random order) needed to copy an angle using a compass and straightedge. Your task is to place the steps in the correct order, then follow the steps to copy the angle below. Steps needed (in correct order): 1. 2. 3. 4. 5. 6. 7. 8. 9. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.20 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Name:______________________________ Geometry M1L3 HW Copy an Angle Date:________________ Period:______________ Directions: Copy each angle below. 1. 2. 3. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Copy and Bisect an Angle 9/29/15 S.21 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.