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COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Name__________________________________________ Period______ Date____________ Lesson 2: Bisect and Copy an Angle Classwork Opening Exercise A 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. Use different colors to show each circle. a. F E What can you conclude about CD and DB? Name two other segment pairs that have the same properties. C D B What can you conclude about all 6 segments? b. What is special about points π«, π¬, and π? Explain how this can be confirmed with the use of a compass. c. Draw DE, EF, and FD. What kind of triangle must β³ DEF be? d. What is special about the four triangles within β³ ABC? e. How many times greater is the area of β³ ABC than the area of β³ CDE? 1 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Discussion Define the terms angle, interior of an angle, and angle bisector. B A C Angle: An angle is Interior: The interior of angle β π΅π΄πΆ is the set of points in the intersection of the half-plane (all points on one side of an infinite straight line) of π΄πΆ that contains π΅ and the half-plane of π΄π΅ that contains πΆ. 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. Angle Bisector: If πΆ is in the interior of β π΄ππ΅, When we say β π¨πΆπͺ = β πͺπΆπ©, we mean that the angle measures are equal and that β π¨πΆπͺ can either refer to the angle itself or its measure when the context is clear. 2 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY 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 |β π΄ππ΅| called the degree or measure of the angle so that 0 < |β π΄ππ΅| < 180. 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 |β π΄ππΆ| + |β πΆππ΅| = |β π΄ππ΅|. (Abbreviation: β π add.) βββββ are opposite rays on a line, and π΄πΆ βββββ is any other ray. In Two angles β π΅π΄πΆ and β πΆπ΄π· form a linear pair if βββββ π΄π΅ and π΄π· 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., |β π΅π΄πΆ| + |β πΆπ΄π·| = 180. (Abbreviation: β π on a line Example 1: Investigate How to Bisect an Angle You will need a compass and a straightedge. Joey and his brother, Jimmy, are working on making a picture frame as a birthday gift for their mother. Although they have the wooden pieces for the frame, they need to find the angle bisector to accurately fit the edges of the pieces together. Using your compass and straightedge, show how the boys bisected the corner angles of the wooden pieces below to create the finished frame on the right. Before After 3 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Letβs try what the carpenter did in the video in the angle below: B A C What steps did you take to bisect an angle? List the steps below: 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. 4 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Example 2: Investigate How to Copy an Angle You will need a compass and a straightedge. As a class we will develop steps needed to copy an angle using a compass and straightedge. B A C 5 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Relevant Vocabulary Midpoint: A point π΅ is called a midpoint of a segment π΄πΆ if π΅ is between π΄ and πΆ, and π΄π΅ = π΄πΆ. Degree: Subdivide the length around a circle into 360 arcs of equal length. A central angle for any of these arcs is called a one-degree angle and is said to have angle measure 1 degree. An angle that turns through π one-degree angles is said to have an angle measure of π degrees. Zero and Straight Angle: A zero angle is just a ray and measures 0Λ. A straight angle is a line and measures 180Λ (the Λ is an abbreviation for βdegreeβ). Problem Set Directions: Bisect each angle below. 1. 2. 6 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY 3. 4. Directions: Copy the angle below. 5. 7 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Creating Parallel Lines by Copying an Angle: Construct a line parallel to a given line through a given point. Given line l and point P, not on l, construct a line through P, parallel to l. 1. Draw a line through P, so that it intersects line l. Label this point A. [figure 1] 2. Open the compass to form an arc below point P, set the compass at A, and make an arc so that it intersects β‘ββββ π΄π (label this point B) and line l (label this point C). Using the same compass opening, make an identical arc with the compass set at P, so that the arc intersects β‘ββββ π΄π Label this point Q. [figure 2] 3. Set the compass at B, and make an arc so that it intersects line l at point C. Using the same compass opening, make an identical arc with the compass set at Q so that it intersects the arc you made in step 3. Label this point R. [figure 3] β‘ββββ . What type of angles did you form? β‘ββββ . Note ππ β‘ββββ || π΄πΆ 4. Draw ππ Fi g ure 1 Fi g ure 2 Q P P B A A l C l Fi g ure 3 Q Figure 4 Q P P R R B B A C l A C l 8 COMMON CORE MATHEMATICS CURRICULUM Bisect and Copy an Angle Lesson 2 M1 GEOMETRY Now try this on your own beginning with the line segment below: 1