Copy and Bisect an Angle
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 𝐷, 𝐸, and 𝐹? Explain how this can be confirmed with the use of a compass. b. Draw DE, EF, and FD. What kind of triangle must △ 𝐷𝐸𝐹 be? c. What is special about the four triangles within △ 𝐴𝐵𝐶? d. How many times greater is the area of △ 𝐴𝐵𝐶 than the area of △ 𝐶𝐷𝐸?
Define the terms angle, interior of an angle, and angle bisector.
ANGLE: An angle is ____________________________________________________
INTERIOR: The interior of ∠𝐵𝐴𝐶 is the set of points in the intersection of the half-plane of AC that contains 𝐵 and the half-plane of AB that contains 𝐶. The interior is easy to identify because it is always the “smaller” 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 𝐶 is in the interior of ∠𝐴𝑂𝐵, _________________________________ When we say 𝑚∠𝐴𝑂𝐶 = 𝑚∠𝐶𝑂𝐵, we mean that the angle measures are equal.
In working with lines and angles, we again make specific assumptions that need to be identified. For example, in the definition of interior of an angle above, we assumed that an angle separated the plane into two disjoint sets. This follows from the assumption: Given a line, the points of the plane that do not lie on the line form two sets called halfplanes, such that (1) each of the sets is convex, and (2) if 𝑃 is a point in one of the sets, and 𝑄 is a point in the other, then the segment 𝑃𝑄 intersects the line.
From this assumption, another obvious fact follows about a segment that intersects the sides of an angle: Given an ∠𝐴𝑂𝐵, then for any point 𝐶 in the interior of ∠𝐴𝑂𝐵, the ray 𝑂𝐶 always intersects the segment 𝐴𝐵.
In this lesson, we move from working with line segments to working with angles, 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 180° angles:
Mathematical Modeling Exercise 1: Investigate How to Bisect an Angle
You 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.
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.
What steps did you take to bisect an angle? List the steps below:
Mathematical Modeling Exercise 2: Investigate How to Copy an Angle
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.
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