Unit Activity Answer Sheet

Transcription

1 Geometry Unit Activity Answer Sheet Unit: Congruence, Proof, and Constructions This Unit Activity will help you meet these educational goals: Mathematical Practices You will reason abstractly and quantitatively and construct viable arguments. STEM You will apply mathematical and technology tools and knowledge to grow in your understanding of mathematics as a creative human activity. Introduction In this Unit Activity, you will extend your understanding of proofs and constructions by examining triangle congruence criteria and performing constructions to inscribe shapes in a circle. Directions and Analysis Task : Criteria for Congruent Triangles In this unit, you learned about three acceptable criteria for identifying congruent triangles: angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS). Let s examine some criteria other than these combinations and determine why they do or do not work as a means for establishing triangle congruence. To show that a criterion does not work, you must find a counterexample in which the criterion fails; that is, two or more noncongruent triangles can be constructed with the given measurements. You will create any counterexamples using the GeoGebra geometry tool. If you need help, follow these instructions for using GeoGebra. a. Determine whether side-side-angle (SSA) is a valid means for establishing triangle congruence. In this case, you know the measure of two adjacent sides and the angle opposite to one of them. If it is a valid criterion, explain why. If it is not valid, use GeoGebra to create a counterexample demonstrating that it doesn t work and give an explanation. (Hint: Try constructing a triangle where the known angle is opposite to the shortest known side.) If you construct a counterexample, paste a screenshot of your work in the space below. In the given triangle, AC > AB, and the known angle is at C. As the diagram shows, I can construct two triangles with the same measurements, ΔEFD and ΔEGD. Only one of them, ΔEGD, is congruent to ΔABC. Therefore, SSA is not enough to prove triangle congruence. 203 EDMENTUM, INC.

2 b. Determine whether angle-angle-angle (AAA) is a valid means for establishing triangle congruence. If it is a valid criterion, explain why. If it is not valid, use GeoGebra to create a counterexample demonstrating that it doesn t work and give an explanation. If you construct a counterexample, paste a screenshot of your work in the space below. As the figure shows, I can create many triangles with the same angle measurements by dilating ΔABC. ΔFEG and ΔIHJ have the same angles as ΔABC but different side lengths. Therefore, the AAA condition does not prove triangle congruence. 2

3 c. Determine whether side-angle-angle (SAA) is a valid means for establishing triangle congruence. In this case, you know the measure of a side, an adjacent angle, and the angle opposite to the side. If it is a valid criterion, explain why. If it is not valid, use GeoGebra to create a counterexample demonstrating that it doesn t work and give an explanation. If you construct a counterexample, paste a screenshot of your work in the space below. The SAA criterion is enough to prove that two triangles are congruent. In two triangles, ΔABC and ΔDEF, I start by assuming that two of the angles are congruent (i.e., have equal measures): m ABC = m DEF, and m BCA = m EFD. By the Triangle Sum Theorem: mabc + mbca + mcab = 80, and mdef + mefd + mfde = 80. By the Substitution Property of Equality: mcab = mfde. Because I know that all three angles and one side of each triangle are congruent, I can now use the ASA criterion to prove that ΔABC and ΔDEF are congruent. Task 2: Geometric Constructions In this unit, you saw how to make basic geometric constructions. One type of construction that arises in geometry is inscribing shapes in circles. For each part of this task, go online to find a procedure for using geometric constructions to inscribe the specified shape in a circle. The procedure should require only a compass and a straightedge. Write down the process for performing the construction. Then open the GeoGebra geometry tool, and create a circle of your choice. Perform the needed steps to inscribe the specified shape inside the circle. The steps you use in GeoGebra might not follow the procedure exactly based on the available tools. a. Outline the process for inscribing an equilateral triangle in a circle. Perform the construction in GeoGebra, and paste a screenshot of the construction below. To inscribe an equilateral triangle in a circle, follow these steps: Set the compass to the radius of the circle. Mark a random point on the circle. Place the needle of the compass on the point, and mark another point on the circle. Perform the previous step using the new point, and repeat until there are six equally spaced points on the circle. Using a straightedge, connect every other point (there will be three points) on the circle to create an equilateral triangle. 3

4 b. Outline the process for inscribing a square in a circle. Perform the construction in GeoGebra, and paste a screenshot of the construction below. To inscribe a square in a circle, follow these steps: Draw a straight line through the center of the circle to create a diameter, and mark the points of intersection on the circle. Set the compass to a length greater than the radius of the circle. Place the needle of the compass on an endpoint of the diameter, and draw a circle. Do the same at the other endpoint of the diameter. Draw a line passing through the two points where the new circles intersect. This will create a line perpendicular to the first diameter. The lines will intersect at the center of the circle. Mark the two points where this new perpendicular line intersects the original circle. There are now four points marked on the original circle. Using a straightedge, connect the four points to create a square. 4

5 c. Outline the process for inscribing a regular hexagon in a circle. Perform the construction in GeoGebra, and paste a screenshot of the construction below. To inscribe a regular hexagon in a circle, follow these steps: Set the compass to the radius of the circle. Mark a random point on the circle. Place the needle of the compass on the point, and mark another point on the circle. Perform the previous step using the new point, and repeat until there are six points on the circle. Using a straightedge, connect the points on the circle to create a regular hexagon. 5

6 Resources Document any references you used for this project below. At minimum, include a title and URL for any Internet resource: Evaluation This project will be evaluated on a rubric that is based on the completeness, clarity, and thinking you exhibit in the Directions and Analysis section above. Total Points: 0 Task : Criteria for Congruent Triangles a. Evaluate SSA with regard to triangle congruence. b. Evaluate AAA with regard to triangle congruence. c. Evaluate SAA with regard to triangle congruence. Task points: 3 Task 2: Geometric Constructions a. Construct an equilateral triangle inscribed in a circle. b. Construct a square inscribed in a circle. c. Construct a regular hexagon inscribed in a circle. Task points: