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Geometry Items to Support Formative Assessment Unit 2: Triangles, Proof, and Similarity Part A: Triangle Relationships, Congruence, and Proof Prove geometric theorems. (This standard will be embedded throughout this unit) G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.C.10 Task In class, we conducted investigations to show you that the Triangle Angle Sum Theorem holds true. Suppose you were to share what you learned about the sum of the angles in a triangle with your family and they ask you whether or not this applies to all triangles. How would you convince them that this is true for every triangle? Write a paragraph proof that explains why the Triangle Angle Sum Theorem is always true. Support each of your claims with valid reasons, using complete sentences. You may also use the sketch below, patty paper, a flowchart, or proof blocks to help develop your reasoning and thought flow. Solution: Student proofs should include the reasoning in the paragraph below in some logical sequence. Auxiliary line FA helps us prove the Triangle Angle Sum Theorem by using what we know about linear pairs and the measure of a straight angle. If you combine ∠1 and ∠2 to form a single angle, that angle forms a linear pair with ∠3. By the Linear Pair Theorem, the sum of their measures is 180 degrees, so m∠1 + m∠2 + m∠3 = 180º. Since line segments AB and AC form transversals between two parallel lines (lines FA and line segment BC), ∠1 is congruent to ∠4 and ∠3 is congruent to ∠ 5 by the Alternate Interior Angles Theorem. By substitution, m∠4 + m∠2 + m∠5 = 180º. Therefore the measures in the angles of any triangle add up to 180º. Note: To add more rigor to this task, consider requiring two different methods of proof rather than suggesting other methods to consider while thinking through it. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. G.CO.C.10 Item 1 Line DE || line AB. Find the measures of the angles in the triangle: Solution: m∠CAB = 63° m∠ABC = 55° m∠BCA = 62° Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. G.CO.C.10 Item 2 Find the sum of the angles marked in the diagram below. Justify your answers. Adapted from Discovering Geometry by Michael Serra (Key Curriculum Press 2008) Solution: Each cluster of 4 angles will have a sum of 360°. Taken together, the angle sum is 1080°. The angles from the triangle then need to be subtracted to get the final answer of 900°. G.CO.C.10 Item 3 Construct ΔPDQ if m∠P = 40°, m∠Q = 55°, and PD = 7 cm. What do you know about the sum of the interior angles of a triangle which helps you construct this triangle? Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Solution: Measure and draw segment PD, and label. Using a protractor sketch ∠P. From the sum of the interior angles theorem, we know m∠D = 85°. Sketch ∠D. The point where the sides intersect will be point Q. G.CO.C.10 Item 4 Look at the two triangles shown below. 1. Make a conjecture about ∠B and ∠ F. 2. Can you prove your conjecture? 3. If you changed one of the congruent angle pairs to different measure (but keeping them congruent), what would change in your answer to part 1? What would stay the same? 4. If you changed the measures of both congruent angle pairs (but keeping the measures in each pair congruent) what would change in your answer to part 1? What would stay the same? 5. Make a conjecture about two triangles that have two pairs of congruent angles. What will always be true about the two triangles? Sample Solutions: 1. ∠B and ∠ F are both acute; ∠B and ∠ F are both 40°.; ∠B and ∠ F are congruent 2. ∠B and ∠ F are both 40°. because in both triangles, if you subtract the other two angles from 180°, you are left with 40°. 3. If the two 30° angles changed to 20°, then ∠ B and ∠F would have to be 50°. They’d still be congruent, just bigger. 4. If the two 30° angles changed to 20° and the two 110° angles changed to 100°, then ∠B and ∠F would have to be 60°. ∠B and ∠ F would still be congruent, but bigger again. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. 5. If two triangles have two pairs of congruent angles, the third angles’ sizes will depend on the first two pairs, but they’ll be congruent. Notes: This item is designed to prove the “Third Angle Theorem.” Students may or may not realize that the triangles are similar. This fact is emphasized in the G.SRT.A.3 Task. G.CO.C.10 Item 5 Will the following sets of side lengths form triangles? Justify your answer using mathematics. 1. 5 ft, 7 ft, 6 ft 2. 2.8 in, 3.1 in, 5.9 in 3. 15 yd, 27 yd, 13 yd 4. 5 1 2 1 cm, 3 cm, 1 cm 2 3 2 Solutions: 1. Yes, because the short sides sum to 11 ft, which is greater than the longest side (7 ft). 2. No, because the short sides sum to 5.9 in, which is equal to the longest side. 3. Yes, because the short sides sum to 28 yd, which is greater than the longest side (27 ft). 1 6 4. No, because the short sides sum to 5 cm, which is shorter than the longest side ( 5 1 2 cm). G.CO.C.10 Item 6 Donald wants to build a triangular fence within a plot of land he has purchased for future development. He will build a new fence along the entire driveway, which is 80 feet in length. The previous owners left behind materials for 100 feet of fencing, which Donald will use as a second side. What is the minimum amount of fencing Donald will need to purchase to create the triangular space? If, instead, Donald would like a larger fenced area, what would be the maximum amount of fencing he could use? Solution: Donald must purchase more than 20 feet of fencing, but he cannot purchase over 180 feet of fencing. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.