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Transcript
Geometry Items to Support Formative Assessment Unit 2: Triangles, Proof, and Similarity Part A: Triangle Relationships, Congruence, and Proof Prove geometric theorems. G.CO.C.9 Prove theorems about lines and angles. Theorems include: points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G.CO.C.9 Task Segment BD is the perpendicular bisector of segment AC. Prove that triangle ABC is an isosceles triangle. Solution: Note: students could use proof blocks, write a paragraph, or use a two column proof. The two column proof is shown below, but students do not have to use this method as long as they have the same content shown in another method. Statements Reasons Segment BD is the perpendicular bisector of segment AC. Given Angle ADB and Angle CDB are right angles Definition of perpendicular bisector Angle ADB is congruent to Angle CDB All right angles are congruent Segment AD is congruent to Segment DC Definition of perpendicular bisector BD is congruent to BD Reflexive Property Triangle ADB is congruent to triangle CDB SAS 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. AB is congruent to BC CPCTC Triangle ABC is an isosceles triangle Two sides are congruent in any isosceles triangle G.CO.C.9 Item 1 Use the diagram below. Verify that the points on the perpendicular bisector are equidistant from the endpoints A and B. Solution: Use a piece of patty paper, a straight edge, or CPCTC to confirm the distance AE is equal to the distance BE. Repeat the process for points C, F, G. G.CO.C.9 Item 2 What do you know about the two triangles created by the perpendicular bisector in the diagram below? Explain your reasoning. 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: The points on the perpendicular bisector are equidistant from the endpoints. Therefore, the corresponding sides are congruent. The perpendicular bisector creates right angles and is congruent to itself. Therefore, the two triangles are right triangles. Prove geometric theorems. (This standard will be embedded throughout this unit) G.CO.C.10 Prove theorems about triangles. (Theorems include: 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.) G.CO.C.10 Task When your geometry teacher asked your class to prove that the base angles of an isosceles triangle are congruent, one group created the poster shown below. Several members of your group have questions about this group’s solution. How would you explain each of these points to your group? How do we know that line DC is the perpendicular bisector of segment AB? How do we know that triangle ACD is congruent to triangle BCD? Possible solution: Since segment AC is congruent to segment BC we know that point C is on the perpendicular bisector of segment AB. So line CD is the perpendicular bisector of segment AB. Triangle ACD is congruent to triangle BCD by SSS. 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.