My Notes. Activity 31 Quadrilaterals and Their Properties 529

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1 Quadrilaterals and Their Properties 4-gon Hypothesis Learning Targets: Develop properties of kites. Prove the Triangle idsegment Theorem. SUGGSTD LRNING STRTGIS: reate Representations, Think-Pair-Share, Interactive Word Wall, Discussion Groups r. ortez, the owner of a tile store, wants to create a database of all of the tiles he sells in his store. ll of his tiles are quadrilaterals, but he needs to learn the properties of different quadrilaterals so he can correctly classify the tiles in his database. r. ortez begins by exploring convex quadrilaterals. The term quadrilateral can be abbreviated quad. 1. Use given quad GO. a. List all pairs of opposite sides. G G and O, G and O b. List all pairs of consecutive sides. G and O, G and G, G and O, O and O O DI VOULRY database is an organized collection of information stored on a computer. database for a tile store could include information about each type of tile the store sells. Investigative ctivity Standards Focus In ctivity 31, students investigate quadrilaterals including kites, trapezoids, parallelograms, rectangles, rhombuses, and squares. They prove and apply theorems including the Triangle idsegment Theorem and theorems about parallelograms, such as opposite sides are congruent, opposite angles are congruent, and the diagonals bisect each other. PLN aterials: Tracing or patty paper Pacing: 1 class period hunking the Lesson #1 # #3 #4 6 #10 Lesson Practice TH 017 ollege oard. ll rights reserved. c. List all pairs of opposite angles. G and O, and d. List all pairs of consecutive angles. G and, G and, and O, O and e. Draw the diagonals, and list them. GO and. heck students drawings. ommon ore State Standards for ctivity 31 HSG-O..3 HSG-O..8 HSG-O..9 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. xplain how the criteria for triangle congruence (S, SS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. ell-ringer ctivity is the midpoint of. For each pair of points and, find the coordinates of. 1. (3, 4) and (1, 6) [(, 5)]. (, 3) and ( 4, 3) [( 1, 0)] 3. ( 5, 6) and (, 1) [( 3.5,.5)] 1 ctivating Prior Knowledge, Think-Pair-Share, Debriefing, reate Representations ll terms and notations should be familiar to students. Review as needed, reminding students of the meanings of the words consecutive and opposite by using them in nonmathematical sentences: onday, Tuesday, and Wednesday are consecutive days in the week. So, consecutive sides of a quadrilateral are next to each other. If you sit opposite someone at a table, you are not sitting next to them. So, opposite sides of a quadrilateral are not next to each other. ctivity 31 Quadrilaterals and Their Properties 59

2 ontinued Developing ath Language any of the new vocabulary words in this activity are names of specific quadrilaterals. The first of these is a kite. If a quadrilateral is a kite, then its four sides can be grouped as exactly two distinct pairs of congruent consecutive sides. LL Support Review the word kite and discuss its different meanings. any students may be familiar with toy kites and see similarities between those and the quadrilateral KIT in Item. While this association can serve as a starting point, emphasize that in this item, a kite is a type of quadrilateral. ctivating Prior Knowledge, Think-Pair-Share, Discussion Groups, Interactive Word Wall, Group Presentations, Debriefing In part a, students are expected to identify the congruent triangles and provide the reason for the congruence. In part b, students write an informal proof (convincing argument) explaining why the diagonals are perpendicular. In part c, students list the properties of a kite for the first time. Debriefing the list of properties will help to direct students thinking later in this activity. 3 ritique Reasoning, Debriefing Suggest that students support their argument with a diagram. Remind them to label their kites. DI VOULRY The word distinct means recognizably different. In a kite, the lengths of one pair of congruent sides are always different than the lengths of the other pair of congruent sides. kite is a quadrilateral with exactly two distinct pairs of congruent consecutive sides. K I X. Given quad KIT with KI K and IT T. a. One of the diagonals divides the kite into two congruent triangles. Draw that diagonal and list the two congruent triangles. xplain how you know the triangles are congruent. KIT KT by SSS b. Draw the other diagonal. xplain how you know the diagonals are perpendicular. KXI KX by SS and IXK XK (PT). If two congruent angles form a linear pair, they are right angles. Perpendicular lines are formed by right angles. c. omplete the following list of properties of a kite. Think about the angles of a kite as well as the sides. 1. xactly two pairs of consecutive sides are congruent.. One diagonal divides a kite into two congruent triangles. 3. The diagonals of a kite are perpendicular. 4. The other diagonal divides a kite into two isosceles triangles. 5. One pair of opposite angles are congruent angles. 6. One diagonal bisects a pair of opposite angles. 3. ritique the reasoning of others. r. ortez says that the diagonals of a kite bisect each other. Is r. ortez correct? Support your answer with a valid argument. No. Sample answer: s a counterexample, suppose a kite is placed on a plane with vertices at (0, 5), (, 7), (, 0), and (4, 5). One diagonal has endpoints at (0, 5) and (4, 5). The other has endpoints at (, 7) and (, 0). The diagonals intersect at (, 5). One segment of the diagonal from (, 7) to (, 0) is 5 units long and one is units long, so that diagonal is not bisected. ommon ore State Standards for ctivity 31 () HSG-O..10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; 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. HSG-O..11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. HSG-SRT..5 Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures. T 017 ollege oard. ll rights reserved. 530 Springoard Integrated athematics I, Unit 6 Triangles and Quadrilaterals

3 Think about a circle. You can rotate the circle around its center and the resulting image will exactly resemble the preimage of the circle. The circle can be mapped onto itself by any rotation around its center. 4. an you map a circle onto itself using a reflection? xplain. Yes; a circle can be reflected across any diameter, and the image will be the same as the pre-image. onsider quadrilaterals GO and KIT, shown below. Use these quadrilaterals for Items 5 6. G I ontinued Developing ath Language ncourage students to build upon their previous understanding of midpoints as they read and analyze the definition of the new vocabulary word midsegment. Differentiating Instruction Some students may benefit from drawing the definition of a midsegment: the segment whose endpoints are the midpoints of two sides of a triangle. Have students draw a triangle, find the midpoints of two sides, and draw a segment connecting them the midsegment! 017 ollege oard. ll rights reserved. O 5. Tell whether you can map quad GO onto itself using the given transformation. a. single reflection No; Sample explanation: The quadrilateral has an irregular shape, so no reflection maps it onto itself. b. single rotation Yes; you can rotate GO around any point using a 360 rotation. 6. Tell whether you can map quad KIT onto itself using the given transformation. a. single reflection Yes; you can reflect KIT over KT. b. single rotation Yes; you can rotate KIT around any point using a 360 rotation. 7. Why is a square not considered a kite? 8. The diagonals of quadrilateral D are perpendicular to each other. One diagonal is longer than the other diagonal. What type of quadrilateral is D? an it be mapped onto itself using a single reflection? xplain. 9. Suppose and D are the diagonals of a kite. What is a formula for the area of the kite in terms of the diagonals? K T TH TIP For Items 5 and 6, you can use tracing paper to copy each quadrilateral. 4 6 Visualization, Use anipulatives, Discussion Groups Suggest that students visualize rotations and reflections involving circles as they answer the questions about rotations and reflections of kites. When rotating the kites, students may think that only points inside the figure or only the intersection point of the diagonals should be used for the 360 rotation, but any point can be used. Students may benefit from turning and/or folding the pieces of paper with their tracings of the kites to further investigate the rotations and reflections. ncourage them to share with other students their strategies to use circles to help them solve problems involving rotations and reflections of other figures. Note that students will revisit reflections and rotations of quadrilaterals in subsequent lessons in ctivity 31 and will come to more general conclusions about transformations of figures (e.g., all figures can be mapped onto themselves by 360 rotations). Later they will look at other figures, such as regular polygons. Debrief students answers to these items to ensure that they understand concepts related to kites. When students answer Item 5, you may also want to have them explain how they arrived at their answer. nswers 7. kite does not have four congruent sides and four congruent angles, so it is not a square. 8. kite or rhombus; if a kite, D can be mapped onto itself using a reflection over the longer diagonal. If a rhombus, D can be mapped onto itself using a reflection over either diagonal. 9. = 1 D ( )( ) ctivity 31 Quadrilaterals and Their Properties 531

4 ontinued 10a Think-Pair-Share, Interactive Word Wall, Debriefing Debriefing this item is essential. Students need to know exactly what they are given and asked to prove in the next item. xplain that they will be completing the proof as they answer the remaining parts of Item b f ctivating Prior Knowledge, reate Representations, Think-Pair- Share, Debriefing Students may have issues with the algebra. irculate as students work to help them with concerns. Debrief students answers to these items to ensure that they understand concepts related to midsegments of triangles. When students answer each item, remind them to explain how they arrived at their answers. SSSS Students answers to the Lesson Practice items will provide a formative assessment of their understanding of kites and the Triangle idsegment Theorem, and of students ability to apply their learning. Short-cycle formative assessment items for are also available in the ssessment section on Springoard Digital. Refer back to the graphic organizer the class created when unpacking mbedded ssessment. sk students to use the graphic organizer to identify the concepts or skills they learned in this lesson. DPT heck students answers to the Lesson Practice to ensure that they understand basic concepts related to kites and triangle midsegments, and are ready to transition to investigating trapezoids. Some students may need clarification of the difference between a midsegment and a midpoint. Remind them that the midsegment is always a segment. nd a midpoint is always a point. See the ctivity Practice on page 547 and the dditional Unit Practice in the Teacher Resources on Springoard Digital for additional problems for this lesson. You may wish to use the Teacher ssessment uilder on Springoard Digital to create custom assessments or additional practice. TH TIP Given (x 1, y 1 ) and (x, y ). idpoint Formula: x1 x y1 y = +, + Slope of : m = ( y y1 ) ( x x1) Distance Formula: = ( x x1) + ( y y1) TH TIP For Item 10f, look at the expression for. egin by rewriting the part of the expression inside the radical symbol. Use this fact: (c a) = [(c a)] = (c a) = 4(c a) The segment whose endpoints are the midpoints of two sides of a triangle is called a midsegment. Triangle idsegment Theorem The midsegment of a triangle is parallel to the third side, and its length is one-half the length of the third side. 10. Use the figure and coordinates below to complete the coordinate proof for the Triangle idsegment Theorem. (a, h) (b, k) N (c, l) a. omplete the hypothesis and conclusion for the Triangle idsegment Theorem. Hypothesis: is the midpoint of. N is the midpoint of. onclusion: N N = 1 b. Find the coordinates of midpoints and N in terms of a, b, c, h, k, and l. (a + b, h + k) and N(b + c, k + l) c. Find the slope of and N. slope of segment : m = l h; slope of segment N: l h c a m = c a d. Simplify your response to part c and explain how your answers to part c show N. m = l c h a for both segments. The segments are parallel because they have the same slope. e. Find and N. = ( c a) + ( h l) ; N = ( c a) + ( h l) f. xplain how you know that = N. Hint: ompare the radicands of the expressions you wrote in part e. The radicand for is 4 times the radicand for N. That means that is times N. 017 ollege oard. ll rights reserved. 53 Springoard Integrated athematics I, Unit 6 Triangles and Quadrilaterals

5 11. re the midsegments of an isosceles triangle congruent? xplain. 1. Given D. Is D a midsegment of triangle? xplain. 5 5 D Triangle PQR with midsegments and is shown. a. Verify the Triangle idsegment Theorem using the coordinates of 4 PQR and midsegment. b. Verify the Triangle idsegment R Theorem using the coordinates of PQR and midsegment. 4 Q P ontinued nswers 11. No. Only the two segments parallel to the legs are congruent. 1. No. D does not bisect sides and. 13. a. Sample answer: Since is the midpoint of RQ, is (4, ). Since is the midpoint of PQ, is (7, 1). The slope of RP = 0 = = The slope of f = = =. 3 3 Since they have the same slope, the segments are parallel. RP = ( 8) + ( 0 ( )) = 40 = ollege oard. ll rights reserved. LSSON 31-1 PRTI 14. XY is a midsegment of triangle DF. Find each measure. XY = DX = X 6 YF = D 4 Y 15. QR is a midsegment of triangle WYZ. Find each measure. W x = WZ = Z QR = x + 7 x.5 Q R Y 16. ake sense of problems. Figure D is a kite with diagonals D and. omplete each statement. D D 17. Use kite D from Item 16. Describe how to map D onto itself using a single reflection. 17 F = ( 4 7) + ( 1) = 10 = 1 RP b. Sample answer: Since is the midpoint of RP, is (5, 1). Since is the midpoint of PQ, is (7, 1). The slope of RQ = = 4 =. The slope of = 1 ( 1 ) = =. Since they have the same slope, the segments are parallel. RQ = ( 6) + ( 0 4) = 3 = 4 = ( 5 7) + ( 1 1) = 8 = = 1 RQ LSSON 31-1 PRTI 14. XY = 8.5; DX = 6; YF = x = 6; WZ = 19; QR = D ; D; D; D; D 17. Reflect the triangle over the segment connecting point and the intersection point of D and. ctivity 31 Quadrilaterals and Their Properties 533