Quadrilaterals and heir Properties 4-gon Hypothesis Lesson 15-1 Kites and riangle Midsegments IVIY 15 Learning argets: evelop properties of kites. Prove the riangle Midsegment heorem. SUGGS LRNING SRGIS: iscussion Groups, Shared Reading, reate Representations, hink-pair-share, Interactive Word Wall, Group Presentations Mr. 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. Mr. ortez begins by exploring convex quadrilaterals. he term quadrilateral can be abbreviated quad. G M O 1. Given quad GOM. a. List all pairs of opposite sides. b. List all pairs of consecutive sides. c. List all pairs of opposite angles. d. List all pairs of consecutive angles. e. raw the diagonals, and list them. ctivity 15 Quadrilaterals and heir Properties 205
IVIY 15 Lesson 15-1 Kites and riangle Midsegments kite is a quadrilateral with exactly two distinct pairs of congruent consecutive sides. I K 2. Given quad KI with KI K and I. a. One of the diagonals divides the kite into two congruent triangles. raw that diagonal and list the two congruent triangles. xplain how you know the triangles are congruent. b. raw the other diagonal. xplain how you know the diagonals are perpendicular. c. omplete the following list of properties of a kite. hink about the angles of a kite as well as the segments. 1. xactly two pairs of consecutive sides are congruent. 2. One diagonal divides a kite into two congruent triangles. 3. he diagonals of a kite are perpendicular. 4. 5. 6. 3. ritique the reasoning of others. Mr. ortez says that the diagonals of a kite bisect each other. Is Mr. ortez correct? Support your answer with a valid argument. heck Your Understanding 4. Why is a square not considered a kite? 5. Suppose and B are the diagonals of a kite. What is a formula for the area of the kite in terms of the diagonals? 206 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-1 Kites and riangle Midsegments IVIY 15 he segment whose endpoints are the midpoints of two sides of a triangle is called a midsegment. riangle Midsegment heorem he midsegment of a triangle is parallel to the third side, and its length is one-half the length of the third side. 6. Use the figure and coordinates below to complete the coordinate proof for the riangle Midsegment heorem. B (b, k) M N (a, h) (c, l) a. omplete the hypothesis and conclusion for the riangle Midsegment heorem. Hypothesis: M is the midpoint of. onclusion: MN N is the midpoint of. MN = b. Find the coordinates of midpoints M and N in terms of a, b, c, h, k, and l. MH IP Given (x 1, y 1 ) and B(x 2, y 2 ). Midpoint Formula: x1 x2 y1 y2 M = +, + 2 2 Slope of B : m = ( y2 y1 ) ( x2 x1) istance Formula: 2 2 1 2 2 1 B = ( x x ) + ( y y ) c. Find the slope of and MN. d. Simplify your response to part c and explain how your answers to part c show MN. ctivity 15 Quadrilaterals and heir Properties 207
IVIY 15 Lesson 15-1 Kites and riangle Midsegments e. Find and MN. f. Simplify your response to part e and explain how your answers to part e show that MN = 1 2. heck Your Understanding 7. re the midsegments of an isosceles triangle congruent? xplain. 8. Given. Is a midsegment of triangle B? xplain. B 5 5 6 6 LSSON 15-1 PRI 9. XY is a midsegment of triangle F. Find each measure. XY = X = X 6 YF = 4 Y 10. QR is a midsegment of triangle WYZ. Find each measure. W x = WZ = Z QR = 2x + 7 2x 2.5 Q R Y 17 F 11. Make sense of problems. Figure B is a kite with diagonals B and. omplete each statement. B B B B B B 208 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-2 rapezoids IVIY 15 Learning argets: evelop properties of trapezoids. Prove properties of trapezoids. SUGGS LRNING SRGIS: Visualization, Shared Reading, reate Representations, hink-pair-share, Interactive Word Wall trapezoid is a quadrilateral with exactly one pair of parallel sides. he parallel sides of a trapezoid are called bases, and the nonparallel sides are called legs. he pairs of consecutive angles that include each of the bases are called base angles. 1. Sketch a trapezoid and label the vertices, R,, and P. Identify the bases, legs, and both pairs of base angles. he median of a trapezoid is the segment with endpoints at the midpoint of each leg of the trapezoid. rapezoid Median heorem he median of a trapezoid is parallel to the bases and its length is the average of the lengths of the bases. F G M Given: rapezoid FGH MN is a median. Prove: MN FG and MN H MN = 1 (FG + H) 2 2. raw one diagonal in trapezoid FGH. Label the intersection of the diagonal with MN as X and explain below how the riangle Midsegment heorem can be used to justify the rapezoid Median heorem. H N ONN O LNGUG he British use the term trapezium for a quadrilateral with exactly one pair of parallel sides and the term trapezoid for a quadrilateral with no parallel sides. hey drive on a different side of the road, too. ctivity 15 Quadrilaterals and heir Properties 209
IVIY 15 Lesson 15-2 rapezoids 3. Given trapezoid FGH and MN is a median. Use the figure in Item 2, properties of trapezoids, and/or the rapezoid Median heorem for each of the following. a. If m GF = 42, then m NM = and m MH =. b. Write an equation and solve for x if FG = 4x + 4, H = x + 5, and MN = 22. c. Find FG if MN = 19 and H = 12. 4. Make use of structure. What property or postulate allowed you to draw the auxiliary line in Item 2? heck Your Understanding 5. How does a trapezoid differ from a kite? 6. an a trapezoid have bases that are congruent? xplain. n isosceles trapezoid is a trapezoid with congruent legs. 7. Given B is isosceles with B = B and =. B a.. xplain. b. xplain why B is isosceles. c.. xplain. d. xplain why quad is an isosceles trapezoid. e.. xplain. 210 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-2 rapezoids IVIY 15 f. omplete the theorem. he base angles of an isosceles trapezoid are. 8. On grid paper, plot quad OL with coordinates (1, 0), O(2, 2), L(5, 3), and (7, 2). a. Show that quad OL is a trapezoid. b. Show that quad OL is isosceles. c. Identify and find the length of each diagonal. d. Based on the results in part c, complete the theorem. he diagonals of an isosceles trapezoid are. 9. t this point, the theorem in Item 8 is simply a conjecture based on one example. Given the figure below, write the key steps for a proof of the theorem. Hint: You may want to use a pair of overlapping triangles and the theorem from Item 8 as part of your argument. Hypothesis: onclusion: Y OR is a trapezoid. O R R O O R ctivity 15 Quadrilaterals and heir Properties 211
IVIY 15 Lesson 15-2 rapezoids heck Your Understanding 10. Given quad PLN is an isosceles trapezoid, use the diagram below and the properties of isosceles trapezoids to find each of the following. P L N a. LPN b. If m PL = 70, then m LPN = and m PN =. c. Write an equation and solve for x if P = x and NL = 3x 8. LSSON 15-2 PRI 11. UV is a midsegment of trapezoid QRS. Find each measure. QU = VS = UV = U 3 Q 11 15 R 4 V S 12. Reason abstractly. F is a midsegment of isosceles trapezoid B. Find each measure. x = y = = = BF = F = B = F = 2y + 3 8x 43 2x 10 B F 6y 1 212 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-3 Parallelograms IVIY 15 Learning argets: evelop properties of parallelograms. Prove properties of parallelograms. SUGGS LRNING SRGIS: Visualization, reate Representations, hink-pair-share, Interactive Word Wall, iscussion Groups parallelogram is a quadrilateral with both pairs of opposite sides parallel. For the sake of brevity, the symbol can be used for parallelogram. 1. Given KY as shown. a. Which angles are consecutive to K? b. Use what you know about parallel lines to complete the theorem. K Y onsecutive angles of a parallelogram are. 2. xpress regularity in repeated reasoning. Use three index cards and draw three different parallelograms. hen cut out each parallelogram. For each parallelogram, draw a diagonal and cut along the diagonal to form two triangles. What do you notice about each pair of triangles? 3. Based upon the exploration in Item 2, complete the theorem. ach diagonal of a parallelogram divides that parallelogram into. I 4. Given parallelogram IG as shown above. omplete the theorems. a. Opposite sides of a parallelogram are. b. Opposite angles of a parallelogram are. c. Prove the theorem you completed in part a. Use the figure in Item 3. G d. Prove the theorem you completed in part b. Use the figure in Item 3. ctivity 15 Quadrilaterals and heir Properties 213
IVIY 15 Lesson 15-3 Parallelograms MH RMS 5. xplain why the theorems in Item 4 can be considered as corollaries to the theorem in Item 3. corollary is a statement that results directly from a theorem. 6. Given LUK, use the figure and the theorems in Items 1, 3, and 4 to find the following. L U K a. KL b. Solve for x if m KU = 10x 15 and m K = 6x + 3. c. Solve for x and y if KL = 2x + y, LU = 7, U = 14, and K = 5y 4x. heorem: he diagonals of a parallelogram bisect each other. 7. a. Rewrite the above theorem in if-then form. b. raw a figure for the theorem, including the diagonals. Label the vertices and the point of intersection for the diagonals. Identify the information that is given and what is to be proved. ONN O P heorems are key to the development of many branches of mathematics. In calculus, two theorems that are frequently used are the Mean Value heorem and the Fundamental heorem of alculus. Given: Prove: c. Write a two-column proof for the theorem. 214 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-3 Parallelograms IVIY 15 heck Your Understanding 8. Why are trapezoids and kites not parallelograms? 9. he measure of one angle of a parallelogram is 68. What are the measures of the other three angles of the parallelogram? 10. he lengths of two sides of a parallelogram are 12 in. and 18 in. What are the lengths of the other two sides? LSSON 15-3 PRI 11. and B are diagonals of parallelogram B. Find each measure. = = = B = 12. Make sense of problems. One of the floor tiles that Mr. ortez sells is shaped like a parallelogram. Find each measure of the floor tile. m W = m X = m Y = m Z = WZ = XY = 4a + 10 2y 3x 13 x + 1 y + 5 Z W X (b + 15) 6a (2b) Y B ctivity 15 Quadrilaterals and heir Properties 215
IVIY 15 Lesson 15-4 Rectangles, Rhombuses, and Squares Learning argets: evelop properties of rectangles, rhombuses, and squares. Prove properties of rectangles, rhombuses, and squares. SUGGS LRNING SRGIS: Visualization, reate Representations, hink-pair-share, Interactive Word Wall, iscussion Groups rectangle is a parallelogram with four right angles. 1. Given quad R is a rectangle. List all right triangles in the figure. xplain how you know the triangles are congruent. R 2. omplete the theorem. he diagonals of a rectangle are. 3. xplain how you know the theorem in Item 2 is true. 4. List all of the properties of a rectangle. Begin with the properties of a parallelogram. 5. Given quad PINK is a rectangle with coordinates P(3,0), I(0,6), and N(8,10). Find the coordinates of point K. 6. Given quad GIF is a rectangle. Use the properties of a rectangle and the figure at right to find the following. F X G I a. If X = 13, then I = and FG =. b. Solve for x if X = 4x + 4 and FX = 7x 23. c. Solve for x if m XF = 6x 4 and m XG = 10x 2. 216 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-4 Rectangles, Rhombuses, and Squares IVIY 15 Indirect proofs can be useful when the conclusion is a negative statement. xample of an Indirect Proof Given: m SR m SI Prove: RIS is not a rectangle. P R S I Statements Reasons 1. RIS is a rectangle. 1. ssumption 2. m SR = m SI = 90 2. efinition of a rectangle 3. m SR m SI 3. Given 4. RIS is not a rectangle. 4. he assumption led to a contradiction between statements 2 and 3. 7. omplete the missing reasons in this indirect proof. Given: W S Prove: Quad WISH is not a. Statements 1. WISH 1. 2. WS and HI bisect each other. Reasons 2. 3. W = S and H = I 3. 4. W S 4. 5. Quad WISH is not a. 5. heck Your Understanding 8. What do rectangles, trapezoids, and kites have in common? How do they differ? 9. ell whether each of the following statements is true or false. a. ll rectangles are parallelograms. b. Some rectangles are trapezoids. c. ll parallelograms are rectangles. d. ll rectangles are quadrilaterals. H W S I MH IP n indirect proof begins by assuming the opposite of the conclusion. he assumption is used as if it were given until a contradiction is reached. Once the assumption leads to a contradiction, the opposite of the assumption (the original conclusion) must be true. ctivity 15 Quadrilaterals and heir Properties 217
IVIY 15 Lesson 15-4 Rectangles, Rhombuses, and Squares rhombus is a parallelogram with four congruent sides. 10. Graph quad USM with coordinates U(1, 1), S(4, 5), M(9, 5), and (6, 1) on the grid below. a. Verify that quad USM is a parallelogram by finding the slope of each side. 10 8 6 4 y 2 b. Verify that USM is a rhombus by finding the length of each side. 2 4 6 8 x c. Find the slopes of the diagonals, MU and S. d. Use the results in part c to complete the theorem. he diagonals of a rhombus are. 11. Given quad FGH is a rhombus. a. List the three triangles that are congruent to HX. H X F b. xplain why FX GFX and HGX FGX. c. omplete the theorem. ach diagonal of a rhombus. G formal proof for the theorem in Item 11 is left as an exercise. 12. List all of the properties of a rhombus. Begin with the properties of a parallelogram. 218 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Lesson 15-4 Rectangles, Rhombuses, and Squares IVIY 15 13. Given quad UH is a rhombus. Use the properties of a rhombus and the figure at right to find each of the following. a. Solve for x if m UP = 4x + 18. U P b. Solve for x and y if U = 5x + 4, = 2x + y, H = 2y 8, and UH = 24. H c. Solve for x if m PH = 8x + 2 and m P = 10x 10. square is a parallelogram with four right angles and four congruent sides. 14. lternate definitions for a square. a. square is a rectangle with S Q. U b. square is a rhombus with. R 15. List all of the properties of a square. 16. Match each region in the Venn diagram below with the correct term in the list. F H G I B kites isosceles trapezoids parallelograms polygons quadrilaterals rectangles rhombi squares trapezoids ctivity 15 Quadrilaterals and heir Properties 219
IVIY 15 Lesson 15-4 Rectangles, Rhombuses, and Squares 17. Model with mathematics. Mr. ortez uses the table below to organize his findings before he enters information in the database. Place a check mark if the polygon has the given property. 4 Sides Opposite Sides Parallel Opposite Sides ongruent Opposite ngles ongruent iagonals Bisect ach Other onsecutive ngles Supplementary iagonals Perpendicular 4 Right ngles 4 ongruent Sides xactly One Pair of Opposite Sides Parallel Quadrilateral Kite rapezoid Parallelogram Rectangle Rhombus Square heck Your Understanding 18. ell whether each statement is true or false. a. ll squares are rectangles. b. ll rhombuses are squares. c. ll squares are parallelograms. d. Some squares are kites. e. No rhombuses are trapezoids. Q Z 30 S B 65 R 19. What do all rectangles, squares, and rhombuses have in common? LSSON 15-4 PRI 20. and B are diagonals of rectangle B. Find each measure. m B = m B = m = m B = m B = m B = m B = 21. QS and R are diagonals of rhombus QRS. Find each measure. m QSR = m QZR = m QS = m QR = m QS = m RZS = 22. Make sense of problems. diagonal of a square tile is 10 mm. What is the area of the tile? 220 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals
Quadrilaterals and heir Properties 4-gon Hypothesis IVIY 15 IVIY 15 PRI Write your answers on notebook paper. Show your work. Lesson 15-1 1. ell whether each statement about kites is always, sometimes, or never true. a. xactly two pairs of consecutive sides are congruent. b. he diagonals divide the kite into four congruent triangles. c. he diagonals are perpendicular. d. kite is a parallelogram. e. One diagonal bisects a pair of opposite angles. f. kite is a rhombus. Lesson 15-2 2. Make a true statement by filling in each blank with always, sometimes, or never. a. trapezoid is isosceles. b. trapezoid is a quadrilateral. c. he length of the median of a trapezoid is equal to the sum of the lengths of the bases. d. rapezoids have a pair of parallel sides. e. rapezoids have two pairs of supplementary consecutive angles. 3. Given quad GHJK is a trapezoid. PQ is the median. H P G a. If HJ = 40 and PQ = 28, find GK. b. If HJ = 5x, PQ = 5x 9, and GK = 3x + 2, then solve for x. J Q K 4. Given quad JON is a trapezoid. O J a. ONJ b. If OJ N, then O. c. If OJ N, then NJ. Lesson 15-3 5. Quadrilateral XN is a parallelogram. is the point of intersection of the diagonals. For each situation, write an equation and solve for y. N S a. N = 5y + 1 and X = 8y 5 b. m NX = 3y 1 and m NX = 2y + 1 c. = y 1 and = 3y 10 d. m N = 7y 5 and m NX = 3y + 5 6. M is the fourth vertex of a parallelogram. he coordinates of the other vertices are (6, 4), (8, 1), and (2, 0). M can have any of the following coordinates except:. (6, 2) B. (12, 5). (4, 3). (0, 3) 7. Given quad QRS with coordinates Q(0, 0), R(2, 6), S(12, 6), and (12, 0). a. What is the best name for quad QRS? xplain. b. Find the coordinates of the midpoint for each side of quad QRS and label them M, N, O, and P. What is the best name for quad MNOP? xplain. X N ctivity 15 Quadrilaterals and heir Properties 221
IVIY 15 Quadrilaterals and heir Properties 4-gon Hypothesis Lesson 15-4 8. Given quad WH with vertices W(2, 4), H(5, 8), (9, 5), and (6, 1). What is the best name for this quadrilateral?. parallelogram B. rhombus. rectangle. square 9. Given quad B is a rhombus and m B = 32. Find the measure of each numbered angle. 10. Given quad RIGH is a rectangle. R 4 2 3 1 I B 12. Write an indirect proof. Given: WIN is not isosceles. Prove: Quad WIN is not a rhombus. W I Y MHMIL PRIS Reason bstractly and Quantitatively 13. Ginger noticed that no matter the height of the adjustable stand for her electric piano, the keyboard remains level and centered over the stand. What has to be true about the legs of the stand? xplain. N H a. If R = 18, then RG =. b. If RG = 4x + 12 and HI = 10x 15, then x =. 11. Given: Parallelogram PQRS with diagonal PR. Prove: PQR RSP G P Q S R 222 SpringBoard Mathematics Geometry, Unit 2 ransformations, riangles, and Quadrilaterals