In an isosceles triangle, the sides and the


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1 51 LSSON Properties of Isosceles and quilateral Triangles Warm Up New oncepts Math Reasoning nalyze oes Theorem 511 also apply to equilateral triangles? How do you know? 1. Vocabulary triangle with two congruent sides is a(n).. In RST, R S and m T = 80. etermine m R. (13) (18) 3. Multiple hoice Which statement is always true? (13) If a triangle is isosceles, then it is equilateral. If a triangle is equilateral, then it is isosceles. oth and. Neither nor. In an isosceles triangle, the sides and the angles of the triangle are classified by their position in relation to the triangle s congruent sides. leg of an isosceles triangle is one of the two congruent sides of the triangle. In the diagram, and are the legs. The verte angle of an isosceles triangle is the angle formed by the legs of the triangle. The verte angle is. The base of an isosceles triangle is the side opposite the verte angle. The base of is. base angle of an isosceles triangle is one of the two angles that have the base of the triangle as a side. In, and are base angles. Theorem 511: Isosceles Triangle Theorem If a triangle is isosceles, then its base angles are congruent. LMN is isosceles. Therefore, M N. L M N orollary If a triangle is equilateral, then it is equiangular. Online onnection ample 1 Proving the Isosceles Triangle Theorem Prove the Isosceles Triangle Theorem. Given: is an isosceles triangle with. is the midpoint of. Prove: 336 Saon Geometry
2 Statements Reasons 1. is isosceles 1. Given.. efinition of isosceles triangle 3. = 3. efinition of midpoint. 5.. efinition of congruent segments 5. Refleive Property SSS Triangle ongruence Postulate PT The onverse of the Isosceles Triangle Theorem is also true, as is the onverse of orollary Theorem 51: onverse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are also congruent. orollary If a triangle is equiangular, then it is equilateral. Hint If you are not sure which side of the triangle is the base and which sides are the legs, sketch a triangle and label the angles and sides with as much information as possible. ample Using the Isosceles Triangle Theorem and Its onverse a. Triangle F is isosceles, and its verte angle is at. If m = 36, determine m and m F. The base angles of F are and, so by the Isosceles Triangle Theorem, F. y the definition of congruent angles, m F = m, so they each measure 36. Therefore, m + m + m F = 180 Triangle ngle Sum Theorem 36 + m + 36 = 180 Substitute m = 108 Solve b. The perimeter of GHJ is 1 inches, and G H. If GH = 5 inches, find GJ. y the onverse of the Isosceles Triangle Theorem, GJ HJ. Since the perimeter is 8 inches and GH = 5 inches, P = GH + HJ + GJ Formula for perimeter 1 = 5 + HJ + GJ Substitute. 1 = 5 + GJ + GJ efinition of congruent segments 1 = 5 + GJ Simplify. GJ = 3.5 in. Solve. Lesson
3 ample 3 Using Relationships in quilateral Triangles triangle is equiangular and has a perimeter of.5 centimeters. etermine the length of each side. y orollary 511, the triangle is equilateral. Let the length of each side be s. The perimeter is the sum of the three sides. P = s + s + s Formula for perimeter.5 = 3s Substitute and simplify. s = 7.5 cm Solve. Math Reasoning onnect How are Theorems 513 and 51 related to each other as conditional statements? Theorem 513 If a line bisects the verte angle of an isosceles triangle, then it is the perpendicular bisector of the base. Theorem 51 If a line is the perpendicular bisector of the base of an isosceles triangle, then it bisects the verte angle. The diagram illustrates both of these theorems. The altitude TU bisects the verte angle and is a perpendicular bisector of the base of the triangle. T ample Proving Theorems 513 and 51 a. Prove Theorem Given: is isosceles, bisects Prove: is the perpendicular bisector of Statements Reasons 1. is isosceles, 1. Given bisects.. efinition of angle bisector efinition of isosceles triangle.. Refleive Property SS Triangle ongruence Postulate PT 7. = 7. efinition of congruent segments PT 9. and form adjacent 9. efinition of adjacent angles angles If lines form congruent adjacent angles, they are perpendicular 11. is bisector of 11. efinition of perpendicular bisector U 338 Saon Geometry
4 b. Write a paragraph proof of Theorem 51. Given: is isosceles, is the perpendicular bisector of Prove: bisects Since is isosceles,. y the Refleive Property,. oth and are right triangles, since is the perpendicular bisector of and forms two right angles at. Therefore, by the HypotenuseLeg Right Triangle ongruence Theorem. y PT,. Therefore, by the definition of an angle bisector, bisects. Math Reasoning Model Picture the telephone pole with cables securing it to the ground from each side. re the cables all coplanar? re any two of them coplanar? ample 5 pplication: Infrastructure This figure shows the north and east view of a telephone pole that is secured by four cables of equal length. a. plain why the base angles, PQ and PRQ, are congruent. In PR, the cable lengths P and RP are equal, so P RP by the definition of congruent segments. Therefore, PR is isosceles by definition. pplying the Isosceles Triangle Theorem, the base angles of PR are congruent, so PQ PRQ. P Q North View P Q ast View R b. Prove that these angles are also congruent to the base angles and. y the Refleive Property of ongruence, PQ PQ. It is given in the problem that P P, so by the HypotenuseLeg Right Triangle ongruence Theorem, PQ PQ. y PT,. Since P is isosceles, by the Isosceles Triangle Theorem. It is given that R, so by the Transitive Property of ongruence, R. Lesson Practice ( ) ( ) ( ) ( 3) a. For the isosceles triangle shown, determine the missing angle measures. b. The perimeter of XYZ is 15. centimeters, and X Z. If XY = 6.3 centimeters, determine XZ. 7 c. If the verte angle of an isosceles triangle measures 0, what are the measures of each of its base angles? d. triangle is equiangular and its perimeter is 7 feet. etermine the length of each side. Lesson
5 ( 5) e. ngineering This diagram shows the sideview profile of a bridge. etermine the angle that each half of the bridge makes with the horizontal. 150 ft ft Practice istributed and Integrated 1. In FGHJ, m H < m FJG, GH =  1, and FG = + 8. Find the range of values for. (Inv ) F G. From the statement, If Fabian s socks are clean, then they are in the dresser, what can you conclude about what will happen when Fabian s socks are clean? (1) J H 3. a. etermine this figure s perimeter. b. etermine its area. (0) 8 in. 15 in. 16 in. *. Write plain how you would find the geometric mean (50) of and 7. 1 in. 17 in. y (Inv 3) 5. What is the measure of each eterior angle in a regular heagon? 6. Using the diagram, find the length of to the nearest hundredth (35) of a centimeter. * 7. In PQR, PR QR and m R = 118. (51) a. Identify the verte angle. b. etermine m P. y (9) 8. lgebra polyhedron has 10 more edges than vertices. How many faces does the polyhedron have? 6 cm 9. Find the unknown length of the side in the triangle shown. (9) * 10. Write plain why each angle of an equilateral triangle measures 60. (51) 13 m 1 m 11. rt In order to design part of her tile pattern correctly, Teresa (3) needs to make the consecutive angles shown supplementary. How can she ensure the angles will be supplementary? y 1. lgebra Two lines are perpendicular. One line has an equation y = 1 m  and the (37) other line has an equation of y = n Find one value for m and n. * 13. onstruction builder needs to position a support brace as shown. (50) What is the length of the support brace, to the nearest tenth? 3 30 Saon Geometry
6 1. Justify plain why, then find. (6) 15. onve pentagon PQRST has vertices P(0, 0), Q(, ), R(1, ), S(, y), (5) and T(, ). QR = ST and RS =. Find (, y) Surveying Kristi, a map surveyor, is using an eastwest baseline to (18) locate various landmarks. She measures the clockwise angle from the baseline to the line passing through her surveying instrument and the landmark. Kristi takes two sightings from a clock tower, as this diagram shows. a. What measure is the angle between the two sightings? b. What theorem did you use in part a? lock Tower 1 80 aseline 17. In the diagram shown at right, what information is needed (30) to conclude that TUV F? U F 18. Generalize Triangle GHI has vertices G(0, 0) and H(a, b). Find (5) coordinates for a point I so that GHI is a right triangle. T V Use the LL ongruence Theorem to show that OPQ RST. (36) T P S O Q R 0. What is the equation of a line perpendicular to y =  + 1, passing through (37) the origin? 1. ell Phone Towers Three cell phone towers are the vertices of (39) a triangle at right. The measure of M is 10 less than the measure of K. The measure of L is one degree greater than the measure of M. Which two towers are closest together? *. nalyze Give a twocolumn proof of the onverse of the (51) Isosceles Triangle Theorem. Given: Prove: is isosceles. Hint: rop an altitude from point to the base of. * 3. Find the value of and y on the triangle shown. (50) Tower L *. MultiStep etermine whether each set of numbers can be the side lengths of a triangle. If they can be a triangle, determine whether it is an acute, obtuse, or right triangle. a. 99, 3 13, 1 b. 7, 8, 3 c. 8, 7, 1 (33, 39) Tower M Tower K y Lesson 51 31
7 5. etermine the perimeter and area of this figure. Give eact answers. (0) 5 in. 10 in If KLM F, what side of F corresponds to side KM? (5) 7. Using the diagram, find PY in terms of if PV = 3 and YU =. Point P is the (38) incenter of the triangle. T Y P U X V Z 8. Multiple hoice The intercepted arc of an inscribed angle is a semicircle (7) if and only if the measure of the angle is * 9. Find the value of in this figure. (51) 6  _ Prove Theorem 39: If one angle of a triangle is larger than another Q (8) angle, then the side opposite the first angle is longer than the side opposite the second angle. Given: m P > m R Prove: QR > QP Hint: There are two cases you must consider. One where QR < QP and one where QR = QP. P R 3 Saon Geometry
8 5 LSSON Properties of Rectangles, Rhombuses, and Squares Warm Up 1. Find the area of the regular quadrilateral. lassify this quadrilateral. (15). Vocabulary quadrilateral with two pairs of parallel sides is a. 3. Find the perimeter of this composite figure. Then, name each quadrilateral in the figure. (19) (0) 1.5 cm 8.7 m New oncepts The diagonals of parallelograms have special properties. Recall that a rhombus is a parallelogram with four congruent sides, a rectangle is a parallelogram with four right angles, and a square shares the properties of both a rectangle and a rhombus. One property of the diagonals of a parallelogram has already been introduced: they bisect each other. Three more are introduced in this lesson. Properties of a Rectangle: ongruent iagonals Math Reasoning nalyze re the diagonals of a square congruent? How do you know? The diagonals of a rectangle are congruent. P Q PR QS If a quadrilateral is a parallelogram, it is a rectangle if and only if the above property is true. S R ample 1 Using iagonals of a Rectangle Online onnection rectangular barn door has diagonal braces. If is 6 feet, what is the length of? = = 6 = 1 = 1 iagonals of a rectangle are congruent iagonals of a parallelogram bisect each other Substitute. Segment ddition Postulate efinition of segment congruence Lesson 5 33
9 ploration Using ontruction Techniques to raw a Rhombus In this eploration, you will use simple construction techniques to construct a quadrilateral, then classify it. You may wish to review onstruction Lab 1 before this eploration. 1. raw JK. Set your compass to JK. Place the compass point at J and draw an arc above JK L. hoose and label a point L on the arc. What is the relationship between JK and JL?. Place the compass point at L and draw an arc to the right of L. L J K 3. Place the compass point at K and draw an arc that intersects the arc you drew in step. Label the point of intersection M. How are JK, KM, ML, and LJ related?. How do you know that the quadrilateral you have L drawn is a rhombus? 5. raw the diagonals JM and LK and label their point of intersection P. Measure LPM. What can you determine about the diagonals? J K 6. y measuring angles, determine the relationship between the diagonals and the angles of the rhombus. J K M Math Language Model Sketch a trapezoid and a kite. oes it seem like either figure has perpendicular diagonals? Properties of a Rhombus: Perpendicular iagonals The diagonals of a rhombus are perpendicular. H K HJ IK I J If a quadrilateral is a parallelogram, it is a rhombus if and only if the above property is true. Since a square is both a rhombus and a rectangle, its diagonals are both perpendicular and congruent. Properties of a Rhombus: iagonals as ngle isectors ach diagonal of a rhombus bisects opposite angles. ecause opposite angles of a rhombus are equal, when they are bisected by a diagonal, four congruent angles result , and If a quadrilateral is a parallelogram, it is a rhombus if and only if the above property is true. 3 Saon Geometry
10 ample Using Properties of iagonals of a Rhombus F is a rhombus. Find the measure of each angle. a. m Since m is 90, then we know that m + m = 90 (3 + 1) + ( + 10) = 90 Substitute. + = 90 Simplify. = 17 Solve. Now substitute the value of to find the measure of. m = m = 3(17) + 1 Substitute for. m = 63 Simplify. b. m Since the diagonals of a rhombus bisect the angles, m = m. m = + 10 m = m = 7 F (3 + 1) ( + 10) Hint It may be possible to classify a parallelogram using more than one of the properties in this lesson. ample 3 Using Properties of Parallelograms UVWX is a parallelogram. ecide what type of parallelogram it is by using the properties of rectangles and rhombuses. a. etermine whether the diagonals are congruent and classify the parallelogram. UW = ( 16) + (  1) = 58 VX = (1  ) + (16) = 58 Since UW = VX, then the diagonals are congruent. y the ongruent iagonals Property of a Rectangle, the shape must be a rectangle. U 6 O y V X 6 W b. etermine whether the diagonals are perpendicular and classify the parallelogram. slope of UW = _ =  3_ slope of VX = _ = _ 7 3 Since  3_ 7 _ 7 = 1, UW is perpendicular to VX. 3 This implies that the parallelogram is a rhombus. Since the shape is both a rectangle and a rhombus, it is also a square. Lesson 5 35
11 ample pplication: rchitecture rectangular building is designed with steel support braces placed diagonally in the interior. etermine the length of the steel brace that will be used for diagonal. a + b = c = c c = 130 ft F = 130 ft F = 130 ft Pythagorean Theorem Substitute Solve Substitute iagonals of a rectangle are congruent Substitute 50 ft 10 ft F Lesson Practice a. In rectangle MNOP, MO = 5. inches. What is the length of NP? ( 1) WXYZ is a rhombus. Using the diagram, answer the questions that follow. b. Find m OXY. X W ( ) (61) c. Find m OYZ. ( ) O Z ( + ) Y d. Quadrilateral RSTU has a center point, V. If RT SU, and RT SU, ( 3) classify the quadrilateral. e. rchitecture building is made with a rhombusshaped courtyard. If the longer diagonal walkway is 50 feet and the shorter one is 0 feet, what is the perimeter of the courtyard to the nearest foot? ( ) Practice istributed and Integrated 1. raw a net for this polyhedron. (Inv 5). MultiStep Find the orthocenter of JKL with vertices J(3, 6), K(3, 9), and L(5, 5). (3) 3. Write ompare a regular octagonal prism and a cylinder. onsider what happens as the number of sides of the base of a prism increases. (9). arpentry cabinetmaker needs to position a support brace as shown. What is the length of the support brace? 7 (50) * 5. rt Joni wants to use a square piece of paper for an art project. (5) plain how she could easily determine if a piece of paper is square. 36 Saon Geometry
12 6. The circumference of a circle is 113 feet. If a sector has an area of 1 ft, what is the measure of the arc to the nearest degree? (35) 7. rror nalysis Henrietta is trying to find the slope of the line that passes through the point (8, 0) and is perpendicular to the line y = + 1. Her answer is y = 1 +. Is she correct? Why or why not? (37) 8. re these two triangles similar? Give a reason to support your answer. (6) P 8 Q 16 U R 0 T 10 S 9. Generalize plain how a parallelogram could be similar to a rectangle. (1) y * 10. lgebra rhombus has two angles that each measure (5) (9 + 1) and two angles that each measure (0 + 5). Find the measure of each of the four angles in the rhombus. * 11. Find the geometric mean of 11 and 5. (50) * 1. hemistry This diagram shows the atoms and bonds in a water molecule. The (51) bond angle at the oygen (O) atom is always The distances from the oygen atom to each hydrogen (H) atom are equal. What are the measures of the other two angles, to the nearest tenth? H O?? H 13. Multiple hoice Which statement has a false converse? (10) If two angles are supplementary, then they sum to 180. If a triangle has two congruent angles, then it is isosceles. If a number is an even prime, then the number is. If two angles are complementary, then they are both acute. 1. nalyze For a triangle with vertices L(0, 0), M(6, 0), and N(3, y) to be equilateral, (5) what must be the value of y? * 15. MultiStep The vertices of a square KLMN are K(0, 1), L(, ), M(1, 6) and (5) N(3, 3). Show that the diagonals are congruent perpendicular bisectors of each other. 16. If a circle has a circumference of 6, and an arc in that circle has a length of, (35) what is the angle measure of the associated sector? y (7) y 17. lgebra In the circle shown, find the measure of I. H 18. In isosceles triangle PQR, PQ = 8, QR = 8, and m Q = 86. (51) a. Identify the verte angle. b. etermine m R. 19. What is the relationship between angles intercepted by the same (7) arc? I 5m 8m 10m F G * 0. Write a paragraph proof showing that if one side of a triangle (31) is longer than another side, then the angle opposite the first side is larger than the angle opposite the second side. Given: RS > RQ Prove: m RQS > m S Q R P S Lesson 5 37
13 1. In isosceles triangle, and each base angle measures 0. (51) etermine m. y. lgebra Write an inequality to show the largest value for that (33) makes the triangle obtuse if the longest side is 9 units In a right triangle, one leg is 36 feet long, and the hypotenuse is (9) 39 feet long. What is the length of the third side? 9 *. Fence onstruction homeowner wants to build a fence gate with reinforcing (5) diagonal braces. How can he make sure the gate is rectangular without measuring the angles? * 5. Predict This figure is formed from three congruent rectangles. (0) a. etermine the perimeter of the figure. b. onsider the figure formed by etending the pattern to four congruent rectangles. etermine the perimeter of the new figure. c. etermine a formula for the perimeter of a figure in the same pattern formed from n congruent rectangles. 3 cm cm 1 cm 1 cm 6. To measure the distance F across the lake, a surveyor at S locates (6) points, F, G, and H as shown. What is the length of F? 7. Write escribe how to circumscribe a circle around an obtuse triangle. (38) 8. is a parallelogram. If is 3 times longer than and the (3) perimeter of is 1, how long is each side? 9. Suppose you are to prove. State the assumption you would (8) make to start an indirect proof. * 30. Hiking Yvette was planning on going around a thick, circular clump of trees, (3) but she found a shortcut through it. If the diameter of the circular area is 100 meters, how much shorter is her direct path, to the nearest meter, than traveling around the trees? F 30 feet G feet 0 feet S 6 feet H 3.5 feet 38 Saon Geometry
14 53 LSSON Right Triangles Warm Up 1. Vocabulary The name given to a triangle with one 90º angle is. (13). Find the area of a right triangle with legs that are 11 inches and 17 inches long. (18) 3. right triangle has legs that are 8 units and units long. What is the ratio of the triangle s hypotenuse to its shortest leg? (9) New oncepts Some right triangles are used so frequently that it is helpful to remember some of their particular properties. These triangles are called special right triangles. The two most common special right triangles are the triangle and the triangle. Since the triangle has two angles with equal measures, it is also an isosceles right triangle. Properties of a Triangle: Side Lengths In a right triangle, both legs are congruent and the length of the hypotenuse is the length of a leg multiplied by. Hint The name of each special right triangle gives the measure of its angles triangles are often used because they are one half of a square. aution When the denominator of a fraction has a square root in it, it must be rationalized. In this eample, multiplying both the top and bottom of the fraction by eliminates the root in the denominator. 5 5 ample 1 Finding the Side Lengths in a Triangle a. Use the properties of a right triangle to find the length of the hypotenuse of the triangle. The length of the hypotenuse is equal to the length of a leg times. Since the leg is inches long, the hypotenuse has a length of inches. b. Use the properties of a right triangle to find the length of a leg of the triangle. The length of the hypotenuse is equal to the length of the leg times. To find the length of a leg when given the hypotenuse, divide by instead. The length of a leg of the triangle is 3 = 3 feet. 5 3 ft 5 in. Lesson 53 39
15 ample Finding the Perimeter of a Triangle with Unknown Measures Find the perimeter of the triangle. 1 yd 5 The length of the hypotenuse is equal to the length of the leg times. Therefore, the hypotenuse is 1 yards long. The perimeter can be found by adding the lengths of the three sides together. P = P = + 1 P 0.97 Therefore, the perimeter is approimately 1 yards. Though it is often faster to use the properties of triangles to find unknown lengths, the Pythagorean Theorem can still be used to determine lengths in special right triangles. ample 3 pplying the Pythagorean Theorem with Right Triangles Find the length of the missing sides to the nearest foot, using the Pythagorean Theorem ft Math Reasoning Verify Use the properties of triangles to find. Is the result the same? Since the legs are congruent, let represent the length of the legs of the triangle. Therefore, a + b = c + = 15 = 1565 = _ 1565 = Therefore, the missing side length is approimately 88 feet. Online onnection Saon Geometry
16 ample pplication: Park onstruction square park is to be fenced around the perimeter with a snow fence for an upcoming outdoor concert. There is a diagonal path that is 30 feet long through the park. How much snow fence is required? Use the Step ProblemSolving Process. Hint good way to check your work is to consider the possible bounds (minimum and maimum) the answer could have. oes your answer fall between the bounds of the minimum and maimum? In this case, any answer over 170 feet or under 0 feet would clearly be incorrect. Understand The fence is to be placed around the perimeter, so the perimeter must be found. The diagonal of the square is given. diagram would be a helpful visual aid to understand this problem. Plan First, draw a diagram. Identify the lengths that need to be found and use the properties of triangles to solve for them. dd the length of each side together to find the perimeter. Solve This involves finding the length of the legs of the triangle created by two adjacent sides of the park and the diagonal path. Path 30 = _ 30 = 30 = Therefore, the length of the side of the square park is 15 feet. To find the perimeter of the park, the formula for the perimeter of a square will be used. P = l P = (15 ) P = 860 P 116. Therefore, the perimeter of the park, and thus the amount of fencing needed, is approimately 116 feet. heck Here, since the diagonal is longer than a side, each side must be less than 30 feet. So the perimeter must be less than 30, or 170 feet. The answer of 116 feet seems to make sense because it is less than 170 feet. 5 Lesson Practice ( 1) ( 1) a. Find the length of this triangle s hypotenuse. b. Find the length of this triangle s missing side. 63 m 5 31 yd 5 Lesson
17 ( ) ( 3) ( ) c. Find the perimeter of the triangle, to the nearest tenth of an inch. d. Find the length of the missing sides to the nearest mile. e. square building has a diagonal length of 150 feet. What would be the square footage of one floor of the building? 5 18 in. 5 8 mi Practice istributed and Integrated 1. onstruction The measurements in the diagram are from the attic space in a new home. If the angle measure of 7 is an inaccurate label on the diagram, what is the range of degrees that the mislabeled angle could be? It is given that the side opposite the mislabeled angle is the longest side of the triangle, and the side opposite the 8 is the shortest side. (39) 5.5 ft 7 7 ft 8. figure has 1 congruent edges and 8 vertices. lassify the figure. (9) b * 3. Find the eact value of the length of leg a in the triangle. (53). enter of Gravity re the orthocenter and the centroid both centers of gravity? If not, which one is? (3) a 5 5 cm * 5. Is the following statement always, sometimes, or never true? (5) parallelogram is a rectangle. T 6. Find the value of and y in the figure at right to the nearest tenth. (50) 7. composite figure is formed by a rectangle with a square removed at one corner. The rectangle measures 6 1 inches by 5 inches. The removed square has side lengths of 3 1 inches. a. etermine the perimeter of the figure. b. etermine the area of the figure. (0) 7 5 y * 8. Write Is every right isosceles triangle a triangle? plain. (53) 9. Identify each line or segment that intersects the circle shown. (3) L y * 10. lgebra The value of the side length in a triangle is inches. (53) What would be the algebraic epression for the length of the triangle s hypotenuse? 11. Write a similarity statement to eplain why the two (6) triangles shown are similar. 100 N M K Saon Geometry
18 1. nalyze What information must be known about two similar triangles in order to () find the ratio of their perimeters? * 13. nalyze Write a flowchart proof showing that if a triangle is equilateral, then (51) it is equiangular. 1. What is the total area of the shaded sectors if the diameter of the circle is (35) 8 meters? Write your answer in terms of π. 15. Find the distance from (5, ) to the line = 13. () y 16. lgebra If in, = (1 + 11), m = 1, m = 1, and (30) = 7 units, and in F, = (5 + 18), m F = 1, m F = 1, and F = 7 units, what is the value of? 17. Justify Give a possible value for the length of XY. plain your answer. 18. What is the value of b in the proportion _ 5 = _ 7 b? (1) (Inv ) X Y H Z G I 19. rt n artist is making a round ceramic plate with a pattern of lines on it. (7) If m KLM = 0, and m MP = 30, find m KNP. 0. Suppose a chord of a circle is 10 inches long, and the radius of the circle is (3) also 10 inches. What is the measure from the chord to the center of the circle? N L * 1. Multiple hoice What would be the perimeter of a triangle with a (53) hypotenuse of 73 feet, to the nearest foot? 5 feet 176 feet 10 feet 198 feet K M P. esign gift bo has the net shown in this figure. Use the net to provide a countereample to the following statement. (1, Inv 5) If the net of a threedimensional figure has all lateral faces congruent, then the figure is closed. * 3. WXYZ is the rectangle shown. Find XY. (5). MultiStep Find the centroid of F with vertices (3, ), (3) (9, 8), and F(3, ). W 10. in. 1.6 in. O X 5. It is given that F. If is 3 units, is 1 units, (1) and is 11 units, what is the length of F? Z Y 6. If two sides of one triangle are proportional to two sides of another triangle, (6) and if their corresponding included angles are congruent to each other, then the triangles are similar by. Lesson
19 * 7. Find the eact value of the length of the hypotenuse in the triangle. (53) 16 yd 5 8. Use the Hypotenusengle ongruence Theorem to prove that F. (36) F * 9. Write plain why the following statement is true. (5) If a quadrilateral is a square, then it is a rhombus. * 30. Surveying Renée takes the bearings of landmarks at,, and from the same (51) position,. Landmark earing from : hurch Steeple 3 : lock Tower : Water Tower Suppose Renée chose point to be equidistant from the church steeple and the water tower. What must be true about the clock tower s distance from each of the other two landmarks? Why? 35 Saon Geometry
20 5 LSSON Representing Solids Warm Up 1. Vocabulary prism with si square faces is called a.. Name each of the pictured solids. If the solid is a prism or pyramid, classify it. 3. ccording to uler s Formula, if a polyhedron has 7 faces and 10 vertices, how many edges does it have? (9) (9) (9) New oncepts In a perspective drawing, nonvertical parallel lines appear to meet at a point called a vanishing point. If you look straight down a highway, it appears that the edges of the highway eventually come together at a vanishing point, Math Language The vanishing point is the point in a perspective drawing on the horizon where parallel lines appear to meet. like point in the diagram. In a perspective drawing, the horizon is the horizontal line that contains the vanishing point(s). drawing with just one vanishing point is called onepoint perspective. ample 1 rawing in OnePoint Perspective raw a rectangular prism in onepoint perspective. Use a pencil with an eraser. Step 1 raw a square and a horizontal line above it representing the horizon. Mark a vanishing point on the horizon. Step raw a dashed line from the vanishing point to each of the four corners of the square. Step 3 Using the dashed lines drawn in Step, draw the sides of a smaller square. Online onnection Step onnect the two squares and erase the reference lines and the horizon that are located behind the prism. This prism is drawn from a onepoint perspective. Lesson 5 355
21 drawing with two vanishing points is said to have twopoint perspective. Look at the following eample to see how a drawing can be made from a twopoint perspective. ample rawing in TwoPoint Perspective raw a rectangular prism in twopoint perspective in which the vanishing points are above the prism. Math Reasoning Model ould you also make a twopoint perspective drawing by placing the vanishing points below the original line segment? Step 1 raw a horizontal line that represents the horizon. Place two vanishing points on the horizon. raw a vertical line segment below the horizontal line and between the two vanishing points, representing the front edge of the prism. Step raw dashed lines from each vanishing point to the top and bottom of the vertical line as shown. Step 3 raw vertical segments between the dashed lines from Step as shown and draw segments to connect them to the first segment. Step raw dashed perspective lines from the segments drawn in Step 3 to each of the vanishing points as shown. Step 5 raw a dashed vertical line between the two intersections of the perspective lines just drawn. Sketch the segments that make the top of the prism. Step 6 rase the horizon line and the dashed perspective lines. Keep the dashed lines inside the prism that represent the edges that are hidden. This prism is drawn from a twopoint perspective. n isometric drawing is a way of drawing a threedimensional figure using isometric dot paper, which has equally spaced dots in a repeating triangular pattern. The drawings can be made by using three aes that intersect to form 10 angles, as shown in the diagram Saon Geometry
22 ample 3 reating Isometric rawings reate an isometric drawing of a rectangular prism. raw the three aes on the isometric dot paper as shown above. Use this verte as the bottom corner of the prism. raw the bo so that the edges of the prism run parallel to the three aes. Shading the top, front, and side of the prism will add the perception of depth. In a twopoint perspective drawing, it appears that one edge of the solid is the front of the diagram. In a onepoint perspective drawing, it appears that a face of the solid is the front. ample pplication: rafting Hint n architecture firm is planning to construct a rectangular building on a corner lot. The client would like a drawing that shows the building as though someone is looking at it from one edge. Should the drawing be from a onepoint or twopoint perspective? Make a sketch of what the drawing should look like. Since the front of the drawing will be an edge of the building, a twopoint perspective drawing is appropriate. The diagram shows a completed view of the building. Lesson Practice ( 1) ( ) ( 3) ( ) a. raw a rectangular prism in onepoint perspective in which the vanishing point is to the left of the square. b. raw a cube in twopoint perspective with the vanishing points and horizon below the vertical line. c. Make an isometric drawing of a triangular prism. d. rafting Morgan wants to make a wooden bookshelf with two shelves. The bookshelf will be 1 meter wide, 1 meter deep, and 1.5 meters tall. To decide how much wood to buy, Morgan will draw his plans for the bookshelf. Should the drawing be from a onepoint or twopoint perspective? Sketch what Morgan s drawing should look like. Lesson 5 357
23 Practice istributed and Integrated * 1. raw a triangular prism in onepoint perspective so that the vanishing point is (5) below the prism.. Write plain why the following statement is true. If a quadrilateral is a square, then it is a rectangle. (5) y (3) y 3. lgebra Find the length of ZP in the diagram.. What is the shortest distance from (5, 3) to the line y =  + 8? () Z P Y 11 X 133 * 5. rchitecture n architect is creating different perspective drawings for a new (5) building. The building is a rectangular prism and the client would like a drawing that focuses on the front façade of the building. Should the architect create the drawing using a onepoint or twopoint perspective? Sketch a sample drawing of the building. 6. figure has a heagonal base and triangular lateral faces. lassify the figure. (9) 7. MultiStep Graph the line and find the slope of the line that passes through the points L(, 1) and M(3, 1). Then find a perpendicular line that passes through point N(, ). (37) 8. Find the value of and y in the triangle shown to the nearest tenth. (50) 13 y 9. What is the sum of the eterior angles of a conve 13sided polygon? (Inv 3) Is the following statement always, sometimes, or never true? (5) parallelogram is a rectangle. * 11. Trace the figure at right on your paper. Then locate the (5) vanishing point and the horizon line. y 1. lgebra In, m = 90, = (37), and (30) m = 60, and in F, m F = 90, = (517), and m F = 60. What value of will make F? 13. The point where three or more lines intersect is the. (3) * 1. Use the Hypotenusengle ongruence Theorem to prove (36) that RST UVW. * 15. Find the eact length of the hypotenuse of a right (53) triangle with a leg that is 57 feet long. S R T V U W 358 Saon Geometry
24 16. Formulate Four congruent circles are cut out of a square as shown. Write an (0) epression for the area of the shaded region in terms of the radius of each circle, r. r 17. viation Four jet aircraft are flying in a triangular formation. Jets,, and (51) form a line perpendicular to the flight heading, while jet is midway between the other two. Jet flies directly in front of jet. If m = 37, what does the verte angle of the triangular formation measure? Which theorem did you use? * 18. Use an indirect proof to prove that if two altitudes, X and Y of are (8) congruent, then the triangle must be isosceles. Given: X Y, X and Y are altitudes. Prove: isosceles. * 19. Find the area, to the nearest hundredth, of a right triangle with a (53) hypotenuse of 17 centimeters. y 0. lgebra If a chord perpendicular to a radius cuts the radius in two pieces that are (3) 7 and inches long, respectively, what are the two possible lengths of the chord to the nearest tenth? * 1. Justify How does a twopoint perspective differ when the vanishing points are (5) located close together compared with when they are located further apart? Justify your reasoning with drawings.. Find the geometric mean of and 5. (50) 3. Multiple hoice If the diagonals of parallelogram JKLM intersect at P, which of (3) the following is true? JP = LP JP = KP JL = KM JM = KM *. onstruction The support of a shelf forms a right triangle, (53) with the shelf and the wall as the legs. actly how long is this support? shelf ( in.) * 5. nalyze NPQ and STV are similar isosceles triangles. How many () of their si sides do you need numerical values for in order find all the other side lengths and the perimeters of both triangles? plain. wall 5 support Lesson 5 359
25 6. Using the diagram on the right, find the length of MP if OP = 5, (38) NO = 8, and MN = 18. M 7. ycling Katya and Sareema start from the same location and bicycle in opposite directions for miles each. Katya turns to her right 90 and continues for another mile. Sareema turns 5 to her left and continues for another mile. t this point, who is closer to the starting point? (Inv ) 8. rror nalysis arius drew this net of a number cube. plain his error. (Inv 5) N O 1 3 P nalyze Square RSTU has vertices at R(0, ) and S(0, 0). What are the possible (5) coordinates of T and U? 30. esign white triangle with vertices at (0, 0), (, 0), and (0, ) is used to create (11) a logo. blue triangle is added to the design so that its vertices are the midpoints of the sides of the white triangle. The blue triangle divides the white triangle into three smaller white triangles. Smaller blue triangles are placed in each small white triangle so that their vertices are the midpoints of the sides of the small white triangles. 3 y 1 O 1 3 a. Find the coordinates of the vertices of the large blue triangle. b. Find the coordinates of the vertices of each small blue triangle. c. Which of the triangles are congruent, if any? Justify your answer. 360 Saon Geometry
26 55 LSSON Triangle Midsegment Theorem Warm Up 1. Vocabulary Two triangles with congruent corresponding angles and corresponding sides that are proportional in length are. (1) etermine if the triangles in each pair are similar. If they are, state the theorem or postulate that proves it. (6). 3.. New oncepts midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. very triangle has three midsegments. Math Language The midpoint of a segment is the point that divides a segment into two congruent segments. M The midsegment is always half the length of the side that does not have a midsegment endpoint on it. N P Theorem 551: Triangle Midsegment Theorem The segment joining the midpoints of two sides of a triangle is parallel to, and half the length of, the third side. RQ PM and RQ = _ 1 PM M Q N R P Online onnection ample 1 Using the Triangle Midsegment Theorem In the diagram, is a midsegment of. Find the values of and y. y 5 7 From the Triangle Midsegment Theorem, = _ 1, so =. = (7) Therefore, = 1. From the definition of a midsegment, =. So, y = 5. Lesson
27 ample Proving the Triangle Midsegment Theorem Given: is the midpoint of and is the midpoint of. and = _ 1 Prove: Statements Reasons 1. is the midpoint of ; is the midpoint of 1. Given. = ; =. efinition of midpoint 3. + = ; 3. Segment ddition Postulate + =. + = ;. Substitute + = 5. = _ 1 ; = _ 1 5. Solve Refleive Property of ongruence SS Triangle Similarity Theorem efinition of similar polygons If corresponding angles are congruent, lines cut by a transversal are parallel 10. = _ efinition of similar polygons and step 5 Math Reasoning nalyze Is Theorem 55 a conditional statement that is related to 551? If so, what is their relationship? Theorem 55 If a line is parallel to one side of a triangle and it contains the midpoint of another side, then it passes through the midpoint of the third side. Since UV RT and RU US, SV VT. R U S V T The measure of QT can be determined using Theorem 55. Since RU = US in triangle QRS, then U is the midpoint of RS. y Theorem 55, since TU QS and U is the midpoint of RS, then T is the midpoint of QR. Since T is the midpoint of QR, then QT = TR. The measure of QT is 13 units. R T U 10 Q S 36 Saon Geometry
28 ample 3 Identifying Midpoints of Sides of a Triangle Triangle MNP has vertices M(, ), N(6, ), and P(, 1). QR is a midsegment of MNP. Find the coordinates of Q and R. R and Q are the midpoints of MN and NP. Use the Midpoint Formula to find the coordinates of Q and R. Q (_ 6 +,_ + (1) ) = Q (, _ 1 ) R (_  + 6,_ + ) = R(, 3) M(, ) y  O   R P(, 1) N(6, ) Q 6 The midsegment of a triangle creates two triangles that are similar by Similarity. In the diagram, since, then and. This shows that. midsegment triangle is the triangle formed by the three midsegments of a triangle. Triangle F is a midsegment triangle. Midsegment triangles are similar to the original triangle and to the triangles formed by each midsegment. In the figure, F F F. F ample pplying Similarity to Midsegment Triangles Triangle STU is the midsegment triangle of PQR. a. Show that STU PQR. Hint It may help to sketch the similar triangles separately so they can be compared more easily. Since STU is the midsegment triangle of PQR, by the definition of midsegment: ST = _ 1 PR; SU = _ 1 QR; TU = _ 1 QP Therefore, STU PQR by SSS similarity.  3 S 19 Q + 7 T b. Find PQ. P U R QR is twice SU, and T is the midpoint of QR, so QT = SU. + 7 = 19 = 6 Since S is the midpoint of PQ, PQ = PS. PQ = (  3) = [(6)  3] = The length of PQ is units. Lesson
29 ample 5 pplication: Maps student determined that Toledo Street is the midsegment of the triangle formed by olumbus venue, Park venue, and William Street. The distance along Park venue between olumbus venue and William Street is 160 meters, and the distance along Toledo Street in the same span is 80 meters. William Street is 0 meters long. Find the distance from the corner of olumbus and William to the corner of William and Toledo ( in the diagram) to the nearest meter. olumbus venue Toledo Street Park venue 160 m 80 m 0 m William Street Since Toledo Street is the midsegment of the triangle, it creates two similar triangles. Use the lengths of Toledo Street and Park venue to find the similarity ratio of 160:80. Name the shorter segment and write a proportion. _ = _ = (80)(0) = 10 m Lesson Practice ( 1) a. TU is a midsegment of QRS. Find the values of and y. y T 8 R 1 U Q S Hint This proof is similar to the proof of Theorem 551, given in ample. Start by showing that the two triangles in the diagram are similar. ( ) b. Prove Theorem 55. Given: and is the midpoint of. Prove: is the midpoint of. 36 Saon Geometry
30 ( 3) ( ) c. Triangle FGH has vertices at F(, ), G(6, ), and H(,1). is a midsegment of FGH. Find the coordinates of and. d. Find the perimeter of the midsegment triangle, XYZ. 5 1 X Z F(, ) O y G(6, ) 6 H(, 1) + Y 3 + ( 5) e. In the diagram, th Street is a midsegment of the triangle formed by aker, Lowry, and 5th Streets. Jeremiah leaves his house at the corner of 5th and aker and walks down 5th Street. He then turns left and walks up Lowry Street until he reaches the corner at Lowery and th Street. How far has Jeremiah walked? aker Street th Street 11 m 5th Street 10 m Lowery Street Practice istributed and Integrated * 1. raw a triangular prism in onepoint perspective so that the vanishing point (5) is below the prism.. Multiple hoice Which of the following statements is not true of a rhombus? ll sides are congruent. The diagonals are perpendicular. The diagonals are congruent. The diagonals bisect the angles. (5) 3. Model Trains Gary is building a track for his model trains. He wants to ensure that the two sides of the tracks run parallel to each other, so he places crossbeams at regular intervals along them. What could Gary check to ensure the tracks are parallel? (Inv 1). rror nalysis fter reading the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side, Ken reasons that the theorem would mean the same if it were written, The sum of the lengths of any two sides of a triangle cannot be less than the length of the third side. Is Ken correct? plain. (39) * 5. Trace the figure shown on your paper. Then draw the vanishing points and the (5) horizon line of the figure. 6. nalyze In which regular polygon is each eterior angle equal to each interior angle? (Inv 3) Lesson
31 7. How many edges does a figure with 8 vertices and 6 faces have? (9) 8. viation pilot determines that after flying 00 kilometers on one leg of a trip and 5 kilometers on a second leg of a trip, that she has enough fuel to fly another 715 kilometers. oes she have enough fuel to get back to her starting point? plain. (39)? 00 km 5 km * 9. nalyze Give a paragraph proof showing that if a triangle is equiangular, then it is (51) equilateral. 10. Golf In his first shot of a golf tournament, illustrated here, Mitch has hit the ball (53) too far to the left from where it should be. How far does he need to hit the ball to get it straight to the green from where it is now? Give your answer to the nearest tenth of a yard. still to go 1st shot green 135 yard 5 tee *11. raw a cube in twopoint perspective. (5) *1. Segment is a midsegment of. (55) Refer to the diagram to determine the coordinates of and. 13. In QRS, m Q = 55, and m R = 86. (6) In TUV, m T = 55, and m V = 39. Is QRS TUV? plain. y (3, ) (, 3) (0, )  1. The ratio of the angle measures in a quadrilateral is 1::5:6. Find the measure (1) of each angle. *15. Segment UT is a midsegment of XYZ. Find the values of and y. X (55) y 16. MultiStep Find the orthocenter of GHI with vertices G(8, 9), H(, 1), (3) and I(, 9). T U Find the geometric mean of 1 and 5. (50) 51 y 18. lgebra Solve the equation ( + 3) =. Provide a justification for 3 () 366 each step. Saon Geometry Z 30 Y
32 19. nalyze triangle has vertices L(0, 0), M(8, 0), and N(, y). What value of (5) y makes LMN equilateral? * 0. Write plain why the median and the altitude from the verte angle of an (51) isosceles triangle are identical. Refer to any theorems you need to justify your eplanation. * 1. elow is the beginning of a paragraph proof of the Triangle Inequality (31) Theorem. Write the rest of the proof. Hint: Use the Isosceles Triangle Theorem. Given: Prove: + >, + >, + > One side of is as long or longer than each of the other sides. Let this side be. Then + > and + >. Therefore what remains to prove is + >. Locate on such that = If a chord is bisected inches from the center of a circle with a diameter of (3) 6 inches, what is the length of half of the chord to the nearest tenth of an inch? y 3. lgebra Find the measure of ST in the circle at right. y (7) *. The midpoints of the sides of are as follows: midpoint of (55) : (, 1); midpoint of : (1, 3); and midpoint of : F(1, 0). raw the midsegment triangle. Find the coordinates of,, and, and then draw. S (7  ) V U T (5 + 0) 5. Travel Marco travels from Yuma to lamo and then from lamo to (53) Gadsden. ach unit on the triangle represents 30 miles. How far does he need to travel to get from Gadsden to Yuma, if he were to travel a straight path between them, rounded to the nearest mile? lamo y 5 units 5 Yuma 6. Landscaping square field is to be hydroseeded. The field has a (53) diagonal length of 5 yards. How many square feet need to be hydroseeded? Gadsden 7. etermine if these two triangles are similar. If so, state the similarity (6) and the reason. 8. Find the geometric mean of 11 and 1.5. (50) 9. Write Sketch this situation or eplain why it is impossible. Two parallel lines are intersected by a transversal so that the sameside interior angles are complementary. (Inv 1) J M 1 N L 8 K 16 y 30. lgebra Points Q, R, and S are midpoints of FGH. If FH = + 1 (55) and QR =  3, what is the length of QR? G R Q F S H Lesson
33 56 LSSON Right Triangles Warm Up 1. Vocabulary (n) triangle has three sides that are congruent.. If a right triangle has a hypotenuse of 13 centimeters and a base of 5 centimeters, what is its height? 3. If an isosceles triangle has a verte angle of 0, what is the measure of a base angle? (51) (9) (51) New oncepts The triangle is another special triangle. Like the triangle, properties of the triangle can be used to find missing measures of a triangle if the length of one side is known. Math Reasoning onnect triangle can be used to prove attributes in a square. What quadrilaterals can be formed using triangles? In the diagram, two triangles are shown net to each other, with the shorter legs aligned Placing the two triangles together so that they share a common leg makes an equilateral triangle. Since all the equilateral triangle s sides are congruent, this shows that the hypotenuse of the triangle is twice the length of the shortest leg. Properties of Triangles In a triangle, the length of the hypotenuse is twice the length of the short leg, and the length of the longer leg is the length of the shorter leg times lgebraically, these relationships can be written as follows. PR = a PQ = a QR = a 3 Q 30 Online onnection ample 1 Finding Side Lengths in a a a Triangle 60 P Find the values of and y. Give your a R answer in simplified radical form. y The shortest leg must be opposite the smallest angle, so the leg with a measure of is the short leg. The hypotenuse is twice the short leg, so y =. The long leg is 3 times the short leg, so = Saon Geometry
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