Indirect proof. Write indirect proof for the following

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1 Indirect proof Write indirect proof for the following 1..

2 Practice C A parallelogram is a quadrilateral with two sets of congruent parallel sides. The opposite angles in a parallelogram are congruent. Consider parallelogram ABCD. 1. Describe what happens to the diagonals if ma and mc are increased without changing any side lengths. The length of BD increases, and the length of AC decreases.. Give the range of lengths for a diagonal in a parallelogram with side lengths of a and b. between zero and (a b) Find the range of values for x.. 6 x 10 x 58 1 (y z) Use the figure for Exercises 5 and 6. BC DC. (Note: The figure is not drawn to scale.) 5. Can BD be longer than DC? If so, find the range of values for x. If not, explain your answer. Yes, BD can be longer than DC ; x Can DC be longer than BC? If so, find the range of values for x. If not, explain your answer. No, DC cannot be longer than BC ; possible answer: the inequalities lead to the contradiction that x must be both less than 4 and greater than. Use the figure for Exercises 7 and 8. The intersection point of the segments is the center of the circle. (Note: The figure is not drawn to scale.) AB, CD, DE, FA, EF, BC 7. Put the segments in order from shortest to longest. 8. Name the segment that is congruent to the radius of the circle. DE 45 Holt Geometry

3 Practice A Fill in the blanks to complete the theorems. 1. If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle.. If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. Compare the given measures AB and DE mi and ml 5. PS and PQ AB DE mi ml PS PQ Complete Exercises 6 10 to find the range of values for x. 6. Compare mutv and mwtv. mutv mwtv 7. Rewrite your answer to Exercise 6 by replacing mutv and mwtv with the values from the figure. 8 x 8. Solve your inequality for Exercise 7 for x. x Any angle in a triangle must have a measure greater than 0. Solve this inequality for x: x 0 x Combine your answers from Exercises 8 and 9 to find the range of values for x. 1 x Find the range of values for z in the figure. 4 z Warren and his dad are preparing to go sailing for the first time this year. The two diagrams show the boat s mast in different positions as they use a winch to raise it. Notice that the length of the mast and the distance from the bottom of the mast to the winch are the same in each diagram. Tell whether the length of the cable from the winch to the top of the mast is longer in Diagram 1 or in Diagram. Diagram 1 Diagram Diagram 1 4 Holt Geometry Practice B Compare the given measures mk and mm. AB and DE. QR and ST mk mm AB DE QR ST Find the range of values for x ( 1) 54 7 x Holt Geometry ( 5) ( 1) x 17 x 10.5 x 4 8. You have used a compass to copy and bisect segments and angles and to draw arcs and circles. A compass has a drawing leg, a pivot leg, and a hinge at the angle between the legs. Explain why and how the measure of the angle at the hinge changes if you draw two circles with different diameters. Possible answer: The legs of a compass and the length spanned by it form a triangle, but the lengths of the legs cannot change. Therefore any two settings of the compass are subject to the Hinge Theorem. To draw a larger-diameter circle, the measure of the hinge angle must be made larger. To draw a smaller-diameter circle, the measure of the hinge angle must be made smaller. Practice C A parallelogram is a quadrilateral with two sets of congruent parallel sides. The opposite angles in a parallelogram are congruent. Consider parallelogram ABCD. 1. Describe what happens to the diagonals if ma and mc are increased without changing any side lengths. The length of BD increases, and the length of AC decreases.. Give the range of lengths for a diagonal in a parallelogram with side lengths of a and b. between zero and (a b) Find the range of values for x.. 10 x Holt Geometry 6 x 1 (y z) Use the figure for Exercises 5 and 6. BC DC. (Note: The figure is not drawn to scale.) 5. Can BD be longer than DC? If so, find the range of values for x. If not, explain your answer. Yes, BD can be longer than DC ; x Can DC be longer than BC? If so, find the range of values for x. If not, explain your answer. No, DC cannot be longer than BC ; possible answer: the inequalities lead to the contradiction that x must be both less than 4 and greater than. Use the figure for Exercises 7 and 8. The intersection point of the segments is the center of the circle. (Note: The figure is not drawn to scale.) 7. Put the segments in order from shortest to longest. AB, CD, DE, FA, EF, BC 8. Name the segment that is congruent to the radius of the circle. DE Theorem Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the included angle that is larger has the longer third side across from it. Example 46 Holt Geometry If K is larger than G, then side LM is longer than side HJ. The Converse of the Hinge Theorem is also true. In the example above, if side LM is longer than side HJ, then you can conclude that K is larger than G. You can use both of these theorems to compare various measures of triangles. Compare NR and PQ in the figure at right. PN QR PR PR mnpr mqrp Since two sides are congruent and NPR is smaller than QRP, the side across from it is shorter than the side across from QRP. So NR PQ by the Hinge Theorem. Compare the given measures. 1. TV and XY. mg and ml TV XY mg ml Reteach. AB and AD mfhe and mhfg AB AD mfhe mhfg 77 Holt Geometry

4 TEKS G.8.C It is recommended that for a height of. Find x, the length of 0 inches, a wheelchair ramp be the weight-lifting incline 19 feet long. What is the value of bench. Round to x to the nearest tenth? the nearest tenth. Problem Solving The Pythagorean Theorem 18.9 ft 1 ft. A ladder 15 feet from the base of a In a wide-screen television, the ratio of building reaches a window that is width to height is 16 : 9. What are the 5 feet high. What is the length of width and height of a television that has the ladder to the nearest foot? a diagonal measure of 4 inches? Round to the nearest tenth. 8 ft width 6.6 in.; height 0.6 in. Choose the best answer. 5. The distance from Austin to San Antonio is about 74 miles, and the distance from San Antonio to Victoria is about 10 miles. Find the approximate distance from Austin to Victoria. A 8 mi B 70 mi C 16 mi D 176 mi 6. What is the approximate perimeter of DEC if rectangle ABCD has a length of 6 centimeters? F 5.1 cm G 6.5 cm H 9.8 cm J 11.1 cm 6 cm 7. The legs of a right triangle measure x 8. A cube has edge lengths and 15. If the hypotenuse measures of 6 inches. What is the x, what is the value of x? approximate length of a A 1 C 6 diagonal d of the cube? B 16 D 1 F 6 in. H 10.4 in. G 8.4 in. J 1 in. 5 Holt Geometry

5 TEKS G.8.C It is recommended that for a height of. Find x, the length of 0 inches, a wheelchair ramp be the weight-lifting incline 19 feet long. What is the value of bench. Round to x to the nearest tenth? the nearest tenth. Problem Solving The Pythagorean Theorem 18.9 ft 1 ft. A ladder 15 feet from the base of a In a wide-screen television, the ratio of building reaches a window that is width to height is 16 : 9. What are the 5 feet high. What is the length of width and height of a television that has the ladder to the nearest foot? a diagonal measure of 4 inches? Round to the nearest tenth. 8 ft width 6.6 in.; height 0.6 in. Choose the best answer. 5. The distance from Austin to San Antonio is about 74 miles, and the distance from San Antonio to Victoria is about 10 miles. Find the approximate distance from Austin to Victoria. A 8 mi B 70 mi C 16 mi D 176 mi 6. What is the approximate perimeter of DEC if rectangle ABCD has a length of 6 centimeters? F 5.1 cm G 6.5 cm H 9.8 cm J 11.1 cm 6 cm 7. The legs of a right triangle measure x 8. A cube has edge lengths and 15. If the hypotenuse measures of 6 inches. What is the x, what is the value of x? approximate length of a A 1 C 6 diagonal d of the cube? B 16 D 1 F 6 in. H 10.4 in. G 8.4 in. J 1 in. 5 Holt Geometry

6 Practice C Multiply and simplify. Assume a and b are nonnegative. 1. ( a b)( a b) a b. (a b )(a b ) a b Find the value of x in each figure. Give your answers in simplest radical form Greg is a modeling enthusiast. He is working on modeling some geometric shapes, but he finds he doesn t have a ruler to take measurements. In Greg s desk drawer, he finds a protractor, a straightedge, and a pencil. For Exercises 9 and 10, use and/or --90 triangles to accomplish each task. 9. Describe how Greg can draw an exact : 1 replica of a --90 triangle. That is, he will draw a triangle that has double the length of each side in the original triangle. (Hint: Look back at Exercise 8.) Possible answer: Use one of the legs of the original --90 triangle as the shorter leg of a triangle. The hypotenuse of the triangle will then have twice the length of one of the legs of the --90 triangle. Then draw a --90 triangle with a leg as the hypotenuse of the triangle. This larger --90 triangle has legs with exactly twice the length of the original --90 triangle. 10. Describe how Greg can draw an exact 1 : replica of a triangle. Sketch an example. Possible answer: Name the length of the longer leg in a triangle x. The shorter leg has length x. Use the shorter leg of the original triangle as the longer leg of another triangle. The shorter leg of this second triangle then has length 1 x. Use that leg as the longer leg of a third triangle. This smallest triangle has sides that are exactly one-third the length of the original. 61 Holt Geometry 1 1

7 Practice A 1. The sum of the angle measures in a triangle is 180. Find the missing angle measure. Then use the Pythagorean Theorem to find the length of the hypotenuse. ; In a --90 triangle, the legs have equal length and the hypotenuse is the length of one of the legs multiplied by. Find the value of x Find the missing angle measure. Then use the Pythagorean Theorem to find the length of the hypotenuse. ; In a triangle, the hypotenuse is the length of the shorter leg multiplied by, and the longer leg is the length of the shorter leg multiplied by. Find the values of x and y x 4 y 4 7. x 7 y x 10 y 0 For Exercises 9 and 10, use a calculator to find each answer. 6. cm cm 9. Andre is building a structure out of playing cards. Each card 90 is 6. centimeters long. He tries leaning the cards against each other so that the angle at the top is 90. Find the distance between the edges of the cards to the nearest tenth. 8.9 cm 10. Andre tries leaning the cards against each other so the angle at the top is. Find the height x of the tops of the cards Holt Geometry cm 11. Tell whether Andre can lay another card across the peaks of the structures he built in Exercises 9 and 10. Possible answer: Andre cannot lay a card across the top of the structure in Exercise 9 because 6. cm 8.9 cm. He can probably not lay a card across the top of the structure in Exercise 10 because 6. cm is the distance between two consecutive peaks, and there should be some overlap for the card to stay. Practice B Find the value of x in each figure. Give your answer in simplest radical form Find the values of x and y. Give your answers in simplest radical form x 0 y 0 5. x 4 y 8 6. x y Lucia is an archaeologist trekking through the jungle of the Yucatan Peninsula. She stumbles upon a stone structure covered with creeper vines and ferns. She immediately begins taking measurements of her discovery. (Hint: Drawing some figures may help.) 7. Around the perimeter of the building, Lucia finds small alcoves at regular intervals carved into the stone. The alcoves are triangular in shape with a horizontal base and two sloped equal-length sides that meet at a right angle. Each of the sloped sides measures inches. Lucia has also found several stone tablets inscribed with characters. The stone tablets measure 1 inches long. Lucia hypothesizes that the alcoves once held the stone 8 tablets. Tell whether Lucia s hypothesis may be correct. Explain your answer. Possible answer: Lucia s hypothesis cannot be correct. The base of the alcove is 57 inches or just over 0 inches long, so a 1 -inch tablet Lucia also finds several statues around the building. The statues measure 9 7 could not fit. 16 inches tall. She wonders whether the statues might have been placed in the alcoves. Tell whether this is possible. Explain your answer. Possible answer: To find the height of a --90 triangle, draw a perpendicular to the hypotenuse. This makes another smaller --90 triangle whose hypotenuse is the length of one of the legs of the larger triangle. The height of the alcove is 57 inches or about 10 inches, so 8 the statues could have been placed in the alcoves. 60 Holt Geometry Practice C Multiply and simplify. Assume a and b are nonnegative. 1. ( a b)( a b) a b. (a b)(a b) a b Find the value of x in each figure. Give your answers in simplest radical form Reteach Theorem --90 Triangle Theorem In a --90 triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg. Example In a --90 triangle, if a leg length is x, then the hypotenuse length is x Greg is a modeling enthusiast. He is working on modeling some geometric shapes, but he finds he doesn t have a ruler to take measurements. In Greg s desk drawer, he finds a protractor, a straightedge, and a pencil. For Exercises 9 and 10, use and/or --90 triangles to accomplish each task. 9. Describe how Greg can draw an exact : 1 replica of a --90 triangle. That is, he will draw a triangle that has double the length of each side in the original triangle. (Hint: Look back at Exercise 8.) Possible answer: Use one of the legs of the original --90 triangle as the shorter leg of a triangle. The hypotenuse of the triangle will then have twice the length of one of the legs of the --90 triangle. Then draw a --90 triangle with a leg as the hypotenuse of the triangle. This larger --90 triangle has legs with exactly twice the length of the original --90 triangle. 10. Describe how Greg can draw an exact 1 : replica of a triangle. Sketch an example. Possible answer: Name the length of the longer leg in a triangle x. The shorter leg has length x. Use the shorter leg of the original triangle as the longer leg of another triangle. The shorter leg of this second triangle then has length 1 x. Use that leg as the longer leg of a third triangle. This smallest triangle has sides that are exactly one-third the length of the original. 61 Holt Geometry 1 1 Use the --90 Triangle Theorem to find the value of x in EFG. Every isosceles right triangle is a --90 triangle. Triangle EFG is a --90 triangle with a hypotenuse of length x Hypotenuse is times the length of a leg. 10 x Divide both sides by. 5 x Rationalize the denominator. Find the value of x. Give your answers in simplest radical form. 1.. x 17 x. 6 Holt Geometry x 4 x 5 81 Holt Geometry

8 TEKS G.5.D Problem Solving For Exercises 1 6, give your answers in simplest radical form. 1. In bowling, the pins are arranged in a pattern based on equilateral triangles. What is the distance between pins 1 and 5? 1 in. or about 0.8 in.. To secure an outdoor canopy, a 64-inch cord is extended from the top of a vertical pole to the ground. If the cord makes a angle with the ground, how tall is the pole? in. or about 55.4 in. Find the length of AB in each quilt pattern.. in. in. in. or about in. 0 4 in. 8 in. or about 6 in. Choose the best answer. 5. An equilateral triangle has an altitude of 6. A shelf is an isosceles right triangle, and 1 inches. What is the side length of the longest side is 8 centimeters. What the triangle? is the length of each of the other two sides? 14 in. 19 cm Use the figure for Exercises 7 and 8. Assume JKL is in the first quadrant, with mk Suppose that JK is a leg of JKL, a triangle. What are possible coordinates of point L? A (6, 5) C (6, ) B (7, ) D (8, 7) 8. Suppose JKL is a triangle and JK is the side opposite the angle. What are the approximate coordinates of point L? F (9, ) H (8.7, ) G (5, ) J (7.1, ) (, 7) (, ) 0 6 Holt Geometry

9 TEKS G.5.D Problem Solving For Exercises 1 6, give your answers in simplest radical form. 1. In bowling, the pins are arranged in a pattern based on equilateral triangles. What is the distance between pins 1 and 5? 1 in. or about 0.8 in.. To secure an outdoor canopy, a 64-inch cord is extended from the top of a vertical pole to the ground. If the cord makes a angle with the ground, how tall is the pole? in. or about 55.4 in. Find the length of AB in each quilt pattern.. in. in. in. or about in. 0 4 in. 8 in. or about 6 in. Choose the best answer. 5. An equilateral triangle has an altitude of 6. A shelf is an isosceles right triangle, and 1 inches. What is the side length of the longest side is 8 centimeters. What the triangle? is the length of each of the other two sides? 14 in. 19 cm Use the figure for Exercises 7 and 8. Assume JKL is in the first quadrant, with mk Suppose that JK is a leg of JKL, a triangle. What are possible coordinates of point L? A (6, 5) C (6, ) B (7, ) D (8, 7) 8. Suppose JKL is a triangle and JK is the side opposite the angle. What are the approximate coordinates of point L? F (9, ) H (8.7, ) G (5, ) J (7.1, ) (, 7) (, ) 0 6 Holt Geometry

8.4 Special Right Triangles

8.4. Special Right Triangles www.ck1.org 8.4 Special Right Triangles Learning Objectives Identify and use the ratios involved with isosceles right triangles. Identify and use the ratios involved with 30-60-90