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1 Triangles 1. A triangle has 2 angles that each measure 71. A. isosceles triangle B. right triangle C. obtuse triangle D. equiangular triangle Copyright 2015 Edmentum All rights reserved. 2. Directions: Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar. Theo is building a garden against his house. He bought two pieces of wood that are each five feet long. He wants to create a triangular garden against his house using the two pieces of wood, without cutting them. His house will be the third side of the triangle. He also wants the perimeter of his garden to be a whole number. There are different triangles he can create that fit these conditions. Of these triangles, there are isosceles, scalene, and equilateral. 3. Which of the following are acute scalene triangles? W. X. Y. Z. A. X and Y only B. Y and Z only C. W, X, and Y only D. X, Y, and Z only 4. A triangle has one angle that measures 68 degrees, one angle that measures 46 degrees, and one angle that measures 66 degrees. A. obtuse triangle B. equilateral triangle practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
2 C. right triangle D. acute triangle 5. How many triangles exist with the given side lengths? A. No triangle exists with the given side lengths. 12 in, 15 in, 18 in B. More than one triangle exists with the given side lengths. C. Exactly one unique triangle exists with the given side lengths. 6. *triangle not drawn to scale 6 units 5 units 9 units What kind of triangle is this? A. scalene triangle B. equilateral triangle C. right triangle D. isosceles triangle 12 units 7. *triangle not drawn to scale 12 units 20 units What kind of triangle is this? A. equilateral triangle B. obtuse triangle C. isosceles triangle D. right triangle 16 units 8. A bridge contains beams that form triangles, as shown below. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
3 Which of the following best describes the triangle with the given measures? A. acute isosceles triangle B. obtuse isosceles triangle C. obtuse scalene triangle D. acute scalene triangle 9. A triangle has 2 angles that measure 60. A. right triangle B. scalene triangle C. obtuse triangle D. equiangular triangle 10. In triangle ABC, the length of side AB is 18 inches and the length of side BC is 23 inches. Which of the following could be the length of side AC? A. 43 inches B. 3 inches C. 46 inches D. 35 inches 11. A triangle has 3 sides that each equal 13 cm. A. obtuse triangle B. right triangle C. equilateral triangle D. scalene triangle 12. A triangle has one angle that measures 27 degrees, one angle that measures 135 degrees, and one angle that measures 18 degrees. A. obtuse triangle B. equilateral triangle C. acute triangle D. right triangle 13. How many triangles exist with the given angle measures? 30, 65, 85 A. More than one triangle exists with the given angle measures. B. No triangle exists with the given angle measures. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
4 C. Exactly one unique triangle exists with the given angle measures. 14. Directions: Select the correct answer from each drop down menu. Shelbie has been asked to construct as many unique triangles as she can that have all sides of integer length, one side of length 5 cm, and a perimeter less than or equal to 16 cm. Shelbie can construct unique triangles of which are strictly isosceles and equilateral. 15. Which of the following is false? A. A triangle can be drawn with exactly one obtuse angle. B. A triangle can be drawn with more than one acute angle. C. A triangle can be drawn with exactly one right angle. D. A triangle can be drawn with exactly one acute angle. 16. Meena has a pair of blue jeans with the design below on its back pockets. Which of the following best describes the triangle with the given measures? A. acute equilateral triangle B. acute isosceles triangle C. obtuse equilateral triangle D. obtuse isosceles triangle 17. How many triangles exist with the given side lengths? A. No triangle exists with the given side lengths. 4 m, 4 m, 7 m B. More than one triangle exists with the given side lengths. C. Exactly one unique triangle exists with the given side lengths. 18. In triangle ABC, the length of side AB is 20 inches and the length of side BC is 29 inches. Which of the following could be the length of side AC? practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
5 A. 44 inches B. 51 inches C. 54 inches D. 7 inches 19. How many triangles exist with the given angle measures? 80, 50, 50 A. No triangle exists with the given angle measures. B. Exactly one unique triangle exists with the given angle measures. C. More than one triangle exists with the given angle measures. 20. How many triangles exist with the given angle measures? 55, 45, 90 A. No triangle exists with the given angle measures. B. Exactly one unique triangle exists with the given angle measures. C. More than one triangle exists with the given angle measures. Answers 1. A A 4. D 5. C 6. A 7. D 8. C 9. D 10. D 11. C 12. A 13. A D 16. A 17. C 18. A 19. C 20. A Explanations 1. Since in any given triangle, the interior angles add together to equal 180, subtract the measure of each of the first 2 angles from 180 to find the measure of the third angle. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
6 = 38 So, this triangle is not equiangular, but has only 2 angles of equal measure. A triangle that has at least 2 equal angles is called an isosceles triangle. 2. First, sketch all the possible triangles Theo can make using pieces of wood that are each five feet long. In order for the perimeter of the triangle to be a whole number, the side of the triangle that is against his house must also be a whole number. No other triangles can be made because the two pieces of wood cannot be spread out anymore and still leave a whole number value for the triangle side that is against his house. So, there are 9 possible triangles he can create that fit the conditions. Look at the sketches. For a triangle to be isosceles, at least two sides must be the same length. Since at least two sides have the same length in every triangle, 9 of the triangles are isosceles. For a triangle to be scalene, none of the sides can have the same length. Since there is no triangle where none of the sides have the same length, 0 of the triangles are scalene. For a triangle to be equilateral, all of the sides must have the same length. Since there is only one triangle where all the sides are the same length, 1 of the triangles is equilateral. 3. Triangle W is a right scalene triangle. Triangle X is an acute scalene triangle. Triangle Y is an acute scalene triangle. Triangle Z is an obtuse scalene triangle. Therefore, X and Y only are acute scalene triangles. 4. An acute triangle is a triangle with all the angles less than According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Since the sum of 12 in and 15 in is greater than 18 in, the side lengths do form a triangle. A triangle with no congruent sides is a scalene triangle. Only one scalene triangle with side lengths of 12 in, 15 in, and 18 in exists. Therefore, exactly one unique triangle exists with the given side lengths. 6. A scalene triangle is a triangle with NO congruent sides and NO congruent angles. 7. A triangle that contains a right angle is called a right triangle. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
7 7. A triangle that contains a right angle is called a right triangle. 8. The triangle has an angle with a measure greater than 90, so the triangle is an obtuse triangle. The triangle does not have any congruent sides, so the triangle is a scalene triangle. Therefore, the triangle is best described as an obtuse scalene triangle. 9. Since in any given triangle, the interior angles add together to equal 180, subtract the measure of each of the first 2 angles from 180 to find the measure of the third angle = 60 So, this triangle has 3 equal angles. A triangle that has 3 equal angles is called an equiangular triangle. 10. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the third side. Since the two given sides have lengths of 18 inches and 23 inches, the third side, AC, must fit the following inequalities. AC < 18 inches + 23 inches = 41 inches and AC > 23 inches 18 inches = 5 inches Thus, 5 inches < AC < 41 inches. Of the answer choices given, 35 inches is the only choice that falls within this range. 11. An equilateral triangle has 3 equal sides. All the angles in an equilateral triangle are equal as well. 12. An obtuse triangle is a triangle that has one angle that is greater than The sum of the angles in a triangle is 180, so the angles do form a triangle. Since all three angles are different, the triangle is a scalene triangle. An infinite number of scalene triangles exist with different side lengths. Therefore, more than one triangle exists with the given angle measures. 14. Based on the restrictions of all sides of integer length, one side of length 5 cm, and a perimeter less than or equal to 16 cm, start by making a list. Keep in mind that that two shorter sides of a triangle must add together to be greater than the third side. Start by looking at the cases in which 5 cm is one of the short sides. If the two short sides are 1 cm and 5 cm, then the remaining side has to be less than 6 cm, which only leaves 5 cm as an option. Since = 10, a triangle with sides of 1 cm, 5 cm, 5 cm is within the perimeter limitations. If the two short sides are 2 cm and 5 cm, then the remaining side has to be less than 7 cm, which leaves 5 cm and 6 cm as options. Since = 9, both of these options are viable. Triangles with sides of 2 cm, 5 cm, 5 cm, and 2 cm, 5 cm, 6 cm are within the perimeter limitations. If the two short sides are 3 cm and 5 cm, then the remaining side has to be less than 8 cm, which leaves 5 cm, 6 cm, and 7 cm as options. Since = 8, all of these options are viable. Triangles with sides of 3 cm, 5 cm, 5 cm; 3 cm, 5 cm, 6 cm; and 3 cm, 5 cm, 7 cm are within the perimeter limitations. If the two short sides are 4 cm and 5 cm, then the remaining side has to be less than 9 cm, which leaves 5 cm, 6 cm, 7 cm, and 8 cm as options. Since = 7, only the triangles with sides of 4 cm, 5 cm, 5 cm; 4 cm, 5 cm, 6 cm; and 4 cm, 5 cm, 7 cm are within the perimeter limitations. If the two short sides are 5 cm and 5 cm, then the remaining side has to be less than 10 cm, which leaves 5 cm, 6 cm, 7 cm, 8 cm, and 9 cm as options. Since = 6, only the triangles with sides of 5 cm, 5 cm, 5 cm and 5 cm, 5 cm, 6 cm are within the perimeter limitations. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
8 If the two short sides are 5 cm and 6 cm, then the remaining side has to be less than 11 cm, which leaves 6 cm, 7 cm, 8 cm, 9 cm, and 10 cm as options. Since = 5, none of the possibilities will be within the perimeter limitations. Now look at situations in which 5 cm would be the long side. Look for integers less than 5 that when added together are greater than 5. There are four possibilities = 6; = 6; = 7; = 8. Therefore 2 cm, 4 cm, 5 cm; 3 cm, 3 cm, 5 cm; 3 cm, 4 cm, 5 cm; and 4 cm, 4 cm, 5 cm are within the given limitations. Make a list of all triangles that were within the given limitations cm, 5 cm, 5 cm 2. 2 cm, 5 cm, 5 cm 3. 2 cm, 5 cm, 6 cm 4. 3 cm, 5 cm, 5 cm 5. 3 cm, 5 cm, 6 cm 6. 3 cm, 5 cm, 7 cm 7. 4 cm, 5 cm, 5 cm 8. 4 cm, 5 cm, 6 cm 9. 4 cm, 5 cm, 7 cm cm, 5 cm, 5 cm cm, 5 cm, 6 cm cm, 4 cm, 5 cm cm, 3 cm, 5 cm cm, 4 cm, 5 cm cm, 4 cm, 5 cm Therefore, Shelbie can construct 15 unique triangles of which 7 are strictly isosceles and 1 is equilateral. 15. An obtuse angle is greater than 90, an acute angle is less than 90, and a right angle is equal to 90. Since the sum of the angles in a triangle is 180, at least two of the angles must be acute. Therefore, it is false that a triangle can be drawn with exactly one acute angle. 16. First, find the measure of the missing angle in the triangle. 180 ( ) = = 60 All the angles in the triangle are less than 90, so the triangle is an acute triangle. Next, the three angles in the triangle each measure 60, so the triangle is equiangular and equilateral. Therefore, the triangle is best described as an acute equilateral triangle. 17. According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Since the sum of 4 m and 4 m is greater than 7 m, the side lengths do form a triangle. A triangle with two equal sides is an isosceles triangle. Only one isosceles triangle with side lengths of 4 m, 4 m, and 7 m exists. Therefore, exactly one unique triangle exists with the given side lengths. 18. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the third side. Since the two given sides have lengths of 20 inches and 29 inches, the third side, AC, must fit the following inequalities. AC < 20 inches + 29 inches = 49 inches and AC > 29 inches 20 inches = 9 inches Thus, 9 inches < AC < 49 inches. Of the answer choices given, 44 inches is the only choice that falls within this range. 19. The sum of the angles in a triangle is 180, so the angles do form a triangle. Since two of the three angles are congruent, the triangle is an isosceles triangle. An infinite number of isosceles triangles exist with different side lengths. Therefore, more than one triangle exists with the given angle measures. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
9 Therefore, more than one triangle exists with the given angle measures. 20. The sum of the angles in a triangle is 180. These angles add up to more than 180, so the angles do not form a triangle. Therefore, no triangle exists with the given angle measures. practice worksheet/9e4d8?cfid=1237a6d7 82b b4e4 a0242c11ee78&cftoken=0&apprnd= /9
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