Multi-Step Angle Problems

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Answer Key & Explanations

Multi-Step Angle Problems · Grade 7 · Worksheet 1

1. Aroha is designing a kite. The kite is shaped like a quadrilateral with one pair of parallel sides. Two angles of the kite are given: one angle measures 7x + 15 degrees, and the other angle (adjacent to it on the same parallel side) measures 3x + 25 degrees. These two angles are supplementary. What is the measure of each of these two angles? Answer: 7x + 15 = 113 degrees, 3x + 25 = 67 degrees Solution: Since the two angles are supplementary, their sum is 180 degrees. Write the equation: (7x + 15) + (3x + 25) = 180. Step 2: Combine like terms: 7x + 3x = 10x, and 15 + 25 = 40.
   Full step-by-step solution

   Step 1: Since the two angles are supplementary, their sum is 180 degrees. Write the equation: (7x + 15) + (3x + 25) = 180. Step 2: Combine like terms: 7x + 3x = 10x, and 15 + 25 = 40. So, 10x + 40 = 180. Step 3: Subtract 40 from both sides: 10x = 140. Step 4: Divide both sides by 10: x = 14. Step 5: Substitute x = 14 into the first angle: 7(14) + 15 = 98 + 15 = 113 degrees. Step 6: Substitute x = 14 into the second angle: 3(14) + 25 = 42 + 25 = 67 degrees. Step 7: Check: 113 + 67 = 180, so the angles are supplementary. The two angles are 113 degrees and 67 degrees.

2. Isabella is designing a triangular sail for a model boat. The sail is a right triangle. One of the acute angles measures 4 times the measure of the other acute angle. What are the measures of the two acute angles? Answer: 18 degrees and 72 degrees Solution: In a right triangle, one angle is 90 degrees. The sum of all angles is 180 degrees, so the two acute angles must sum to 180 - 90 = 90 degrees. Let the smaller acute angle be x degrees.
   Full step-by-step solution

   Step 1: In a right triangle, one angle is 90 degrees. The sum of all angles is 180 degrees, so the two acute angles must sum to 180 - 90 = 90 degrees. Step 2: Let the smaller acute angle be x degrees. Then the larger acute angle is 4x degrees (since it is 4 times the smaller). Step 3: Write an equation: x + 4x = 90 Step 4: Combine like terms: 5x = 90 Step 5: Divide both sides by 5: x = 18 Step 6: The smaller acute angle is 18 degrees. The larger acute angle is 4 * 18 = 72 degrees. The answer is 18 degrees and 72 degrees.

3. Liam is designing a triangular garden with angles that have a ratio of 2:3:4. He needs to determine the measure of the largest angle to ensure proper sunlight exposure for his plants. What is the measure of the largest angle in degrees? Answer: 80 Solution: The angles of the triangle are in the ratio 2 : 3 : 4. Let the angles be \( 2x \), \( 3x \), and \( 4x \) degrees. The sum of angles in any triangle is 180 degrees.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Understand the problem** The angles of the triangle are in the ratio 2 : 3 : 4. Let the angles be \( 2x \), \( 3x \), and \( 4x \) degrees. --- **Step 2: Use the triangle angle sum property** The sum of angles in any triangle is 180 degrees. So: \[ 2x + 3x + 4x = 180 \] --- **Step 3: Combine like terms** \[ 9x = 180 \] --- **Step 4: Solve for x** \[ x = 180 / 9 \] \[ x = 20 \] --- **Step 5: Find each angle** First angle: \( 2x = 2 \times 20 = 40 \) degrees Second angle: \( 3x = 3 \times 20 = 60 \) degrees Third angle: \( 4x = 4 \times 20 = 80 \) degrees --- **Step 6: Identify the largest angle** The largest angle is \( 4x = 80 \) degrees. --- **Final Answer:** 80

4. Emma is designing a triangular garden for her backyard. One angle of the triangle measures 41 degrees, and another angle measures 73 degrees. She wants to build a fence around the garden and needs to know all three angles to cut the fence pieces correctly. What is the measure of the third angle in Emma's triangular garden? Answer: 66 degrees Solution: Recall that the sum of the interior angles of any triangle is always 180 degrees. Add the two known angles: 41 + 73 = 114 degrees.
   Full step-by-step solution

   Step 1: Recall that the sum of the interior angles of any triangle is always 180 degrees. Step 2: Add the two known angles: 41 + 73 = 114 degrees. Step 3: Subtract the sum of the two known angles from the total sum of 180 degrees to find the third angle: 180 - 114 = 66 degrees. The measure of the third angle is 66 degrees.

5. In triangle ABC, angle A is 3x + 15, angle B is 2x - 10, and angle C is 5x + 25. Find the measure of each angle. Answer: Angle A = 60°, Angle B = 20°, Angle C = 100° Solution: Write the sum of the angles: (3x + 15) + (2x - 10) + (5x + 25) = 180. Combine like terms: 3x + 2x + 5x = 10x, and 15 - 10 + 25 = 30. So 10x + 30 = 180.
   Full step-by-step solution

   Step 1: Write the sum of the angles: (3x + 15) + (2x - 10) + (5x + 25) = 180. Step 2: Combine like terms: 3x + 2x + 5x = 10x, and 15 - 10 + 25 = 30. So 10x + 30 = 180. Step 3: Subtract 30 from both sides: 10x = 150. Step 4: Divide by 10: x = 15. Step 5: Find each angle: Angle A = 3(15) + 15 = 45 + 15 = 60°. Angle B = 2(15) - 10 = 30 - 10 = 20°. Angle C = 5(15) + 25 = 75 + 25 = 100°. Step 6: Check: 60 + 20 + 100 = 180. Correct. Final answer: Angle A = 60°, Angle B = 20°, Angle C = 100°.

6. Noah is building a wooden frame for a triangular window. One angle of the triangle measures 61°, and another angle measures 76°. Noah needs to know the measure of the third angle to cut the final piece of wood. What is the measure of the third angle in degrees? Answer: 43 Solution: The sum of all angles in a triangle is always 180°. Add the two known angles: 61° + 76° = 137°. Subtract the sum from 180° to find the third angle: 180° - 137° = 43°.
   Full step-by-step solution

   Step 1: The sum of all angles in a triangle is always 180°. Step 2: Add the two known angles: 61° + 76° = 137°. Step 3: Subtract the sum from 180° to find the third angle: 180° - 137° = 43°. The third angle measures 43°.

7. Liam is designing a triangular garden with sides measuring 18 feet, 24 feet, and 30 feet. He wants to create a similar triangular flower bed where the longest side is 20 feet. What will be the perimeter of the smaller flower bed? Answer: 48 Solution: We have two similar triangles. The larger triangle (garden) has sides: 18 ft, 24 ft, 30 ft. The smaller triangle (flower bed) is similar to it, with longest side = 20 ft.
   Full step-by-step solution

   Step 1: Understand the problem. We have two similar triangles. The larger triangle (garden) has sides: 18 ft, 24 ft, 30 ft. The smaller triangle (flower bed) is similar to it, with longest side = 20 ft. We need the perimeter of the smaller triangle. Step 2: Identify the longest side of the larger triangle. The sides are 18, 24, 30. The longest side is 30 ft. Step 3: Find the scale factor from the larger to the smaller triangle. The longest sides correspond in similar triangles. Scale factor k = (longest side of smaller) / (longest side of larger) k = 20 / 30 = 2/3. Step 4: Apply the scale factor to all sides of the larger triangle to get the smaller triangle's sides. First side: 18 × (2/3) = 36/3 = 12 ft Second side: 24 × (2/3) = 48/3 = 16 ft Third side: 30 × (2/3) = 60/3 = 20 ft (matches given longest side) Step 5: Find the perimeter of the smaller triangle. Perimeter = 12 + 16 + 20 = 48 ft. Step 6: Conclusion. The perimeter of the smaller flower bed is 48 feet. Answer: 48

8. Emma is designing a triangular patio with sides measuring 18 feet, 24 feet, and 30 feet. She wants to create a similar triangular flower bed where the shortest side is only 6 feet. What will be the perimeter of the smaller flower bed? Answer: 24 Solution: Identify the scale factor using the shortest sides. The original triangle's shortest side is 18 feet, and the similar triangle's shortest side is 6 feet. Scale factor = 6 / 18 = 1/3.
   Full step-by-step solution

   Step 1: Identify the scale factor using the shortest sides. The original triangle's shortest side is 18 feet, and the similar triangle's shortest side is 6 feet. Scale factor = 6 / 18 = 1/3. Step 2: Apply the scale factor to find the other two sides of the smaller triangle. Second side = 24 * (1/3) = 8 feet. Third side = 30 * (1/3) = 10 feet. Step 3: Calculate the perimeter of the smaller triangle. Perimeter = 6 + 8 + 10 = 24 feet. The perimeter of the smaller flower bed is 24 feet.