Multi-Step Angle Problems

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Answer Key & Explanations

Multi-Step Angle Problems · Grade 7 · Worksheet 1

1. In triangle ABC, angle A = 2x, angle B = 3x, and angle C = 4x. Find the value of x and the measure of each angle. Answer: x = 20, angle A = 40°, angle B = 60°, angle C = 80° Solution: Write the equation for the sum of angles in a triangle. Angle A + Angle B + Angle C = 180 2x + 3x + 4x = 180 Combine like terms. 9x = 180 Solve for x.
   Full step-by-step solution

   Step 1: Write the equation for the sum of angles in a triangle. Angle A + Angle B + Angle C = 180 2x + 3x + 4x = 180 Step 2: Combine like terms. 9x = 180 Step 3: Solve for x. x = 180 ÷ 9 x = 20 Step 4: Find each angle. Angle A = 2x = 2 × 20 = 40° Angle B = 3x = 3 × 20 = 60° Angle C = 4x = 4 × 20 = 80° Step 5: Check the sum. 40° + 60° + 80° = 180° ✓ Final answer: x = 20, angle A = 40°, angle B = 60°, angle C = 80°

2. 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

3. In triangle ABC, angle A is 3 times angle B, and angle C is 45 degrees more than angle B. Find the measure of each angle. Answer: Angle A = 81°, Angle B = 27°, Angle C = 72° Solution: Let angle B = x. Then angle A = 3x (three times angle B). Angle C = x + 45 (45 degrees more than angle B).
   Full step-by-step solution

   Let angle B = x. Then angle A = 3x (three times angle B). Angle C = x + 45 (45 degrees more than angle B). The sum of angles in a triangle is 180 degrees: x + 3x + (x + 45) = 180 Combine like terms: 5x + 45 = 180 Subtract 45 from both sides: 5x = 135 Divide both sides by 5: x = 27 So angle B = 27 degrees. Angle A = 3 × 27 = 81 degrees. Angle C = 27 + 45 = 72 degrees. Check: 81 + 27 + 72 = 180. Correct. Final answer: Angle A = 81°, Angle B = 27°, Angle C = 72°.

4. Emma is designing a triangular patio for her backyard. The angles of the triangle form a geometric sequence where the smallest angle is 20 degrees and the common ratio is 2. What is the measure of the largest angle in her triangular patio design? Answer: 80 Solution: Let the three angles be a, ar, and ar², where a is the smallest angle and r is the common ratio. We know a = 20 and r = 2. Check that these angles sum to 180: 20 + 40 + 80 = 140 + 40 = 180.
   Full step-by-step solution

   Step 1: Let the three angles be a, ar, and ar², where a is the smallest angle and r is the common ratio. Step 2: We know a = 20 and r = 2. Step 3: So the angles are: 20, 20×2 = 40, and 20×2² = 20×4 = 80. Step 4: Check that these angles sum to 180: 20 + 40 + 80 = 140 + 40 = 180. Step 5: The largest angle is 80 degrees. The answer is 80.

5. Charlotte is designing a triangular banner for a school event. The angles of the triangle are in the ratio 4:5:9. She needs to know the measure of each angle to cut the fabric correctly. What is the measure of the largest angle in Charlotte's banner? Answer: 90 Solution: The sum of angles in any triangle is 180 degrees. Step 2: The ratio is 4:5:9, so the total number of parts is 4 + 5 + 9 = 18. Step 3: Each part is worth 180 / 18 = 10 degrees.
   Full step-by-step solution

   Step 1: The sum of angles in any triangle is 180 degrees. Step 2: The ratio is 4:5:9, so the total number of parts is 4 + 5 + 9 = 18. Step 3: Each part is worth 180 / 18 = 10 degrees. Step 4: The largest angle corresponds to the largest ratio number, which is 9. So the largest angle is 9 * 10 = 90 degrees. The answer is 90.

6. Aisha 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 feet Solution: Identify the shortest side of the original triangle: 18 feet Find the scale factor between the triangles: 6 feet (new shortest side) ÷ 18 feet (original shortest side) = 1/3 Apply the scale factor to find the other sides of the smaller triangle: - Second side: 24 feet × 1/3 = 8 feet - Third…
   Full step-by-step solution

   Step 1: Identify the shortest side of the original triangle: 18 feet Step 2: Find the scale factor between the triangles: 6 feet (new shortest side) ÷ 18 feet (original shortest side) = 1/3 Step 3: Apply the scale factor to find the other sides of the smaller triangle: - Second side: 24 feet × 1/3 = 8 feet - Third side: 30 feet × 1/3 = 10 feet Step 4: Calculate the perimeter of the smaller triangle: 6 feet + 8 feet + 10 feet = 24 feet The answer is 24 feet.

7. Two supplementary angles have measures (4x + 15)° and (2x + 45)°. Find the value of x. Answer: 20 Solution: Supplementary angles sum to 180°.
   Full step-by-step solution

   Step 1: Supplementary angles sum to 180°. Step 2: Write the equation: (4x + 15) + (2x + 45) = 180 Step 3: Combine like terms: 4x + 2x + 15 + 45 = 180 → 6x + 60 = 180 Step 4: Subtract 60 from both sides: 6x = 120 Step 5: Divide both sides by 6: x = 20 Step 6: Check: First angle = 4(20) + 15 = 80 + 15 = 95°; Second angle = 2(20) + 45 = 40 + 45 = 85°; 95 + 85 = 180°. The answer is 20.

8. Emma is designing a rectangular garden in her backyard. She draws the garden on a coordinate plane with corners at (0, 0), (40, 0), (40, 25), and (0, 25). She wants to install a diagonal path from corner (0, 0) to corner (40, 25). Along this path, she will place a small circular fountain with its center at the midpoint of the diagonal. The fountain has a radius of 5 units. What is the total area of the garden that is NOT covered by the fountain? (Use π = 3.14) Answer: 921.5 Solution: Find the total area of the rectangular garden. Length = 40 - 0 = 40 units. Width = 25 - 0 = 25 units.
   Full step-by-step solution

   Step 1: Find the total area of the rectangular garden. Length = 40 - 0 = 40 units. Width = 25 - 0 = 25 units. Area = 40 * 25 = 1000 square units. Step 2: Find the midpoint of the diagonal from (0,0) to (40,25). Midpoint = ((0+40)/2, (0+25)/2) = (20, 12.5). This is the center of the fountain. Step 3: Find the area of the circular fountain. Radius = 5 units. Area = π * radius^2 = 3.14 * (5 * 5) = 3.14 * 25 = 78.5 square units. Step 4: Subtract the fountain area from the garden area to find the uncovered area. 1000 - 78.5 = 921.5 square units. The answer is 921.5.