Multi-Step Angle Problems

lessonbunny.com

Answer Key & Explanations

Multi-Step Angle Problems · Grade 7 · Worksheet 2

1. Liam is designing a triangular garden plot with angles that form an arithmetic sequence. The smallest angle is 30 degrees. What are the measures of all three angles in his garden plot? Answer: 30, 60, 90 Solution: We have a triangle with angles in arithmetic sequence. The smallest angle is 30°. Let the angles be: \( a, a + d, a + 2d \), where \( a = 30 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a triangle with angles in arithmetic sequence. The smallest angle is 30°. Let the angles be: \( a, a + d, a + 2d \), where \( a = 30 \). --- **Step 2: Use triangle angle sum** The sum of angles in a triangle is 180°. So: \[ 30 + (30 + d) + (30 + 2d) = 180 \] --- **Step 3: Simplify and solve for \( d \)** \[ 90 + 3d = 180 \] \[ 3d = 90 \] \[ d = 30 \] --- **Step 4: Find all three angles** First angle: \( a = 30 \) Second angle: \( 30 + d = 30 + 30 = 60 \) Third angle: \( 30 + 2d = 30 + 60 = 90 \) --- **Step 5: Check** Sum: \( 30 + 60 + 90 = 180 \) ✓ Arithmetic sequence: \( 30, 60, 90 \) — common difference 30 ✓ --- **Final answer:** 30, 60, 90

2. Two supplementary angles have measures (7x + 12)° and (5x + 18)°. Find the value of x. Answer: 12.5 Solution: Write the equation for supplementary angles: (7x + 12) + (5x + 18) = 180. Combine like terms: 7x + 5x = 12x, and 12 + 18 = 30, so 12x + 30 = 180. Subtract 30 from both sides: 12x = 150.
   Full step-by-step solution

   Step 1: Write the equation for supplementary angles: (7x + 12) + (5x + 18) = 180. Step 2: Combine like terms: 7x + 5x = 12x, and 12 + 18 = 30, so 12x + 30 = 180. Step 3: Subtract 30 from both sides: 12x = 150. Step 4: Divide both sides by 12: x = 150 / 12 = 12.5. The value of x is 12.5.

3. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (12, 1), (12, 8), and (2, 8). A diagonal path is drawn from (2, 1) to (12, 8), dividing the garden into two triangular sections. What is the area of one of these triangular sections? Answer: 35 Solution: A = (2, 1) B = (12, 1) C = (12, 8) D = (2, 8) The diagonal from A = (2, 1) to C = (12, 8) divides the rectangle into two congruent triangles.
   Full step-by-step solution

   Let's solve step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (12, 1) C = (12, 8) D = (2, 8) The diagonal from A = (2, 1) to C = (12, 8) divides the rectangle into two congruent triangles. --- **Step 2: Find the area of the rectangle** Length along x-axis: from x = 2 to x = 12 → length = 12 - 2 = 10 Height along y-axis: from y = 1 to y = 8 → height = 8 - 1 = 7 Area of rectangle = length × height = 10 × 7 = 70 --- **Step 3: Area of one triangle** The diagonal splits the rectangle into two triangles of equal area. Area of one triangle = (Area of rectangle) / 2 = 70 / 2 = 35 --- **Step 4: Conclusion** The area of one triangular section is 35. --- **Final answer:** 35

4. Two supplementary angles are in the ratio 4:5. Find the measure of each angle. Answer: 80° and 100° Solution: Let the two angles be 4x and 5x, since they are in the ratio 4:5. Supplementary angles sum to 180°, so 4x + 5x = 180. Combine like terms: 9x = 180.
   Full step-by-step solution

   Step 1: Let the two angles be 4x and 5x, since they are in the ratio 4:5. Step 2: Supplementary angles sum to 180°, so 4x + 5x = 180. Step 3: Combine like terms: 9x = 180. Step 4: Divide both sides by 9: x = 20. Step 5: First angle = 4x = 4 × 20 = 80°. Step 6: Second angle = 5x = 5 × 20 = 100°. Step 7: Check: 80° + 100° = 180°, which is correct. Final answer: The angles are 80° and 100°.

5. Sophia is designing a triangular banner for a school event. Two of the angles in the banner measure 47° and 82°. She needs to know the measure of the third angle to ensure the banner is cut correctly. What is the measure of the third angle in degrees? Answer: 51 Solution: Recall that the sum of the interior angles of any triangle is 180°. Add the two given angles: 47° + 82° = 129°. Subtract the sum from 180° to find the third angle: 180° - 129° = 51°.
   Full step-by-step solution

   Step 1: Recall that the sum of the interior angles of any triangle is 180°. Step 2: Add the two given angles: 47° + 82° = 129°. Step 3: Subtract the sum from 180° to find the third angle: 180° - 129° = 51°. The third angle measures 51°.

6. Emma is designing a triangular garden with angles that have a ratio of 3:5:7. She needs to determine the measure of the largest angle to select the right type of plants for that sun exposure. What is the measure of the largest angle in degrees? Answer: 84 Solution: The angles are in ratio 3:5:7, so let the angles be 3x, 5x, and 7x degrees. The sum of angles in a triangle is 180 degrees, so 3x + 5x + 7x = 180. Combine like terms: 15x = 180.
   Full step-by-step solution

   Step 1: The angles are in ratio 3:5:7, so let the angles be 3x, 5x, and 7x degrees. Step 2: The sum of angles in a triangle is 180 degrees, so 3x + 5x + 7x = 180. Step 3: Combine like terms: 15x = 180. Step 4: Solve for x: x = 180 ÷ 15 = 12. Step 5: The largest angle is 7x = 7 × 12 = 84 degrees. The measure of the largest angle is 84 degrees.

7. In triangle Aroha, angle A is 3 times angle B, and angle C is 15° more than angle B. Find the measure of each angle. Answer: Angle A = 99°, Angle B = 33°, Angle C = 48° Solution: Let angle B = x. Then angle A = 3x (three times angle B). Angle C = x + 15 (15° more than angle B).
   Full step-by-step solution

   Let angle B = x. Then angle A = 3x (three times angle B). Angle C = x + 15 (15° more than angle B). The sum of angles in a triangle is 180°: x + 3x + (x + 15) = 180 Combine like terms: 5x + 15 = 180 Subtract 15 from both sides: 5x = 165 Divide both sides by 5: x = 33 So angle B = 33°. Angle A = 3 × 33 = 99°. Angle C = 33 + 15 = 48°. Check: 99 + 33 + 48 = 180. Correct. Final answer: Angle A = 99°, Angle B = 33°, Angle C = 48°.

8. Aroha is building a wooden frame in the shape of a triangle. One angle of the triangle measures 63 degrees. The second angle is three times the measure of the third angle. Find the measure of the second and third angles. Answer: second angle = 87.75 degrees, third angle = 29.25 degrees Solution: The sum of the angles in any triangle is 180 degrees. Let the third angle = x degrees. The second angle = 3x degrees.
   Full step-by-step solution

   Step 1: The sum of the angles in any triangle is 180 degrees. Step 2: Let the third angle = x degrees. Step 3: The second angle = 3x degrees. Step 4: The first angle is given as 63 degrees. Step 5: Write the equation: 63 + 3x + x = 180. Step 6: Combine like terms: 63 + 4x = 180. Step 7: Subtract 63 from both sides: 4x = 180 - 63 = 117. Step 8: Divide both sides by 4: x = 117 / 4 = 29.25. Step 9: So the third angle = 29.25 degrees. Step 10: The second angle = 3 * 29.25 = 87.75 degrees. Step 11: Check: 63 + 87.75 + 29.25 = 180. Correct. The answer is: second angle = 87.75 degrees, third angle = 29.25 degrees.