Multi-Step Angle Problems

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Answer Key & Explanations

Multi-Step Angle Problems · Grade 7 · Worksheet 2

1. Olivia is designing a triangular window for a treehouse. The three angles of the triangle are in the ratio 2:3:5. She needs to know the measure of each angle to cut the glass correctly. What is the measure of the largest angle? Answer: 90 degrees Solution: Let the common multiplier be x. The angles are 2x, 3x, and 5x. Step 2: The sum of angles in a triangle is 180 degrees, so 2x + 3x + 5x = 180.
   Full step-by-step solution

   Step 1: Let the common multiplier be x. The angles are 2x, 3x, and 5x. Step 2: The sum of angles in a triangle is 180 degrees, so 2x + 3x + 5x = 180. Step 3: Combine like terms: 10x = 180. Step 4: Divide both sides by 10: x = 18. Step 5: The largest angle is 5x = 5 * 18 = 90 degrees. The answer is 90 degrees.

2. 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

3. Mere is building a triangular wooden frame for a kite. One angle of the triangle measures 48 degrees. The other two angles are such that one is 6 degrees more than twice the other. What are the measures of all three angles in Mere's triangle? Answer: 48 degrees, 42 degrees, 90 degrees Solution: Let the smaller of the two unknown angles be x degrees. Then the other unknown angle is 2x + 6 degrees. The sum of all three angles in a triangle is 180 degrees.
   Full step-by-step solution

   Step 1: Let the smaller of the two unknown angles be x degrees. Then the other unknown angle is 2x + 6 degrees. Step 2: The sum of all three angles in a triangle is 180 degrees. So: 48 + x + (2x + 6) = 180 Step 3: Combine like terms: 48 + 6 + x + 2x = 180 → 54 + 3x = 180 Step 4: Subtract 54 from both sides: 3x = 126 Step 5: Divide by 3: x = 42 Step 6: So the smaller unknown angle is 42 degrees. The larger unknown angle is 2(42) + 6 = 84 + 6 = 90 degrees. Step 7: Check: 48 + 42 + 90 = 180. The three angles are 48 degrees, 42 degrees, and 90 degrees.

4. 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

5. In triangle PQR, angle P is 3x + 7, angle Q is 5x - 3, and angle R is 7x - 4. Find the measure of each angle. Answer: Angle P = 43°, Angle Q = 57°, Angle R = 80° Solution: Write the sum of the angles equal to 180. (3x + 7) + (5x - 3) + (7x - 4) = 180 Combine like terms. 3x + 5x + 7x = 15x 7 - 3 - 4 = 0 So the equation becomes: 15x = 180 Solve for x.
   Full step-by-step solution

   Step 1: Write the sum of the angles equal to 180. (3x + 7) + (5x - 3) + (7x - 4) = 180 Step 2: Combine like terms. 3x + 5x + 7x = 15x 7 - 3 - 4 = 0 So the equation becomes: 15x = 180 Step 3: Solve for x. x = 180 ÷ 15 x = 12 Step 4: Find each angle. Angle P = 3(12) + 7 = 36 + 7 = 43° Angle Q = 5(12) - 3 = 60 - 3 = 57° Angle R = 7(12) - 4 = 84 - 4 = 80° Step 5: Check that they add to 180. 43 + 57 + 80 = 180 ✓ Final answer: Angle P = 43°, Angle Q = 57°, Angle R = 80°

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 ABC, angle A is 7 times angle B, and angle C is 45° more than angle B. Find the measure of each angle. Answer: Angle B = 15°, Angle A = 105°, Angle C = 60° Solution: Let angle B = x. Then angle A = 7x (7 times angle B). Angle C = x + 45 (45° more than angle B).
   Full step-by-step solution

   Let angle B = x. Then angle A = 7x (7 times angle B). Angle C = x + 45 (45° more than angle B). The sum of angles in a triangle is 180°: x + 7x + (x + 45) = 180 Combine like terms: 9x + 45 = 180 Subtract 45 from both sides: 9x = 135 Divide both sides by 9: x = 15 So angle B = 15°. Angle A = 7 × 15 = 105°. Angle C = 15 + 45 = 60°. Check: 15 + 105 + 60 = 180. Correct. Final answer: Angle B = 15°, Angle A = 105°, Angle C = 60°.

8. 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