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Angle Relationships

Grade 7 · Geometry · Worksheet 1

  1. Ava is creating a geometric pattern for a quilt. She draws two adjacent angles on a straight line of fabric. One angle is 38 degrees less than twice the other angle. What is the measure of the smaller angle in degrees? Answer: ______________
  2. A city park designer creates a triangular garden with two supplementary angles. One angle measures 115° and is formed by the intersection of the main walkway and a diagonal path. The other supplementary angle is split into two smaller angles by a decorative fountain placed along the diagonal path. If one of these smaller angles measures 42°, what is the measure of the other smaller angle? Answer: ______________
  3. Mere is setting up two floodlights in her backyard so they form a straight beam of light along a fence line. The two angles formed where the lights meet are supplementary. One angle measures 38 degrees more than twice the other angle. What is the measure of the smaller angle in degrees? Answer: ______________
  4. Two supplementary angles have measures (9x + 18)° and (7x - 12)°. Find x. Answer: ______________
  5. Emma is designing a rectangular garden with two intersecting pathways that form an X shape. At the center of the garden, the pathways create four angles. One of the angles measures 65°, and the angle directly opposite it (vertical angle) measures (5x - 10)°. The other two angles are vertical to each other and are supplementary to the first two. Find the value of x. Answer: ______________
  6. If ∠A and ∠B are supplementary and ∠A = 115°, then ∠B = ? Answer: ______________
  7. Two supplementary angles have measures (6x + 14)° and (4x + 36)°. Find x. Answer: ______________
  8. If ∠A and ∠B are supplementary angles and ∠A = 125°, then ∠B = ? Answer: ______________
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Answer Key & Explanations

Angle Relationships · Grade 7 · Worksheet 1

  1. Ava is creating a geometric pattern for a quilt. She draws two adjacent angles on a straight line of fabric. One angle is 38 degrees less than twice the other angle. What is the measure of the smaller angle in degrees? Answer: 72.67 Solution: Let x represent the measure of the smaller angle in degrees. The larger angle is 38 degrees less than twice the smaller angle, so it can be written as 2x - 38.
    Full step-by-step solution

    Step 1: Let x represent the measure of the smaller angle in degrees. Step 2: The larger angle is 38 degrees less than twice the smaller angle, so it can be written as 2x - 38. Step 3: Since the angles are supplementary (they form a straight line), their sum is 180 degrees: x + (2x - 38) = 180 Step 4: Combine like terms: 3x - 38 = 180 Step 5: Add 38 to both sides: 3x = 218 Step 6: Divide both sides by 3: x = 218 / 3 = 72.666... Step 7: The smaller angle measures approximately 72.67 degrees. The answer is 72.67.

  2. A city park designer creates a triangular garden with two supplementary angles. One angle measures 115° and is formed by the intersection of the main walkway and a diagonal path. The other supplementary angle is split into two smaller angles by a decorative fountain placed along the diagonal path. If one of these smaller angles measures 42°, what is the measure of the other smaller angle? Answer: 23 Solution: We have two supplementary angles. Supplementary angles add up to 180°. One of them is given as 115°.
    Full step-by-step solution

    Step 1: Understand the problem. We have two supplementary angles. Supplementary angles add up to 180°. One of them is given as 115°. Step 2: Find the other supplementary angle. Let the other supplementary angle be A. 115° + A = 180° A = 180° - 115° A = 65° Step 3: Interpret the problem. The 65° angle is split into two smaller angles by a decorative fountain. One of these smaller angles is 42°. Let the other smaller angle be x. Step 4: Set up the equation. 42° + x = 65° Step 5: Solve for x. x = 65° - 42° x = 23° Step 6: Conclusion. The other smaller angle is 23°. Final answer: 23

  3. Mere is setting up two floodlights in her backyard so they form a straight beam of light along a fence line. The two angles formed where the lights meet are supplementary. One angle measures 38 degrees more than twice the other angle. What is the measure of the smaller angle in degrees? Answer: 47.33 Solution: Let x represent the smaller angle in degrees. Step 2: The larger angle is 38 degrees more than twice the smaller, so it is 2x + 38. Step 3: Since the angles are supplementary, x + (2x + 38) = 180.
    Full step-by-step solution

    Step 1: Let x represent the smaller angle in degrees. Step 2: The larger angle is 38 degrees more than twice the smaller, so it is 2x + 38. Step 3: Since the angles are supplementary, x + (2x + 38) = 180. Step 4: Combine like terms: 3x + 38 = 180. Step 5: Subtract 38 from both sides: 3x = 142. Step 6: Divide both sides by 3: x = 142 / 3 = 47.33 (rounded to two decimal places). The smaller angle measures 47.33 degrees.

  4. Two supplementary angles have measures (9x + 18)° and (7x - 12)°. Find x. Answer: 10.875 Solution: Write the equation for supplementary angles: (9x + 18) + (7x - 12) = 180 Combine like terms: 9x + 7x + 18 - 12 = 180 Simplify: 16x + 6 = 180 Subtract 6 from both sides: 16x = 174 Divide both sides by 16: x = 174/16 = 10.875 The answer is 10.875.
    Full step-by-step solution

    Step 1: Write the equation for supplementary angles: (9x + 18) + (7x - 12) = 180 Step 2: Combine like terms: 9x + 7x + 18 - 12 = 180 Step 3: Simplify: 16x + 6 = 180 Step 4: Subtract 6 from both sides: 16x = 174 Step 5: Divide both sides by 16: x = 174/16 = 10.875 The answer is 10.875.

  5. Emma is designing a rectangular garden with two intersecting pathways that form an X shape. At the center of the garden, the pathways create four angles. One of the angles measures 65°, and the angle directly opposite it (vertical angle) measures (5x - 10)°. The other two angles are vertical to each other and are supplementary to the first two. Find the value of x. Answer: 15 Solution: Identify the relationship. Vertical angles are equal. The angle measuring 65° and the angle measuring (5x - 10)° are vertical angles.
    Full step-by-step solution

    Step 1: Identify the relationship. Vertical angles are equal. The angle measuring 65° and the angle measuring (5x - 10)° are vertical angles. Step 2: Set up the equation: 65 = 5x - 10. Step 3: Add 10 to both sides: 75 = 5x. Step 4: Divide both sides by 5: x = 15. Therefore, the value of x is 15.

  6. If ∠A and ∠B are supplementary and ∠A = 115°, then ∠B = ? Answer: 65 Solution: Supplementary angles are two angles whose measures add up to 180 degrees. Write the relationship between ∠A and ∠B. ∠A + ∠B = 180° Substitute the given value of ∠A into the equation.
    Full step-by-step solution

    Step 1: Understand the meaning of supplementary angles. Supplementary angles are two angles whose measures add up to 180 degrees. Step 2: Write the relationship between ∠A and ∠B. ∠A + ∠B = 180° Step 3: Substitute the given value of ∠A into the equation. 115° + ∠B = 180° Step 4: Solve for ∠B. ∠B = 180° - 115° Step 5: Perform the subtraction. ∠B = 65° Final Answer: ∠B = 65

  7. Two supplementary angles have measures (6x + 14)° and (4x + 36)°. Find x. Answer: 13 Solution: Write the equation for supplementary angles: (6x + 14) + (4x + 36) = 180 Combine like terms: 6x + 4x = 10x and 14 + 36 = 50, so 10x + 50 = 180 Subtract 50 from both sides: 10x = 130 Divide both sides by 10: x = 13 The answer is 13.
    Full step-by-step solution

    Step 1: Write the equation for supplementary angles: (6x + 14) + (4x + 36) = 180 Step 2: Combine like terms: 6x + 4x = 10x and 14 + 36 = 50, so 10x + 50 = 180 Step 3: Subtract 50 from both sides: 10x = 130 Step 4: Divide both sides by 10: x = 13 The answer is 13.

  8. If ∠A and ∠B are supplementary angles and ∠A = 125°, then ∠B = ? Answer: 55° Solution: Supplementary angles are two angles whose measures add up to 180 degrees. Write the relationship between ∠A and ∠B. ∠A + ∠B = 180° Substitute the given value of ∠A into the equation.
    Full step-by-step solution

    Step 1: Understand the definition of supplementary angles. Supplementary angles are two angles whose measures add up to 180 degrees. Step 2: Write the relationship between ∠A and ∠B. ∠A + ∠B = 180° Step 3: Substitute the given value of ∠A into the equation. 125° + ∠B = 180° Step 4: Solve for ∠B. ∠B = 180° − 125° Step 5: Perform the subtraction. 180 − 125 = 55 Step 6: State the final answer. ∠B = 55°