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Angle Relationships

Grade 7 · Geometry · Worksheet 3

  1. Mason is examining a diagram of two intersecting lines. One of the angles formed at the intersection is 127°. The angle adjacent to it on the same straight line is represented as (3x + 2)°. These two angles are supplementary. What is the value of x? Answer: ______________
  2. Hana is designing a decorative garden path using rectangular paving stones. Two adjacent angles formed by the path and the garden edge are supplementary. The larger angle measures 62 degrees more than twice the measure of the smaller angle. What is the measure of the smaller angle in degrees? Answer: ______________
  3. Mere is designing a geometric art piece using two intersecting metal rods. The rods create four angles at their intersection point. One of the angles measures 108°. The angle directly opposite this angle is marked as (3x + 18)°. These two angles are vertical angles. Find the value of x. Answer: ______________
  4. Hana is designing a large geometric sculpture for a park. Two intersecting steel beams form four angles at their intersection point. One angle measures 112°, and its adjacent angle on the same side of one beam measures (3x - 14)°. These two angles are supplementary. Find the value of x. Answer: ______________
  5. (5x + 25)° + (3x - 15)° = 180° Answer: ______________
  6. Two supplementary angles have measures (3x + 14)° and (5x - 30)°. Find x. Answer: ______________
  7. Kaia is helping design a geometric art installation where two long metal beams cross each other. At their intersection point, four angles are formed. One of the angles measures 108°. The angle directly opposite it (vertical angle) is equal to (3x + 18)°. Find the value of x. Answer: ______________
  8. Emma is building a decorative wooden frame for a mirror. Two adjacent wooden strips meet to form a straight line, creating two supplementary angles. The larger angle measures 15 degrees more than four times the smaller angle. What is the measure of the smaller angle in degrees? Answer: ______________
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Answer Key & Explanations

Angle Relationships · Grade 7 · Worksheet 3

  1. Mason is examining a diagram of two intersecting lines. One of the angles formed at the intersection is 127°. The angle adjacent to it on the same straight line is represented as (3x + 2)°. These two angles are supplementary. What is the value of x? Answer: 17 Solution: Identify the relationship. The two angles are supplementary, meaning their sum is 180°. Write the equation: 127° + (3x + 2)° = 180° Combine the constant terms: 127 + 2 = 129, so the equation is 129 + 3x = 180.
    Full step-by-step solution

    Step 1: Identify the relationship. The two angles are supplementary, meaning their sum is 180°. Step 2: Write the equation: 127° + (3x + 2)° = 180° Step 3: Combine the constant terms: 127 + 2 = 129, so the equation is 129 + 3x = 180. Step 4: Subtract 129 from both sides: 3x = 180 - 129 = 51. Step 5: Divide both sides by 3: x = 51 / 3 = 17. Therefore, the value of x is 17.

  2. Hana is designing a decorative garden path using rectangular paving stones. Two adjacent angles formed by the path and the garden edge are supplementary. The larger angle measures 62 degrees more than twice the measure of the smaller angle. What is the measure of the smaller angle in degrees? Answer: 39.333 Solution: Let x represent the measure of the smaller angle in degrees. The larger angle is 62 degrees more than twice the smaller angle, so it is 2x + 62.
    Full step-by-step solution

    Step 1: Let x represent the measure of the smaller angle in degrees. Step 2: The larger angle is 62 degrees more than twice the smaller angle, so it is 2x + 62. Step 3: Since the angles are supplementary, their sum is 180 degrees: x + (2x + 62) = 180 Step 4: Combine like terms: 3x + 62 = 180 Step 5: Subtract 62 from both sides: 3x = 118 Step 6: Divide both sides by 3: x = 118 / 3 = 39.333... Step 7: The smaller angle measures 39.333 degrees (or 118/3 degrees). The answer is 39.333.

  3. Mere is designing a geometric art piece using two intersecting metal rods. The rods create four angles at their intersection point. One of the angles measures 108°. The angle directly opposite this angle is marked as (3x + 18)°. These two angles are vertical angles. Find the value of x. Answer: 30 Solution: Identify the relationship. Vertical angles are opposite angles formed by two intersecting lines. Since the given angle (108°) and the angle marked (3x + 18)° are vertical angles, they are equal: 108 = 3x + 18.
    Full step-by-step solution

    Step 1: Identify the relationship. Vertical angles are opposite angles formed by two intersecting lines. Vertical angles are always equal in measure. Step 2: Set up the equation. Since the given angle (108°) and the angle marked (3x + 18)° are vertical angles, they are equal: 108 = 3x + 18. Step 3: Solve for x. Subtract 18 from both sides: 108 - 18 = 3x, so 90 = 3x. Step 4: Divide both sides by 3: 90 / 3 = x, so x = 30. The answer is 30.

  4. Hana is designing a large geometric sculpture for a park. Two intersecting steel beams form four angles at their intersection point. One angle measures 112°, and its adjacent angle on the same side of one beam measures (3x - 14)°. These two angles are supplementary. Find the value of x. Answer: 27.33 Solution: Identify the relationship. The two angles are supplementary, meaning their measures add to 180°. Write the equation: 112° + (3x - 14)° = 180°.
    Full step-by-step solution

    Step 1: Identify the relationship. The two angles are supplementary, meaning their measures add to 180°. Step 2: Write the equation: 112° + (3x - 14)° = 180°. Step 3: Combine the constant terms on the left: 112 - 14 = 98. So the equation becomes 98 + 3x = 180. Step 4: Subtract 98 from both sides: 3x = 180 - 98 = 82. Step 5: Divide both sides by 3: x = 82 / 3 = 27.333... Step 6: Rounding to two decimal places (if needed): x = 27.33. Therefore, the value of x is 27.33.

  5. (5x + 25)° + (3x - 15)° = 180° Answer: 21.25 Solution: Write the equation: (5x + 25) + (3x - 15) = 180 Combine like terms: 5x + 3x + 25 - 15 = 180 → 8x + 10 = 180 Subtract 10 from both sides: 8x = 170 Divide both sides by 8: x = 170 ÷ 8 Calculate: 170 ÷ 8 = 21.25 The answer is 21.25.
    Full step-by-step solution

    Step 1: Write the equation: (5x + 25) + (3x - 15) = 180 Step 2: Combine like terms: 5x + 3x + 25 - 15 = 180 → 8x + 10 = 180 Step 3: Subtract 10 from both sides: 8x = 170 Step 4: Divide both sides by 8: x = 170 ÷ 8 Step 5: Calculate: 170 ÷ 8 = 21.25 The answer is 21.25.

  6. Two supplementary angles have measures (3x + 14)° and (5x - 30)°. Find x. Answer: 24.5 Solution: Write the equation for supplementary angles: (3x + 14) + (5x - 30) = 180 Combine like terms: 3x + 5x + 14 - 30 = 180 → 8x - 16 = 180 Add 16 to both sides: 8x = 196 Divide both sides by 8: x = 24.5 The answer is 24.5.
    Full step-by-step solution

    Step 1: Write the equation for supplementary angles: (3x + 14) + (5x - 30) = 180 Step 2: Combine like terms: 3x + 5x + 14 - 30 = 180 → 8x - 16 = 180 Step 3: Add 16 to both sides: 8x = 196 Step 4: Divide both sides by 8: x = 24.5 The answer is 24.5.

  7. Kaia is helping design a geometric art installation where two long metal beams cross each other. At their intersection point, four angles are formed. One of the angles measures 108°. The angle directly opposite it (vertical angle) is equal to (3x + 18)°. Find the value of x. Answer: 30 Solution: Recall that vertical angles are equal in measure. Set up the equation: 108° = (3x + 18)°. Remove the degree symbols for simplicity: 108 = 3x + 18.
    Full step-by-step solution

    Step 1: Recall that vertical angles are equal in measure. Step 2: Set up the equation: 108° = (3x + 18)°. Step 3: Remove the degree symbols for simplicity: 108 = 3x + 18. Step 4: Subtract 18 from both sides: 108 - 18 = 3x + 18 - 18, so 90 = 3x. Step 5: Divide both sides by 3: 90 / 3 = 3x / 3, so 30 = x. Step 6: Therefore, x = 30. The answer is 30.

  8. Emma is building a decorative wooden frame for a mirror. Two adjacent wooden strips meet to form a straight line, creating two supplementary angles. The larger angle measures 15 degrees more than four times the smaller angle. What is the measure of the smaller angle in degrees? Answer: 33 Solution: Let x represent the measure of the smaller angle in degrees. The larger angle is 15 degrees more than four times the smaller angle, so it can be written as 4x + 15.
    Full step-by-step solution

    Step 1: Let x represent the measure of the smaller angle in degrees. Step 2: The larger angle is 15 degrees more than four times the smaller angle, so it can be written as 4x + 15. Step 3: Since the angles are supplementary, their sum is 180 degrees: x + (4x + 15) = 180 Step 4: Combine like terms: 5x + 15 = 180 Step 5: Subtract 15 from both sides: 5x = 165 Step 6: Divide both sides by 5: x = 33 Step 7: The smaller angle measures 33 degrees.