Angle Relationships

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Angle Relationships · Grade 7 · Worksheet 2

1. 2x + 15 = 180 - (x + 30) Answer: 45 Solution: 2x + 15 = 180 - (x + 30) 180 - (x + 30) = 180 - x - 30 2x + 15 = 180 - x - 30 180 - 30 = 150 2x + 15 = 150 - x 2x + x + 15 = 150 - x + x 3x + 15 = 150 Subtract 15 from both sides: 3x + 15 - 15 = 150 - 15 3x = 135 Divide both sides by 3: x = 135 / 3 x = 45 Substitute x = 45 into the original…
   Full step-by-step solution

   Let's solve step by step. We start with the equation: 2x + 15 = 180 - (x + 30) **Step 1: Distribute the negative sign on the right side** 180 - (x + 30) = 180 - x - 30 So the equation becomes: 2x + 15 = 180 - x - 30 **Step 2: Simplify the right side** 180 - 30 = 150 So: 2x + 15 = 150 - x **Step 3: Get all x terms on one side** Add x to both sides: 2x + x + 15 = 150 - x + x 3x + 15 = 150 **Step 4: Isolate the x term** Subtract 15 from both sides: 3x + 15 - 15 = 150 - 15 3x = 135 **Step 5: Solve for x** Divide both sides by 3: x = 135 / 3 x = 45 **Final check:** Substitute x = 45 into the original equation: Left side: 2(45) + 15 = 90 + 15 = 105 Right side: 180 - (45 + 30) = 180 - 75 = 105 Both sides match. **Answer:** x = 45

2. A construction company is building a large bridge where two support beams meet. The angle between Beam A and Beam B measures 154°. What is the measure of the supplementary angle to this angle? Answer: 26 Solution: Supplementary angles are two angles whose measures add up to 180°. The given angle measures 154°. To find the supplementary angle, subtract the given angle from 180°.
   Full step-by-step solution

   Step 1: Supplementary angles are two angles whose measures add up to 180°. Step 2: The given angle measures 154°. Step 3: To find the supplementary angle, subtract the given angle from 180°. Step 4: 180° - 154° = 26° Step 5: The supplementary angle measures 26°. The answer is 26.

3. Aroha is building a wooden frame for a rectangular window. Two adjacent wooden beams form a straight line along the top edge. The larger angle between the beams measures 48 degrees more than twice the smaller angle. If these two angles are supplementary, what is the measure of the smaller angle in degrees? Answer: 44 Solution: Let x represent the measure of the smaller angle in degrees. The larger angle is 48 degrees more than twice the smaller angle, so it can be written as 2x + 48.
   Full step-by-step solution

   Step 1: Let x represent the measure of the smaller angle in degrees. Step 2: The larger angle is 48 degrees more than twice the smaller angle, so it can be written as 2x + 48. Step 3: Since the angles are supplementary, their sum is 180 degrees: x + (2x + 48) = 180 Step 4: Combine like terms: 3x + 48 = 180 Step 5: Subtract 48 from both sides: 3x = 132 Step 6: Divide both sides by 3: x = 44 Step 7: The smaller angle measures 44 degrees. The answer is 44.

4. A construction company is building two straight sections of a new highway that will intersect. The angle between the northbound and eastbound lanes measures 143°. What is the measure of the supplementary angle to this intersection angle? Answer: 37 Solution: Supplementary angles are two angles whose measures add up to 180°. The given angle measures 143°. To find the supplementary angle, subtract the given angle from 180°.
   Full step-by-step solution

   Step 1: Supplementary angles are two angles whose measures add up to 180°. Step 2: The given angle measures 143°. Step 3: To find the supplementary angle, subtract the given angle from 180°. Step 4: 180° - 143° = 37° The supplementary angle measures 37°.

5. If ∠A and ∠B are supplementary angles and ∠A = 127°, then ∠B = ? Answer: 53° 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. Since they are supplementary: ∠A + ∠B = 180° Step 3: Substitute the given value of ∠A into the equation. ∠A = 127°, so: 127° + ∠B = 180° Step 4: Solve for ∠B. Subtract 127° from both sides: ∠B = 180° − 127° Step 5: Perform the subtraction. 180 − 127 = 53 Step 6: State the final answer. ∠B = 53°

6. Emma is designing a stained glass window with two adjacent angles that form a straight line. One angle measures (5x + 40)° and the other measures (3x + 20)°. What is the measure of the larger angle in degrees? Answer: 115 Solution: Since the angles form a straight line, they are supplementary and their sum is 180°.
   Full step-by-step solution

   Step 1: Since the angles form a straight line, they are supplementary and their sum is 180°. Step 2: Write the equation: (5x + 40) + (3x + 20) = 180 Step 3: Combine like terms: 5x + 3x + 40 + 20 = 180 → 8x + 60 = 180 Step 4: Subtract 60 from both sides: 8x = 120 Step 5: Divide both sides by 8: x = 15 Step 6: Calculate the first angle: 5(15) + 40 = 75 + 40 = 115° Step 7: Calculate the second angle: 3(15) + 20 = 45 + 20 = 65° Step 8: Compare the angles: 115° > 65°, so the larger angle is 115°. The answer is 115.

7. (3x + 15) + (2x - 25) = 180 Answer: 38 Solution: (3x + 15) + (2x - 25) = 180 Since it's all addition between the parentheses, we can just drop them: 3x + 15 + 2x - 25 = 180 Combine the x terms: 3x + 2x = 5x Combine the constants: 15 - 25 = -10 5x - 10 = 180 Add 10 to both sides: 5x - 10 + 10 = 180 + 10 5x = 190 Divide both sides by 5: x = 190…
   Full step-by-step solution

   Let's solve the equation step-by-step. We start with: (3x + 15) + (2x - 25) = 180 **Step 1: Remove the parentheses** Since it's all addition between the parentheses, we can just drop them: 3x + 15 + 2x - 25 = 180 **Step 2: Combine like terms** Combine the x terms: 3x + 2x = 5x Combine the constants: 15 - 25 = -10 So now we have: 5x - 10 = 180 **Step 3: Isolate the x term** Add 10 to both sides: 5x - 10 + 10 = 180 + 10 5x = 190 **Step 4: Solve for x** Divide both sides by 5: x = 190 / 5 x = 38 **Final check:** Substitute x = 38 into the original equation: (3*38 + 15) + (2*38 - 25) = (114 + 15) + (76 - 25) = 129 + 51 = 180 ✓ **Answer:** x = 38

8. (2x + 15)° + (3x - 5)° = 180° Answer: x = 34 Solution: We are given the equation: (2x + 15)° + (3x - 5)° = 180° Remove the degree symbols for calculation purposes. (2x + 15) + (3x - 5) = 180 Combine like terms on the left side.
   Full step-by-step solution

   We are given the equation: (2x + 15)° + (3x - 5)° = 180° Step 1: Remove the degree symbols for calculation purposes. The equation becomes: (2x + 15) + (3x - 5) = 180 Step 2: Combine like terms on the left side. 2x + 3x = 5x 15 - 5 = 10 So the left side becomes: 5x + 10 Now the equation is: 5x + 10 = 180 Step 3: Subtract 10 from both sides to isolate the term with x. 5x + 10 - 10 = 180 - 10 5x = 170 Step 4: Divide both sides by 5 to solve for x. 5x / 5 = 170 / 5 x = 34 Step 5: Conclusion. The solution is x = 34.

9. A city park is designing a decorative tile pattern using two intersecting straight paths. The paths form four angles around their intersection point. One angle measures 78° and its adjacent angle on the same side of one path measures (2x + 12)°. These two angles are supplementary. Find the value of x. Answer: 45 Solution: Identify that the two angles are supplementary, meaning they sum to 180° Write the equation: 78° + (2x + 12)° = 180° Combine like terms: 78 + 2x + 12 = 180 Simplify: 90 + 2x = 180 Subtract 90 from both sides: 2x = 90 Divide both sides by 2: x = 45 Therefore, the value of x is 45.
   Full step-by-step solution

   Step 1: Identify that the two angles are supplementary, meaning they sum to 180° Step 2: Write the equation: 78° + (2x + 12)° = 180° Step 3: Combine like terms: 78 + 2x + 12 = 180 Step 4: Simplify: 90 + 2x = 180 Step 5: Subtract 90 from both sides: 2x = 90 Step 6: Divide both sides by 2: x = 45 Therefore, the value of x is 45.