Angle Relationships

Grade 7 · geometry · 100 practice problems · read aloud

🔊 Listen to this explanation

Angle Relationships: Unlocking the Secrets of Shapes

Angle relationships describe how angles interact with each other, especially when lines cross or when a line cuts through parallel lines. Understanding these rules helps you find unknown angles in geometric figures, which is essential for everything from simple puzzles to complex design and construction! 🧩

Key Angle Relationships to Know

  • Adjacent Angles: Angles that share a vertex and a side, but don't overlap.
  • Vertical Angles: Angles opposite each other when two lines cross. They are always equal.
  • Supplementary Angles: Two angles that add up to 180°. They often form a straight line.
  • Complementary Angles: Two angles that add up to 90°. They often form a corner.

How to Solve Angle Problems: A Step-by-Step Guide

  1. Identify: Look for lines that cross (intersecting lines) or parallel lines with a transversal (a line that cuts across them).
  2. Label: Mark any angles you know on the diagram.
  3. Relationship: Figure out which relationship the angles have (e.g., vertical, supplementary).
  4. Set Up & Solve: Write an equation and solve for the missing angle.

Worked Examples

Example 1: Intersecting Lines

Two lines intersect. One angle is 75°. Find the other three angles.

  1. The angle vertical to 75° is also 75°.
  2. The other two angles are supplementary to 75°. 180° - 75° = 105°.
  3. So, the four angles are 75°, 105°, 75°, and 105°.

Example 2: Supplementary Angles

Two angles are supplementary. One angle is 122°. Find its supplement.

Supplementary angles sum to 180°. So, 180° - 122° = 58°.

⚠️ Common Mistakes to Avoid

  • Mixing up Supplementary and Complementary: Remember, Supplementary is for Straight lines (both start with 'S'). Complementary is for Corners (both start with 'C').
  • Assuming all angles are equal: Angles are only equal if they are vertical angles or in other specific situations. Don't guess!
  • Forgetting the sum: Always double-check that your angles on a straight line add up to 180°.

💡 Tips & Tricks

  • Use color coding! Use one color to highlight all vertical angles and another for supplementary angles.
  • Look for "Z" and "F" shapes formed by parallel lines and a transversal—the angles in these shapes are often equal.
  • When you find one angle, you often unlock several others. Start with the angle you know.

How to Practice

To master angle relationships, try these activities:

  • Use online math games that focus on angle puzzles.
  • Draw your own sets of intersecting and parallel lines, measure the angles with a protractor, and confirm the relationships.
  • Solve at least 5 practice problems every day, checking your work as you go.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 2x + 5 = 17

Hint: To solve for the variable, first isolate the term containing the variable by moving the constant to the other side of the equation, then divide both sides by the coefficient of the variable.

Show the answer

Answer: 6

  1. Subtract 5 from both sides of the equation. This is done to isolate the term with the variable x. 2x + 5 - 5 = 17 - 5 2x = 12
  2. Divide both sides by 2. This is done to solve for x. 2x / 2 = 12 / 2 x = 6
  3. Check the solution. Substitute x = 6 into the original equation: 2(6) + 5 = 12 + 5 = 17 Since 17 = 17 is true, the solution is correct. Final answer: x = 6

We are solving the equation: 2x + 5 = 17

2 2x + 15 = 31

Hint: Isolate the variable by performing inverse operations on both sides of the equation.

Show the answer

Answer: 8

  1. Subtract 15 from both sides of the equation. This is done to isolate the term with the variable (2x) on one side. 2x + 15 - 15 = 31 - 15 2x = 16
  2. Divide both sides by 2. This is done to solve for x. 2x / 2 = 16 / 2 x = 8
  3. Check the solution. Substitute x = 8 into the original equation: 2(8) + 15 = 16 + 15 = 31 Since both sides are equal, the solution is correct. ANSWER: 8

We are solving the equation: 2x + 15 = 31

3 2x + 15 = 180

Hint: To solve for the variable, first isolate the term containing the variable by performing the inverse operation on the constant term. Then, divide both sides by the coefficient of the variable.

Show the answer

Answer: 82.5

  1. Subtract 15 from both sides to isolate the term with x. Reasoning: We want to get the term with x by itself on one side. Calculation: 2x + 15 - 15 = 180 - 15 2x = 165
  2. Divide both sides by 2 to solve for x. Reasoning: Since 2x means 2 times x, dividing by 2 will give x alone. Calculation: x = 165 / 2
  3. Simplify the fraction. 165 ÷ 2 = 82.5 Final answer: x = 82.5

We start with the equation: 2x + 15 = 180

4 2x + 15 = 180 - (x + 30)

Hint: Combine like terms and solve for the variable using inverse operations.

Show the answer

Answer: 45

  1. Distribute the negative sign on the right side** 180 - (x + 30) = 180 - x - 30 So the equation becomes: 2x + 15 = 180 - x - 30 **
  2. Simplify the right side** 180 - 30 = 150 So: 2x + 15 = 150 - x **
  3. Get all x terms on one side** Add x to both sides: 2x + x + 15 = 150 - x + x 3x + 15 = 150 **
  4. Isolate the x term** Subtract 15 from both sides: 3x + 15 - 15 = 150 - 15 3x = 135 **
  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

Let's solve step by step. We start with the equation: 2x + 15 = 180 - (x + 30) **

5 (3x + 15) + (2x - 25) = 180

Hint: Combine like terms and solve for the variable using inverse operations.

Show the answer

Answer: 38

  1. Remove the parentheses** Since it's all addition between the parentheses, we can just drop them: 3x + 15 + 2x - 25 = 180 **
  2. Combine like terms** Combine the x terms: 3x + 2x = 5x Combine the constants: 15 - 25 = -10 So now we have: 5x - 10 = 180 **
  3. Isolate the x term** Add 10 to both sides: 5x - 10 + 10 = 180 + 10 5x = 190 **
  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

Let's solve the equation step-by-step. We start with: (3x + 15) + (2x - 25) = 180 **

6 (2x + 15)° + (3x - 5)° = 180°

Hint: When two angles form a straight line, their measures add up to 180 degrees. Combine like terms and solve for the variable.

Show the answer

Answer: x = 34

  1. Remove the degree symbols for calculation purposes. The equation becomes: (2x + 15) + (3x - 5) = 180
  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
  3. Subtract 10 from both sides to isolate the term with x. 5x + 10 - 10 = 180 - 10 5x = 170
  4. Divide both sides by 5 to solve for x. 5x / 5 = 170 / 5 x = 34
  5. Conclusion. The solution is x = 34.

We are given the equation: (2x + 15)° + (3x - 5)° = 180°

Practise this topic — 10 free problems, no signup →