Multi-Step Angle Problems

Grade 7 · geometry · 101 practice problems · read aloud

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Multi-Step Angle Problems

🔍 What Is It & Why Is It Useful?

Multi-step angle problems require you to use several angle rules together to find an unknown angle. You'll often need to solve for one angle before you can find another. This is super useful in real life for design, construction, and navigation, where shapes and angles are everywhere!

🧩 How to Solve: A Step-by-Step Guide

  1. Identify Known Angles: Label all the angles you already know on the diagram.
  2. Find Your Target: Clearly mark the angle you are trying to find.
  3. Choose Your Tools: Decide which angle rules to use (see Tips & Tricks!).
  4. Solve Step-by-Step: Work through the problem, finding one missing angle at a time. Write each new angle you find on the diagram.
  5. Check Your Work: See if your final answer makes sense with the rules you used.

📐 Visual Examples

Example 1: Intersecting Lines

Two lines intersect. One angle is 35°. Find the other three angles.

  1. Vertical angles are equal. The angle opposite 35° is also 35°.
  2. Adjacent angles form a straight line (180°). So, 180° - 35° = 145°.
  3. The angle opposite 145° is also 145° (vertical angles).

Example 2: Triangle within a Triangle

In a large triangle, one angle is 60°. Inside it, a smaller triangle has angles of 70° and 50°. Find the missing angle (x) in the small triangle.

  1. First, find the third angle of the small triangle: 180° - 70° - 50° = 60°.
  2. Notice this 60° and the large triangle's 60° might be part of a straight line or another shape, leading to the next step in a larger problem.

⚠️ Common Mistakes to Avoid

  • Mixing up rules: Remember, complementary angles add to 90°. Supplementary angles add to 180°. Triangles always sum to 180°.
  • Assuming parallel lines: Only use corresponding or alternate interior angles if you KNOW the lines are parallel.
  • Not writing on the diagram: As you find new angles, write them down! This prevents confusion and helps you see the next step.

💡 Tips & Tricks

  • Your Angle Toolbox:
    • Straight Line: 180°
    • Full Rotation: 360°
    • Triangle Sum: 180°
    • Vertical Angles: Are EQUAL
  • Look for "Angle Bridges": A missing angle in one triangle might help you find an angle in another shape that shares a side.
  • Color Code: Use different colors to highlight different sets of angles (e.g., all vertical angles in blue).

🎯 How to Practice

  • Start with simple problems using just one rule, then combine them.
  • Draw your own complex diagrams with intersecting lines and triangles and challenge a friend to solve them.
  • Look for real-world angles—in a pizza slice (isosceles triangle), a window pane (right angles), or a scissors (intersecting lines).

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 (3/4 + 1/2) × 24 - 15 = ?

Hint: First, find a common denominator to add the fractions inside the parentheses. Then multiply the result by the whole number, and finally subtract the remaining number.

Show the answer

Answer: 15

  1. Handle the parentheses first: 3/4 + 1/2 To add these, find a common denominator. The common denominator of 4 and 2 is 4. 3/4 stays as 3/4. 1/2 = 2/4. So 3/4 + 2/4 = 5/4.
  2. Multiply 5/4 by 24. 5/4 × 24 = (5 × 24) / 4 = 120 / 4 = 30.
  3. Subtract 15 from the result. 30 - 15 = 15. Final answer: 15

Let's solve step by step.

2 (3/4 × 48) - (2/3 × 36) = ?

Hint: First perform the multiplication operations inside the parentheses, then subtract the results. Remember to simplify fractions before multiplying.

Show the answer

Answer: 12

  1. Calculate 3/4 × 48** 3/4 × 48 means: multiply 48 by 3, then divide by 4. First: 48 × 3 = 144 Then: 144 ÷ 4 = 36 So, 3/4 × 48 = 36. --- **
  2. Calculate 2/3 × 36** 2/3 × 36 means: multiply 36 by 2, then divide by 3. First: 36 × 2 = 72 Then: 72 ÷ 3 = 24 So, 2/3 × 36 = 24. --- **
  3. Subtract the results** (3/4 × 48) - (2/3 × 36) = 36 - 24 = 12 --- **Final Answer:** 12

Let's solve step by step. We have: (3/4 × 48) - (2/3 × 36) --- **

3 (3/4 × 2/3) ÷ (1/2 + 1/6) = ?

Hint: First simplify the multiplication of fractions by canceling common factors, then find a common denominator for the addition in the denominator before performing the final division.

Show the answer

Answer: 3/4

  1. Simplify the numerator (3/4 × 2/3)** Multiply the numerators: 3 × 2 = 6 Multiply the denominators: 4 × 3 = 12 So 3/4 × 2/3 = 6/12 Simplify 6/12: divide numerator and denominator by 6 → 1/2 So numerator = 1/2 --- **
  2. Simplify the denominator (1/2 + 1/6)** Find a common denominator for 1/2 and 1/6. The least common denominator of 2 and 6 is 6. 1/2 = 3/6 1/6 = 1/6 Add: 3/6 + 1/6 = 4/6 Simplify 4/6: divide numerator and denominator by 2 → 2/3 So denominator = 2/3 --- **
  3. Perform the division (1/2) ÷ (2/3)** Dividing by a fraction is the same as multiplying by its reciprocal: (1/2) ÷ (2/3) = (1/2) × (3/2) Multiply numerators: 1 × 3 = 3 Multiply denominators: 2 × 2 = 4 Result = 3/4 --- **Final Answer:** 3/4

Let's solve step-by-step. We have: (3/4 × 2/3) ÷ (1/2 + 1/6) --- **

4 Two supplementary angles are in the ratio 4:5. Find the measure of each angle.

Hint: Remember that supplementary angles add up to 180°. Use the ratio to set up an equation with a variable, then solve for the variable and multiply to find each angle.

Show the answer

Answer: 80° and 100°

  1. Let the two angles be 4x and 5x, since they are in the ratio 4:5.
  2. Supplementary angles sum to 180°, so 4x + 5x = 180.
  3. Combine like terms: 9x = 180.
  4. Divide both sides by 9: x = 20.
  5. First angle = 4x = 4 × 20 = 80°.
  6. Second angle = 5x = 5 × 20 = 100°.
  7. Check: 80° + 100° = 180°, which is correct. Final answer: The angles are 80° and 100°.

5 Two supplementary angles have measures (3x + 17)° and (5x - 9)°. Find the value of x.

Hint: Supplementary angles add up to 180 degrees. Write an equation by adding the two expressions and setting the sum equal to 180. Then solve for x by combining like terms and isolating the variable.

Show the answer

Answer: 21.5

  1. Since the angles are supplementary, their sum is 180°. (3x + 17) + (5x - 9) = 180
  2. Combine like terms. 3x + 5x + 17 - 9 = 180 8x + 8 = 180
  3. Subtract 8 from both sides. 8x = 172
  4. Divide both sides by 8. x = 172 ÷ 8 x = 21.5

The answer is 21.5.

6 Two supplementary angles have measures (7x + 12)° and (5x + 18)°. Find the value of x.

Hint: Recall that supplementary angles add up to 180°. Set up an equation by adding the two angle expressions and setting the sum equal to 180. Then solve for x by combining like terms and isolating the variable.

Show the answer

Answer: 12.5

  1. Write the equation for supplementary angles: (7x + 12) + (5x + 18) = 180.
  2. Combine like terms: 7x + 5x = 12x, and 12 + 18 = 30, so 12x + 30 = 180.
  3. Subtract 30 from both sides: 12x = 150.
  4. Divide both sides by 12: x = 150 / 12 = 12.5. The value of x is 12.5.
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