Finding Unknown Angles
Combining all angle properties to solve problems
Summary of Angle Properties
Angles on a Straight Line
Sum = 180°
∠a + ∠b = 180°
Vertically Opposite
Equal
Angles at a Point
Sum = 360°
Problem-Solving Strategy
Step-by-Step Approach:
- Identify what information is given
- Look for straight lines, intersecting lines, or angles at a point
- Apply the correct property
- Calculate step by step
- Check your answer makes sense
Multi-Step Problems
Sometimes you need to use more than one property to find all the unknown angles.
Worked Example: Two intersecting lines
Two straight lines cross. One angle is 72°. Find ∠b, ∠c and ∠d.
Solution:
Step 1: Use vertically opposite angles
(vertically opposite)
Step 2: Use angles on a straight line
Step 3: Use vertically opposite angles again
(vertically opposite to ∠b)
Answer: ∠b = 108°, ∠c = 72°, ∠d = 108°
Multiple Solution Methods
Some problems can be solved in more than one way. Both ways should give the same answer!
Worked Example: Using Both Properties
Two straight lines cross. One angle is 55°. Find ∠d.
Way 1: Vertically opposite, then straight line
∠c = 55° (vertically opposite to 55°)
∠c and ∠d are angles on a straight line:
∠d = 180° − 55° = 125°
Way 2: Straight line only
The 55° angle and ∠d are angles on a straight line:
∠d = 180° − 55° = 125°
Practice Problems
Practice 1: Find angles b, c and d
Two straight lines cross. One angle is 65°. Find ∠b, ∠c and ∠d.
ShowHide solution
∠c = 65° (vertically opposite)
∠b = 180° − 65° = 115° (angles on straight line)
∠d = 115° (vertically opposite to ∠b)
Answer: ∠b = 115°, ∠c = 65°, ∠d = 115°
Practice 2: Four angles on a straight line
The angles are on a straight line. Find ∠q.
ShowHide solution
Angles on a straight line sum to 180°:
Practice 3: Find the unknown angle at a point
Four angles meet at a point. Find ∠x.
ShowHide solution
Angles at a point sum to 360°:
Key Takeaways
- Angles on a straight line sum to 180°
- Vertically opposite angles are always equal
- Angles at a point sum to 360°
- Complex problems often need multiple properties
- Always check your answer by verifying the sum or equality