Angle Relationships • Topic 2 of 3

Vertical & Adjacent Angles

Vertical Angles

When two lines intersect, they form two pairs of vertical angles.

Vertical angles are ALWAYS EQUAL!

        ┌─────┐
        │     │
     ───┼─────┼───
        │     │
        └─────┘

Angle 1 = Angle 3 (vertical)
Angle 2 = Angle 4 (vertical)

Adjacent Angles on a Line

When two angles share a vertex and a side, and together form a straight line, they are adjacent and supplementary (sum to 180°).

        /  
       / 1│
      /   │
     /    │
    └─────┘
    2

Angles 1 and 2 are adjacent and form a linear pair
1 + 2 = 180°

VERTICAL ANGLES - INTERSECTING LINES:

              │
              │
        1─────┼─────2
              │
              │
    
    Angles: 1 and 3 are vertical → equal
           2 and 4 are vertical → equal
    
    Adjacent: 1 and 2 are supplementary (180°)


FULL DIAGRAM WITH LABELS:

                    l₁
                    ↑
                    │
            ∠2      │      ∠1
          ←─────────┼─────────→
                    │
            ∠3      │      ∠4
                    │
                    ↓
                    l₂
    
    ∠1 = ∠3 (vertical)
    ∠2 = ∠4 (vertical)
    
    ∠1 + ∠2 = 180° (adjacent on line)
    ∠2 + ∠3 = 180° (adjacent on line)
    ∠3 + ∠4 = 180° (adjacent on line)
    ∠4 + ∠1 = 180° (adjacent on line)


FINDING ALL ANGLES:

    Given: ∠1 = 70°
    
    Then: ∠3 = 70° (vertical)
          ∠2 = 180° - 70° = 110° (supplementary)
          ∠4 = 110° (vertical)

Solved Examples

Worked Example

Two intersecting lines form angles. One angle is 55°. Find the other three.

Solution

Vertical angle = also 55°
Adjacent angles = 180° - 55° = 125°
Other vertical angle = 125°
Answer: 55°, 125°, 125°

Worked Example

In the diagram, ∠1 = 40°. Find ∠2, ∠3, ∠4.

Solution

∠3 = ∠1 = 40° (vertical)
∠2 = 180° - 40° = 140° (supplementary)
∠4 = ∠2 = 140° (vertical)
Answer: ∠2=140°, ∠3=40°, ∠4=140°

Worked Example

Vertical angles are always ________.

Solution

Vertical angles are always equal
Answer: Equal

Key Points

Vertical angles are opposite each other
Vertical angles are ALWAYS EQUAL
Adjacent angles on a line sum to 180°
When lines intersect, 4 angles are formed
Opposite angles are equal, adjacent are supplementary

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Vertically opposite angles are:

Explanation: Equal.

Q2.Adjacent angles share a common vertex and:

Explanation: Common arm.

Q3.A linear pair sums to:

Explanation: $180^\circ$.

Q4.If one angle of a linear pair is $70^\circ$, the other is:

Explanation: $180-70$.

Practice more MCQs → 📝 Assignment · 35 marks