4.1.3 ANGLES ON A TRANSVERSAL
Definition:
A transversal is a line that cuts two lines at two distinct points.
Properties of Angles Formed by a Transversal and Two Parallel Lines
| Angle Type | Definition | Angle Relationships |
|---|---|---|
| Corresponding Angles | Angles formed at the same relative position at each intersection. | ∠a = ∠e, ∠c = ∠g, ∠b = ∠f, ∠d = ∠h |
| Vertically Opposite Angles | Angles formed opposite each other at an intersection. | ∠a = ∠d, ∠b = ∠c, ∠e = ∠h, ∠f = ∠g |
| Alternate Interior Angles | Angles formed inside the two parallel lines on opposite sides of the transversal. | ∠c = ∠f, ∠d = ∠e |
| Alternate Exterior Angles | Angles formed outside the two parallel lines on opposite sides of the transversal. | ∠a = ∠h, ∠b = ∠g |
| Consecutive Interior (Co-interior) Angles | Angles formed inside the two parallel lines on the same side of the transversal. These angles are supplementary. | ∠c + ∠e = 180°, ∠d + ∠f = 180° |
Example 4.5
Find the value of each indicated angle.
Summary
- Corresponding angles are equal.
- Vertically opposite angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Co-interior angles add up to 180°.
Exercise
(a)
Find x.
7x +61=180
7x = 180-61
7x =119
x=17
(b)
Find x.
(x − 4 ) + 135 = 180
x -4 = 180-135
x - 4 = 45
x=49
(e)
Find x.
Corresponding angles are equal.
x − 9 = 42
x = 51
(d)
Find x.
3x + 2 = 107
3x = 105
x = 35