Lines and Angles
This chapter studies the angles formed when lines meet or are crossed by a transversal. You will learn the types of angles and angle pairs, the linear-pair and vertically-opposite-angle relationships, the properties of parallel lines cut by a transversal, and the angle sum and exterior-angle results for triangles. These angle rules are used in every later geometry chapter.
Learning objectives
- Classify angles and identify angle pairs.
- Apply the linear pair and vertically opposite angle properties.
- Use the angle relationships of parallel lines and a transversal.
- Apply the angle sum property of a triangle.
- Use the exterior angle theorem.
Key concepts
Types of angles and pairs
Angles are acute (< 90°), right (90°), obtuse (between 90° and 180°), straight (180°) or reflex (between 180° and 360°). Two angles are complementary if they add to 90°, and supplementary if they add to 180°.
Linear pair and vertically opposite angles
When two lines intersect, adjacent angles on a straight line form a linear pair and add to 180°. The angles opposite each other (vertically opposite angles) are equal.
Parallel lines and a transversal
When a transversal cuts two parallel lines: corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles are supplementary (add to 180°).
Angles of a triangle
The three interior angles of a triangle add up to 180° (angle sum property). An exterior angle of a triangle equals the sum of the two interior opposite angles (exterior angle theorem).
Important formulas
Linear pair
∠1 + ∠2 = 180°
Complementary / supplementary
sum = 90° / sum = 180°
Angle sum of a triangle
∠A + ∠B + ∠C = 180°
Exterior angle theorem
exterior angle = sum of two interior opposite angles
Key definitions
- Linear pair
- Two adjacent angles formed on a straight line, summing to 180°.
- Vertically opposite angles
- The equal angles formed opposite each other when two lines intersect.
- Transversal
- A line that crosses two or more other lines.
- Complementary angles
- Two angles whose measures add up to 90°.
Solved examples
Q1. Two angles form a linear pair and one of them is 110°. Find the other.
Solution: Linear pair angles sum to 180°, so the other angle = 180° − 110° = 70°.
Q2. Two parallel lines are cut by a transversal. If one alternate interior angle is 65°, find the other.
Solution: Alternate interior angles are equal, so the other angle is also 65°.
Q3. Two angles of a triangle are 50° and 60°. Find the third angle.
Solution: Third angle = 180° − (50° + 60°) = 70°.
Common mistakes to avoid
- Confusing complementary (90°) with supplementary (180°).
- Assuming alternate angles are supplementary — they are equal for parallel lines.
- Forgetting the angle sum of a triangle is 180°, not 360°.
- Mixing up corresponding and co-interior angle relationships.
Lines and Angles — MCQ Quiz
12 questions with instant feedback. Use number keys 1–4 to answer.
Two angles that add up to 180° are called:
Practice questions
Short answer
What is the supplement of 120°?
180° − 120° = 60°.
Define alternate interior angles.
The pair of equal angles on opposite sides of a transversal, between two parallel lines.
Can a triangle have two right angles? Why?
No — two right angles already total 180°, leaving nothing for the third angle.
Long answer
Two parallel lines are cut by a transversal. If one of the eight angles is 70°, find all the others.
The angles come in two groups. All angles equal to the 70° one (its vertically opposite, corresponding and alternate angles) are 70°. The remaining angles are its supplements, each 110°. So four angles are 70° and four are 110°.
Prove using the angle sum property that the angles of an equilateral triangle are each 60°.
All three angles are equal (say x each). By the angle sum property, x + x + x = 180°, so 3x = 180° and x = 60°. Hence each angle is 60°.
HOTS (Higher Order Thinking)
Two lines are each parallel to a third line. What can you conclude about the two lines?
They are parallel to each other, since lines parallel to the same line are parallel to one another.
The angles of a triangle are in the ratio 1 : 2 : 3. Find them.
Let the angles be x, 2x, 3x. Then x + 2x + 3x = 180° ⇒ 6x = 180° ⇒ x = 30°. The angles are 30°, 60° and 90°.
Quick revision
Revision notes
- Complementary = 90°; supplementary = 180°.
- Linear pair = 180°; vertically opposite angles are equal.
- Parallel + transversal: corresponding equal, alternate equal, co-interior supplementary.
- Triangle: angle sum 180°; exterior angle = sum of opposite interiors.
Key takeaways
- Alternate and corresponding angles are equal (not supplementary).
- Use the exterior angle theorem to find angles quickly.
- The 180° triangle rule solves most angle-chasing problems.
Frequently asked questions
What is the difference between complementary and supplementary angles?
Complementary angles add to 90°; supplementary angles add to 180°.
What are alternate interior angles?
Equal angles on opposite sides of a transversal that lies between two parallel lines.
What is the exterior angle theorem?
An exterior angle of a triangle equals the sum of the two interior angles opposite to it.