Whenever two lines meet, an angle is born. This chapter teaches the secret rules angles obey — how they pair up, add up to 180°, and how a pair of parallel lines cut by a third line creates a whole family of equal angles. Master these and most geometry proofs become easy!
Angle pairs
Complementary add to 90°, supplementary add to 180°.
Linear pair
Two adjacent angles on a straight line sum to 180°.
Vertically opposite
When two lines cross, opposite angles are equal.
Parallel lines
A transversal makes equal corresponding & alternate angles.
1. Basic terms: point, line, ray, segment
A point has no length, breadth or thickness — just a position. A line extends endlessly in both directions and has no end points. A line segment is a part of a line with two end points, so it has a definite length (like AB). A ray has one fixed end point and extends endlessly in one direction. When two rays start from the same end point, they form an angle, and the common end point is called the vertex. The two rays are the arms of the angle. Angles are measured in degrees (°).
2. Types of angles
Angles are named by their size. An acute angle is more than 0° but less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° but less than 180°. A straight angle is exactly 180°. A reflex angle is more than 180° but less than 360°. A complete angle is exactly 360°. Knowing these names helps you describe figures precisely in proofs.
3. Complementary and supplementary angles
Two angles are complementary if their measures add up to 90°. For example, 30° and 60° are complementary. Two angles are supplementary if their measures add up to 180°. For example, 110° and 70° are supplementary. The complement of an angle x is (90° − x), and the supplement of x is (180° − x). A handy memory link: Complementary comes before Supplementary in the alphabet, just as 90 comes before 180.
4. Adjacent angles and linear pair
Adjacent angles share a common vertex and a common arm, and their non-common arms lie on opposite sides of the common arm. They do not overlap. A linear pair is a special pair of adjacent angles whose non-common arms form a straight line. The Linear Pair Axiom says: if a ray stands on a line, the sum of the two adjacent angles so formed is 180°. The converse is also true: if the sum of two adjacent angles is 180°, their non-common arms form a straight line. This pair of statements is used in almost every angle proof.
5. Angles at a point and vertically opposite angles
The sum of all the angles formed around a single point is 360°. When two straight lines intersect, four angles are formed. The angles that are opposite to each other (not adjacent) are called vertically opposite angles. The Vertically Opposite Angles Theorem states that when two lines intersect, the vertically opposite angles are equal. This is proved using linear pairs: each pair of vertically opposite angles is supplementary to the same angle, so they must be equal.
6. Parallel lines and a transversal
Two lines in the same plane that never meet, no matter how far they are extended, are parallel lines (written l ∥ m). A line that cuts two or more lines at distinct points is called a transversal. When a transversal cuts two lines, eight angles are formed. These have special names: corresponding angles (in matching positions, like top-left with top-left), alternate interior angles (between the two lines, on opposite sides of the transversal), alternate exterior angles (outside the two lines, on opposite sides), and co-interior (or consecutive interior) angles (between the two lines, on the same side).
7. Properties when the lines are parallel
If a transversal cuts two parallel lines, then three powerful results follow. (a) Corresponding angles are equal — this is taken as an axiom. (b) Alternate interior angles are equal. (c) Co-interior angles are supplementary (their sum is 180°). The converses are equally important and are used to prove that two lines are parallel: if a transversal makes equal corresponding angles, or equal alternate angles, or supplementary co-interior angles, then the two lines must be parallel.
8. Lines parallel to the same line
An important corollary states: lines which are parallel to the same line are parallel to each other. So if line l ∥ m and line m ∥ n, then l ∥ n. This transitive property is frequently used in figures with several parallel lines.
9. Angle sum property of a triangle
Using parallel-line results we can prove that the sum of the three angles of a triangle is 180°. A related result is the Exterior Angle Theorem: if a side of a triangle is produced (extended), the exterior angle so formed equals the sum of the two interior opposite angles. For example, if the two remote interior angles are 50° and 60°, the exterior angle is 110°. The exterior angle is always greater than either of the interior opposite angles.
- Complementary: two angles add to 90°
- Supplementary: two angles add to 180°
- Linear pair: adjacent angles on a line sum to 180°
- Angles at a point: total = 360°
- Vertically opposite angles are equal
- Parallel lines: corresponding equal, alternate equal, co-interior supplementary
- Triangle angle sum = 180°
- Exterior angle = sum of two interior opposite angles
In the figure, a ray OC stands on a line AOB. If ∠AOC = (2x + 10)° and ∠BOC = (3x − 30)°, find the value of x and both angles.
- Since OC stands on the line AOB, ∠AOC and ∠BOC form a linear pair, so their sum is 180°.
- Write the equation: (2x + 10) + (3x − 30) = 180.
- Combine like terms: 5x − 20 = 180.
- Add 20 to both sides: 5x = 200, so x = 40.
- Find the angles: ∠AOC = 2(40) + 10 = 90° and ∠BOC = 3(40) − 30 = 90°.
In the figure, l ∥ m and a transversal t cuts them. One of the angles is 65°. Find its co-interior angle and the corresponding angle.
- Let the given angle be an interior angle equal to 65°.
- Since l ∥ m, co-interior (same-side interior) angles are supplementary, so co-interior angle = 180° − 65° = 115°.
- Since l ∥ m, corresponding angles are equal, so the corresponding angle = 65°.
- (Check: the alternate interior angle is also 65°, which agrees with the rule that alternate angles are equal.)
For the F, Z and U shapes formed by a transversal: F = corresponding angles (equal), Z = alternate angles (equal), and U (or C) = co-interior angles (supplementary, sum 180°). Trace the letter on the figure and you instantly know the rule!
Do not mix up complementary (90°) with supplementary (180°) — this is the most common slip. Also, co-interior angles are supplementary, NOT equal; only corresponding and alternate angles are equal. In proofs, always state the reason (e.g. “alternate angles, l ∥ m”) — marks are given for reasons, not just the final number.
Q1. Two supplementary angles are in the ratio 2 : 3. Find both angles.
Answer: Let the angles be 2x and 3x. Since they are supplementary, 2x + 3x = 180°, so 5x = 180° and x = 36°. Therefore the angles are 2x = 72° and 3x = 108°. (Check: 72 + 108 = 180. ✓)
Q2. Two lines AB and CD intersect at O. If ∠AOC = 50°, find ∠BOD, ∠AOD and ∠BOC.
Answer: ∠BOD is vertically opposite to ∠AOC, so ∠BOD = 50°. ∠AOD forms a linear pair with ∠AOC, so ∠AOD = 180° − 50° = 130°. ∠BOC is vertically opposite to ∠AOD, so ∠BOC = 130°.
Q3. In a triangle, the angles are in the ratio 1 : 2 : 3. Find each angle and name the triangle.
Answer: Let the angles be x, 2x and 3x. By the angle sum property, x + 2x + 3x = 180°, so 6x = 180° and x = 30°. The angles are 30°, 60° and 90°. Since one angle is 90°, it is a right-angled triangle.
Q4. The side BC of ∆ABC is produced to D. If ∠ABC = 55° and ∠ACD = 120°, find ∠BAC.
Answer: ∠ACD is the exterior angle. By the Exterior Angle Theorem, the exterior angle equals the sum of the two interior opposite angles: ∠ACD = ∠ABC + ∠BAC. So 120° = 55° + ∠BAC, giving ∠BAC = 120° − 55° = 65°.
- ✅ Complementary = 90°, Supplementary = 180°
- ✅ Linear pair sums to 180°; angles round a point = 360°
- ✅ Vertically opposite angles are always equal
- ✅ Parallel lines: corresponding & alternate equal, co-interior supplementary (F, Z, U)
- ✅ Triangle angles sum to 180°; exterior angle = sum of remote interior angles
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