Lines & Angles
Basic Terms and Types of Angles
A ray starts at a point and goes on forever one way; a line segment has two endpoints; an angle is formed by two rays from a common vertex.
By size: acute (< 90°), right (= 90°), obtuse (between 90° and 180°), straight (= 180°), reflex (between 180° and 360°) and complete (= 360°).
Example 1: Classify a 145° angle.
It is between 90° and 180°, so it is obtuse.
Example 2: What is a 180° angle called?
A straight angle.
Quick recap
- Ray: one endpoint; segment: two endpoints.
- Acute < 90° < right < obtuse < 180° (straight) < reflex < 360°.
✓ Quick check
In ΔABC, the angle bisector of ∠A meets BC at D. If ∠B = 70° and ∠C = 30°, find ∠ADC.
Total sum of angles in ΔABC = 180° → ∠A = 180° − (70° + 30°) = 80°. Since AD bisects ∠A, ∠BAD = ∠CAD = 40°. In ΔADC, ∠ADC = 180° − (∠CAD + ∠C) = 180° − (40° + 30°) = 110°.
The angle whose reflex angle is three times its direct acute/obtuse value must satisfy which measure?
Let the angle be x. Its reflex angle is 360° − x. Given 360° − x = 3x → 4x = 360° → x = 90°.
Pairs of Angles
What are Intersecting and Non-Intersecting Lines?
- Intersecting lines are lines that cross each other at exactly one point (called the point of intersection)
- Non-intersecting lines (parallel lines) are lines that never meet, no matter how far they are extended
What are Different Angle Pairs?
When two lines intersect or when lines are combined, various angle pairs are formed:
| Angle Pair | Definition | Diagram Description | Property |
|---|---|---|---|
| **Supplementary Angles** | Two angles whose sum is 180° | 110° + 70° = 180° | Each is supplement of the other |
| **Adjacent Angles** | Two angles sharing a common vertex and common arm, with non-common arms on opposite sides | Angles next to each other | They do not overlap |
| **Linear Pair** | Adjacent angles formed when two lines intersect; their non-common arms are opposite rays | ∠AOC and ∠COB sharing ray OC | Sum = 180° |
| **Vertically Opposite Angles** | Angles opposite each other when two lines intersect | ∠AOC and ∠BOD | They are always EQUAL |
Real-Life Examples:
- Intersecting lines: Crossroads, scissors, window grills
- Parallel lines: Railway tracks, opposite edges of a notebook
- Complementary angles: Acute angles in a right triangle, clock hands at 3:00 (90°)
- Linear pair: Door opening, book pages
Example 1: The supplement of 110° is?
180° − 110° = 70°.
Example 2: Two lines cross; one angle is 40°. Its vertically opposite angle?
Also 40°, since vertically opposite angles are equal.
Quick recap
- Complementary add to 90°; supplementary add to 180°.
- Linear pair ⇒ supplementary; vertically opposite angles are equal.
✓ Quick check
An angle is equal to one-third of its supplement. Find the measure of its supplement.
Let the supplement angle be x. The angle itself is x/3. Since they are supplementary, x + x/3 = 180° → 4x/3 = 180° → 4x = 540° → x = 135°.
An angle is 14° more than its complementary angle. The measure of this angle is:
Let the angle be x. Its complement is 90° − x. According to the question, x = (90° − x) + 14° → 2x = 104° → x = 52°.
Parallel Lines, Transversals and Triangles
What is a Transversal?
A transversal is a line that intersects two or more lines at distinct points.
Angle Relationships When a Transversal Cuts Two Lines:
When a transversal intersects two lines, eight angles are formed with special relationships:
| Angle Pair | Position | Property (if lines are parallel) |
|---|---|---|
| **Alternate interior angles** | Inside, opposite sides of transversal | EQUAL |
| **Alternate exterior angles** | Outside, opposite sides of transversal | EQUAL |
| **Consecutive interior angles** (co-interior) | Inside, same side of transversal | SUM = 180° |
Lines Parallel to the Same Line:
If two lines are both parallel to the same line, then they are parallel to each other.
- If l ∥ m and m ∥ n, then l ∥ n
Real-Life Applications:
- Building construction (ensuring walls are parallel)
- Railway tracks (transversal by cross ties)
- Map grids and road networks
Example 1: Two parallel lines, transversal; a corresponding angle is 65°. The matching one?
Also 65° — corresponding angles are equal.
Example 2: A triangle has angles 50° and 60°. The third?
180° − 50° − 60° = 70°.
Quick recap
- Parallel + transversal: corresponding and alternate angles equal; co-interior supplementary.
- Triangle angles sum to 180°; exterior = sum of remote interior angles.
✓ Quick check
An express train track switches onto a loop line. The main line and the loop line form an obtuse angle of 135° at the point of divergence. What is the measure of the linear pair partner angle formed by this track layout?
The straight track line implies a linear pair layout. 180° − 135° = 45°.
On a scale map, three straight highways intersect to enclose a triangular market zone. If highway A and B meet at 72°, and highway B and C meet at 58°, at what angle do highway A and C meet?
The three highways form a triangle. The sum of the interior angles is 180°. Third angle = 180° − (72° + 58°) = 180° − 130° = 50°.
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