Lines and Angles

A line extends without end in both directions and has no endpoints, while a line segment is a fixed part of a line with two endpoints. A ray starts at one endpoint and extends endlessly in only one direction. Points that lie on the same line are called collinear points; points that do not all lie on one line are non-collinear points.

An angle is formed when two rays start from the same point. This common starting point is called the vertex, and the two rays are called the arms of the angle. Angles are measured in degrees using a protractor, and every angle can be classified by its measure.

• Acute angle: greater than 0° and less than 90°
• Right angle: exactly 90°
• Obtuse angle: greater than 90° and less than 180°
• Straight angle: exactly 180°
• Reflex angle: greater than 180° and less than 360°
• Complete angle: exactly 360° (one full turn)

Worked example: An angle measures three-fifths of a right angle. Classify the angle.

A right angle is 90°, so the angle = (3/5) × 90° = 54°. Since 0° < 54° < 90°, this is an acute angle.

Two angles whose measures add up to 90° are called complementary angles, and two angles whose measures add up to 180° are called supplementary angles. These pairs need not be next to each other.

Adjacent angles have the same vertex, share one common arm, and their non-common arms lie on either side of the common arm. When two adjacent angles are formed by a ray standing on a line so that their non-common arms form a straight line, they are called a linear pair. By the Linear Pair Axiom: if a ray stands on a line, the two adjacent angles formed always add up to 180°. The converse is also an axiom: if the sum of two adjacent angles is 180°, their non-common arms lie along one straight line.

When two lines cross each other at a point, four angles are formed. The pair of angles opposite each other at that point are called vertically opposite angles, and they are always equal.

Worked example (linear pair): Two adjacent angles forming a linear pair measure 4x° and 5x°. Find x and the two angles.

4x + 5x = 180° ⇒ 9x = 180° ⇒ x = 20°. So the angles are 4(20°) = 80° and 5(20°) = 100°.

Proof that vertically opposite angles are equal: Let lines AB and CD intersect at O, forming ∠AOC, ∠COB, ∠BOD and ∠DOA. Since ray OC stands on line AB, ∠AOC + ∠COB = 180° ... (1). Since ray OB stands on line CD, ∠COB + ∠BOD = 180° ... (2). Comparing (1) and (2): ∠AOC + ∠COB = ∠COB + ∠BOD, which gives ∠AOC = ∠BOD. In the same way, ∠COB = ∠AOD. So each pair of vertically opposite angles is equal.

A line that intersects two or more lines at distinct points is called a transversal. When a transversal cuts two lines, eight angles are formed, and these are grouped into pairs based on their position.

• Corresponding angles occupy the same relative position at each intersection (for example, both above the line and on the same side of the transversal).
• Alternate interior angles lie between the two lines, on opposite sides of the transversal.
• Alternate exterior angles lie outside the two lines, on opposite sides of the transversal.
• Co-interior angles (also called consecutive interior or allied angles) lie between the two lines, on the same side of the transversal.

When the two lines cut by the transversal are parallel, these pairs follow fixed rules: each pair of corresponding angles is equal (Corresponding Angles Axiom); each pair of alternate interior angles is equal; and each pair of co-interior angles is supplementary, adding up to 180°.

Worked example: Lines l and m are parallel and are cut by a transversal t. One interior angle on line l measures 65°. Find (a) the alternate interior angle on line m, and (b) the co-interior angle on line m on the same side of the transversal.

(a) Since l ∥ m, alternate interior angles are equal, so the alternate interior angle on m is also 65°.

(b) Co-interior angles are supplementary, so the co-interior angle on m = 180° − 65° = 115°.

The angle relationships formed by a transversal also work in reverse — they give tests to prove that two lines are parallel, without measuring the lines themselves.

• If a transversal cuts two lines so that a pair of corresponding angles is equal, the two lines are parallel.
• If a transversal cuts two lines so that a pair of alternate interior angles is equal, the two lines are parallel.
• If a transversal cuts two lines so that a pair of co-interior angles is supplementary (adds to 180°), the two lines are parallel.

Another useful result: if two lines are each parallel to a third line, then they are parallel to each other. That is, if l ∥ m and m ∥ n, then l ∥ n.

Worked example: A transversal cuts lines p and q, forming a pair of co-interior angles measuring (2x + 10)° and (3x − 5)°. If p ∥ q, find x and the two angles.

Since p ∥ q, the co-interior angles must add up to 180°: (2x + 10) + (3x − 5) = 180 ⇒ 5x + 5 = 180 ⇒ 5x = 175 ⇒ x = 35.

The angles are 2(35) + 10 = 80° and 3(35) − 5 = 100°. Check: 80° + 100° = 180°. ✓

Angle sum property: The sum of the three interior angles of any triangle is always 180°.

Proof sketch: In triangle ABC, draw a line l through vertex A parallel to side BC. Since l ∥ BC, using AB as a transversal, the alternate interior angle to ∠B equals ∠B; using AC as a transversal, the alternate interior angle to ∠C equals ∠C. These two alternate angles together with ∠BAC lie on the straight line l at A, so they add up to 180°. Therefore ∠B + ∠BAC + ∠C = 180°.

When one side of a triangle is extended (produced), the angle formed outside the triangle is called an exterior angle. It forms a linear pair with the interior angle of the triangle right next to it, so the two are supplementary.

Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two interior opposite (remote) angles.

Proof sketch: In triangle ABC, extend BC to D, forming exterior angle ∠ACD. Since ∠ACB and ∠ACD form a linear pair, ∠ACB + ∠ACD = 180°. Also, by the angle sum property, ∠A + ∠B + ∠ACB = 180°. Subtracting these two equal totals gives ∠ACD = ∠A + ∠B.

Worked example: In triangle PQR, side QR is produced to a point S. The exterior angle ∠PRS = 115° and ∠Q = 50°. Find ∠P and ∠PRQ.

By the exterior angle theorem: ∠PRS = ∠P + ∠Q ⇒ 115° = ∠P + 50° ⇒ ∠P = 65°.

∠PRQ and ∠PRS form a linear pair, so ∠PRQ = 180° − 115° = 65°. Check with the angle sum property: 65° + 50° + 65° = 180°. ✓