Lesson 2: Parallel Lines and Transversal

• Lines in the same plane that do not intersect are parallel lines. ‘Line l ∥ line m’ is read ‘line l is parallel to line m’.
• A line that intersects two given lines in two distinct points is a transversal of those lines.
• A transversal makes 8 angles: 4 at each of the two points of intersection. Each angle has one arm on the transversal and the other on one of the given lines.
• Four of the angles lie between (inside) the two lines (interior) and four lie outside (exterior).

• Line l ∥ line m. The transversal t cuts them at two distinct points M and N and forms 8 angles.

• The strip between l and m is shaded: the angles c, d, e and f lie inside it (interior) and a, b, g and h lie outside (exterior).

• Corresponding angles: arms on the transversal in the same direction and the other arms on the same side of the transversal. Four pairs. In the lesson figure: ∠AMP and ∠MNR; ∠PMN and ∠RNT; ∠AMQ and ∠MNS; ∠QMN and ∠SNT.
• Interior angles: on the same side of the transversal and inside the given lines. Pairs: ∠PMN and ∠MNR; ∠QMN and ∠MNS.
• Alternate angles: on opposite sides of the transversal, arms on the transversal in opposite directions. Interior alternate: ∠PMN and ∠MNS; ∠QMN and ∠RNM. Exterior alternate: ∠AMP and ∠TNS; ∠AMQ and ∠RNT.
• With the letters a, b, c, d (at the first point) and e, f, g, h (at the second point): corresponding (a, e), (b, f), (c, g), (d, h); interior alternate (d, f), (c, e); exterior alternate (a, g), (b, h); interior (d, e), (c, f).

• Corresponding angles hold the same position at the two points: (a, e), (b, f), (c, g) and (d, h) — each pair has one colour.

• Interior angles lie on the same side of t, between the lines: the pairs are (d, e) in gold and (c, f) in pink.

• Alternate angles lie on opposite sides of t: interior alternate (d, f) and (c, e); exterior alternate (a, g) and (b, h).

When two parallel lines are cut by a transversal:

• each pair of corresponding angles is congruent;
• each pair of alternate angles (interior and exterior) is congruent;
• each pair of interior angles is supplementary (sum 180°), e.g. m∠PMN + m∠MNR = 180°.
• Paper activity: the angles shown by part I and part II form a linear pair; ∠b ≅ ∠d ≅ ∠f ≅ ∠h and ∠a ≅ ∠c ≅ ∠e ≅ ∠g. If one of the eight angles is known, the other seven can be found.
• Vertically opposite angles are equal, and a linear pair adds up to 180°.

• Shapes to remember (l ∥ m): F-shape corresponding angles are equal, Z-shape alternate angles are equal, C-shape interior angles add up to 180°.

• If one of the eight angles is 65°, four angles are 65° (gold) and the other four are 180° − 65° = 115° (pink).

• Solved example 1: AB ∥ PQ, LM transversal, m∠MNQ = 70°. Method I: m∠ONP = m∠MNQ = 70° (opposite angles); ∠AON and ∠ONP are interior angles, so m∠AON = 180° − 70° = 110°. Method II: m∠NOB = 70° (corresponding), so m∠AON = 180° − 70° = 110°.
• Solved example 2: m ∥ n, m∠b = (x + 15)°, m∠e = (2x + 15)°. ∠b ≅ ∠f (corresponding), and ∠f + ∠e = 180° (linear pair). So x + 15 + 2x + 15 = 180, 3x = 150, x = 50°.
• Method: identify the pair of angles, decide whether they are equal or add up to 180°, form the equation and solve.
• Practice answers: 3x and x interior → x = 45°.

• Example 1: ∠MNQ = 70° and ∠ONP is vertically opposite to it, so ∠ONP = 70°. The interior angles ∠AON and ∠ONP add up to 180°, so ∠AON = 110°.

• Example 2: b = (x + 15)° and e = (2x + 15)°. Since b = f (corresponding) and f + e = 180° (linear pair), 3x + 30 = 180, so x = 50.

• Method: name the pair of angles, decide whether they are equal or add up to 180°, write the equation, solve it and check the answer.

• Converse: if a transversal cuts two lines and a pair of corresponding angles is congruent, or a pair of alternate angles is congruent, or a pair of interior angles is supplementary, then the lines are parallel.
• Construction I, Method I (two set-squares): draw line l, take a point P outside it, place two set-squares touching each other; keep one edge of set-square A close to P and draw a line along the edge of B; this line m ∥ l.
• Method II: draw PM ⊥ l, take N on l, draw NQ ⊥ l with NQ = MP; line PQ ∥ l.
• Construction II (parallel at a given distance, e.g. 2.5 cm): take A and B on l, draw perpendiculars, mark P and Q at 2.5 cm from A and B, join PQ. Line PQ ∥ l at distance 2.5 cm.

• Converse: the corresponding angles 72° and 72° are equal, so l ∥ m. Equal alternate angles, or interior angles adding up to 180°, also prove that the lines are parallel.

• Two set-squares: keep B fixed and slide A along its edge from line l up to the point P. The line m drawn along the edge of A through P is parallel to l.

• Equal perpendiculars: PM ⊥ l and NQ ⊥ l with PM = NQ = 2.5 cm. The line PQ is parallel to l, at a distance of 2.5 cm from it.