Section 03

Proving that two straight lines are parallel

Two straight lines cut by a transversal are parallel whenever one of three conditions holds. Combine a condition with a clean, justified write-up and the proof is complete.

The three conditions

Two corresponding angles are equal in measure.

Two alternating angles (both interior or both exterior) are equal in measure.

Two interior angles on the same side of the transversal are supplementary.

Also useful: if two straight lines are parallel to a third line, they are parallel to each other. If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other. Two lines perpendicular to the same line in a plane are parallel.

How to write a proof in geometry

Step 1
Read carefully
Separate what is given from what is required. Every word in the statement is data.
Step 2
Write the given
List the given facts as short bullet points, using symbols (∥, ⊥, m(∠A) = …).
Step 3
Write the required
State exactly what must be found or proved.
Step 4
Plan the proof
Sketch the essential steps that connect the given to the required before writing.
Step 5
Write statements + reasons
Each mathematical statement needs a justification: a definition, postulate, theorem, given fact or property.
Step 6
Check
Confirm the required question is answered and the numbers are consistent.

Notation: the symbol ∵ is short for “since” and introduces a given fact or theorem; ∴ is short for “therefore” and introduces a derived statement.

Worked proofs

Example 2 — find x and y

Given

CD ∥ BA
BC ∥ DE
m(∠ABC) = 70°
m(∠CDE) = (x + 10)°
y = m(∠BCD)

Required

Find the values of x and y.

Proof

∵ BA ∥ CD and BC is a transversal
∴ m(∠ABC) = m(∠BCD) = 70°	alternating interior angles are equal
∴ y = 70
∵ BC ∥ DE and CD is a transversal
∴ m(∠CDE) + m(∠BCD) = 180°	interior angles on the same side are supplementary
∴ m(∠CDE) = 180° − 70° = 110°
∴ x + 10 = 110 ⟹ x = 100

Example 3a — prove BA ∥ CD

Given

m(∠ABC) = 58°
m(∠ECD) = 29°
CE bisects ∠BCD

Required

Prove that BA ∥ CD.

Proof

∵ CE bisects ∠BCD
∴ m(∠BCD) = 2 × 29° = 58°	definition of an angle bisector
∵ m(∠ABC) = m(∠BCD) = 58°	and they are alternating interior angles
∴ BA ∥ CD	equal alternating interior angles

Example 3b — prove AB ∥ CD

Given

m(∠EFB) = 54°
m(∠CMF) = 126°

Required

Prove that AB ∥ CD.

Proof

∵ AB ∩ EN = {F}
∴ m(∠EFB) = m(∠AFM) = 54°	vertically opposite angles
∵ m(∠AFM) + m(∠CMF) = 54° + 126° = 180°	and they are interior angles on the same side
∴ AB ∥ CD	co-interior angles are supplementary

Example 4 — the auxiliary parallel line

Given

DE ∥ BA
m(∠EDC) = 120°
m(∠ABC) = 130°

Required

Find m(∠DCB) with proof.

Proof

Construction: draw CF ∥ DE and ∥ BA through C	two lines parallel to a third are parallel
∵ DE ∥ CF and DC is a transversal
∴ m(∠DCF) = 180° − 120° = 60°	interior angles on the same side
∵ BA ∥ CF and BC is a transversal
∴ m(∠BCF) = 180° − 130° = 50°	interior angles on the same side
∴ m(∠DCB) = 60° + 50° = 110°	angle addition