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When a line crosses two parallel lines, it makes eight angles but only two different sizes. Knowing which angles are equal and which add to 180° turns a diagram into equations you can solve. This lesson shows how to match each pair of angles to its rule and write a complete, reasoned solution.
The condition before the equation
Two lines looking parallel is not enough. Here AB ∥ CD is stated, and EF is the transversal: it cuts AB at P and CD at Q.
Track both crossings
Parallel lines AB and CD with transversal EF. Open boundary through the listed points in order. Equal drawing scales, y increasing down the page. A (-45, 0); B (45, 0); C (-25, 58); D (65, 58); E (-10, -29); F (30, 87); P (0, 0); Q (20, 58); Line 1 start (-25, 58); Line 1 end (65, 58); Line 2 start (-10, -29); Line 2 end (30, 87); Marked opening a vertex (0, 0); Marked opening a first arm (45, 0); Marked opening a second arm (30, 87); Marked opening b vertex (0, 0); Marked opening b first arm (30, 87); Marked opening b second arm (-45, 0); Marked opening c vertex (0, 0); Marked opening c first arm (-45, 0); Marked opening c second arm (-10, -29); Marked opening d vertex (0, 0); Marked opening d first arm (-10, -29); Marked opening d second arm (45, 0); Marked opening e vertex (20, 58); Marked opening e first arm (65, 58); Marked opening e second arm (30, 87); Marked opening f vertex (20, 58); Marked opening f first arm (30, 87); Marked opening f second arm (-25, 58); Marked opening g vertex (20, 58); Marked opening g first arm (-25, 58); Marked opening g second arm (-10, -29); Marked opening h vertex (20, 58); Marked opening h first arm (-10, -29); Marked opening h second arm (65, 58).
Scroll across to read every label.
Figure data
This table is included in print while it is open.
| Label | Information |
|---|---|
| Parallel condition | AB ∥ CD |
| Intersections | EF crosses AB at P and CD at Q |
| a | ∠BPF |
| b | ∠FPA |
| c | ∠APE |
| d | ∠EPB |
| e | ∠DQF |
| f | ∠FQC |
| g | ∠CQE |
| h | ∠EQD |
Follow the rays that bound each marked opening. Corresponding angles occupy matching positions at the two crossings. Alternate interior angles are between the parallel lines, on opposite sides of the transversal. Same-side interior angles are between the parallel lines, on the same side of the transversal.
Check your understanding 1
Trace the partners of a
Check your understanding 1
Trace the partners of a
Write one corresponding equality, one alternate-interior equality and one same-side interior sum involving angle a. Include the condition that makes the relations valid.
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Because AB ∥ CD, corresponding angles give a = e, alternate interior angles give a = g, and same-side interior angles give a + h = 180°. The position of each opening decides the relation; equal-looking sizes alone do not justify it.
Keep the lists complete
There are four matching-position pairs, two alternate-interior pairs and two same-side interior pairs in this diagram:
| Relation | Complete equations |
|---|---|
| Corresponding angles | a = e, b = f, c = g, d = h |
| Alternate interior angles | a = g, b = h |
| Same-side interior angles | a + h = 180°, b + g = 180° |
Count each pair once. For example, e = a is the same pair as a = e. An exterior angle can belong to a corresponding pair, but it cannot belong to an interior pair.
You will also use linear-equation steps when the given angles contain an unknown parameter. The geometry chooses the equation; the algebra solves it.
Worked example 1
When the angles are expressions
Worked example 1
When the angles are expressions
Suppose a = (2t + 9)° and e = (3t-22)° in the diagram with AB ∥ CD. Find t and all eight marked angles.
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Choose the equation from the corresponding positions
Method
Set the two angle expressions equal, then isolate t.Reason
AB ∥ CD is stated, and a and e occupy corresponding positions. That geometric condition justifies equality before the algebra begins.Working
Answer
2t + 9 = 3t-22; 31 = t.Return to the angle values
Method
Substitute into both givens, then use the crossing and parallel-line relations for the other openings.Reason
The parameter is 31; it is not itself an angle. Checking both expressions tests the equation against the original geometry.Working
Answer
a = 2(31) + 9 = 71°,; e = 3(31)-22 = 71°.Both values agree, as corresponding angles must. Vertically opposite and corresponding relations give c = g = 71° too. Each adjacent straight-line partner is 180°-71° = 109°, so b = d = f = h = 109°.
Quick check
The two original expressions both give 71°, and every neighbouring pair sums to 180°. Giving only t = 31 would leave the requested angles unanswered.
Worked example 2
Choose the sum from the positions
Worked example 2
Choose the sum from the positions
For a different parameter, suppose b = (2u + 9)° and g = (u + 21)° in the same diagram. Find u, b and g.
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Use the same-side interior sum
Method
Add the two angle expressions and set the sum to 180°.Reason
Both openings lie between the stated parallel lines on the same side of EF. This pair is supplementary, rather than a pair of equal angles.Working
Answer
(2u + 9) + (u + 21) = 180; 3u + 30 = 180; u = 50.Check the two original expressions
Method
Substitute u = 50 into both expressions and test their sum.Reason
An algebraic solution must satisfy the geometric relation that produced its equation.Working
Answer
The values are b = 2(50) + 9 = 109° and g = 50 + 21 = 71°. Their sum is 109° + 71° = 180°.
Quick check
Setting the expressions equal would use the wrong geometric relation. Check the angle values and the relation after solving, even when the algebra itself is correct.
Check your understanding 2
Remove one piece of information
Check your understanding 2
Remove one piece of information
A new drawing has two crossings in the same arrangement, but the two lines are not stated to be parallel. The marked corresponding-position expressions are 2t + 9 and 3t-22. A learner writes 2t + 9 = 3t-22 and obtains t = 31. What has the learner assumed, and is the conclusion justified?
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The learner assumed the two lines were parallel. Matching positions in a drawing do not establish equality without that condition or a separate valid argument proving it. The algebra solves the written equation, but the stated geometry does not justify writing that equation. The given information therefore does not establish t = 31.
Use the same sequence with fresh diagrams: establish the condition, identify the pair, form the equation, then check the resulting angles. A labelled relation is a bounded piece of reasoning; a general geometric proof may need further steps.
Common mistakes
- Using a parallel-line rule without parallel lines. The lines must be stated or marked parallel.
- Mixing up the pair types. Corresponding and alternate angles are equal; co-interior angles add to 180°.
- Reading the wrong angle at a vertex. Follow the two rays that bound the marked angle.
- Giving the value without the reason. Name the angle fact for each equation.
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Syllabus and review details
- G1 Mathematics 2027 · 2027
- G2 Mathematics 2027 · 2027
- G3 Mathematics 2027 · 2027
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