In a plane or space, two lines can meet at a point, run side by side, cross at a right angle, or pass as skew lines.
Two railway rails that never touch however far they run are parallel, and a beam across them acts as a transversal.
Where a line crosses parallel lines, corresponding and alternate angles appear, equal while the lines stay parallel.
Slide to switch the relation, then turn the transversal to compare the angles.
Two-Line Positional Relations
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Intersect · Parallel · Perpendicular · Skew
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👀 Four Positional Relations
①Intersect: meet at a single point
②Parallel: same plane, never meet (∥)
③Perpendicular: meet at 90° (⊥)
④Skew: not in the same plane, neither meeting nor parallel
Parallel Lines and Angles
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Corresponding (blue) and alternate (red) angles — equal when lines are parallel
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🔍 Corresponding & Alternate Angles
①Corresponding: angles in the same position
②Alternate: angles in opposite positions
③Parallel ⇒ corresponding angles equal
④Parallel ⇒ alternate angles equal
Conditions for Parallel
Parallel Test (Corresponding)
Equal corresponding angles ⇒ parallel lines
Conversely, parallel ⇒ corresponding angles are equal
Parallel Test (Alternate)
Equal alternate angles ⇒ parallel lines
Equivalent to the corresponding-angle condition
💡 Distance from Point to Line
①The foot of the perpendicular from the point to the line
②This is the shortest distance
③Distance between parallel lines is constant
Work It Out
Example 1
Two parallel lines are crossed by a transversal; one corresponding angle is 70°. Find the angle that corresponds to it.
1
With parallel lines, corresponding angles are equal.
parallel ⇒ corresponding angles equal
2
So it equals the given angle.
corresponding angle = 70°
▸ 70°
If two lines are parallel, corresponding and alternate angles are each equal.
Example 2
For two parallel lines, one co-interior angle is 110°. Find the other co-interior angle.
1
Co-interior angles on the same side sum to 180°.
co-interior angles sum = 180°
2
Subtract from 180°.
180° − 110° = 70°
▸ 70°
Co-interior angles are not equal; they are supplementary (sum 180°).
Exam Key Points
Positional Essentials
Parallel ↔ corresponding equal ↔ alternate equal
The three are equivalent
Grade-7 school exam type
A transversal crosses two parallel lines; two corresponding angles are (2x + 10)° and (x + 40)°. What is x?
①20
②25
③30
④35
⑤40
▸ ③ 30
1
Corresponding angles are equal, so set the two expressions equal.
2x + 10 = x + 40
2
Solve for x.
2x − x = 40 − 10 ⇒ x = 30
🎯 Exam Key Points
①Use ∥ for parallel, ⊥ for perpendicular
②Identify corresponding vs alternate by position
③If lines aren't parallel, those angles are NOT equal
④Skew exists only in 3D space
⑤Foot of the perpendicular = right-angle intersection