Two lines in a plane either meet at one point or are parallel, and when they meet at a right angle they are called perpendicular.
In space there is one more case: two lines that neither meet nor are parallel are called skew.
When a third line crosses two parallel lines, it makes corresponding angles in matching positions and alternate angles in opposite positions, and when the two lines are parallel, corresponding angles are equal and so are alternate angles.
On this page you see the four relationships in pictures, then tilt the crossing line to check that the corresponding and alternate angles stay equal.
How two lines can sit
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Two lines: meeting at a point, parallel, perpendicular, skew
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👀 How two lines can sit
①They meet at one point, called the point of intersection
②They are parallel: in the same plane, they never meet however far they are extended, written ℓ ∥ m
③They are perpendicular: they meet at a right angle, written ℓ ⊥ m, and each is a perpendicular to the other
④They are skew: this only happens in space, like edges AB and CG in the picture, which neither meet nor are parallel
Parallel lines, corresponding and alternate angles
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Parallel lines and a transversal: corresponding angles (blue) are equal, and so are alternate angles (red)
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🔍 Corresponding and alternate angles
①When a line crosses two other lines, the two angles in the same position at the two crossing points are corresponding angles: the two blue angles in the picture
②The two angles between the lines on opposite sides of the transversal are alternate angles: the two red angles
③If the two lines are parallel, corresponding angles are equal and alternate angles are equal
④In the picture, a blue angle and a red angle sit side by side on one straight line, so they add up to 180°
When are two lines parallel?
Parallel lines: property and test
two lines are parallel ⇔ corresponding angles are equal
Turned around, equal corresponding angles mean the lines are parallel
Testing with alternate angles
two lines are parallel ⇔ alternate angles are equal
One equal pair of corresponding or alternate angles is enough to make the lines parallel
💡 Distance from a point to a line
①Draw a line from a point to a given line so that it meets it at a right angle; the meeting point is the foot of the perpendicular
②The distance between the point and the foot is the distance from the point to the line, and it is shorter than the distance to any other point on the line
③The distance between two parallel lines is the same wherever you measure it
Check with examples
Example 1
Two parallel lines are crossed by a third line, and one of the angles is 70°. Find the angle corresponding 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°
Set the transversal to 70° in the picture and both blue angles show 70°.
Example 2
Two parallel lines are crossed by a third line. The angle corresponding to ∠x is 70°. Find ∠y, which sits beside ∠x on a straight line.
1
Corresponding angles are equal with parallel lines, so ∠x = 70°.
∠x = 70°
2
∠x and ∠y lie side by side on a straight line, so they add up to 180°.
∠y = 180° − 70° = 110°
▸ 110°
Set the transversal to 70° in the picture: the blue angle shows 70° and the red angle 110°, side by side on one line.