grade 7 Positional Relations

Positional Relations

In a plane or space, two lines can meet at a point (intersecting), run side by side (parallel), cross at a right angle (perpendicular), 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. Change the relation and the transversal angle, and the corresponding and alternate angles update.

Two-Line Positional Relations

Relation type0

👀 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

Transversal angle60

🔍 Corresponding & Alternate Angles

①Corresponding: angles in the same position

②Alternate: angles in opposite positions

③Parallel ⇒ corresponding angles equal

④Parallel ⇒ alternate angles equal

⑤Same-side interior: the two inner angles on the same side of the transversal. Parallel ⇒ they sum to 180°

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.

With parallel lines, corresponding angles are equal.

parallel ⇒ corresponding angles equal

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.

Co-interior angles on the same side sum to 180°.

co-interior angles sum = 180°

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

Corresponding angles are equal, so set the two expressions equal.

2x + 10 = x + 40

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

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Points, Lines, Planes & Angles

Construction & Congruence

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