Lines & Angles — Class 7 IMO

Complementary & Supplementary Angles

Two angles are complementary if they add up to 90°, and supplementary if they add up to 180°. So the complement of 30° is 60°, and the supplement of 110° is 70°.

Example 1: What is the complement of 30°?

90 − 30 = 60°.

Example 2: What is the supplement of 110°?

180 − 110 = 70°.

Quick recap

Complementary angles add to 90°.
Supplementary angles add to 180°.

✓ Quick check

What is the complement of 45°?

90 − 45 = 45°.

What is the supplement of 90°?

180 − 90 = 90°.

Linear Pair & Vertically Opposite Angles

Angles in a linear pair are next to each other on a straight line and add up to 180°. When two lines cross, the vertically opposite angles are equal.

Example 1: Two angles form a linear pair and one is 120°. What is the other?

180 − 120 = 60°.

Example 2: Two lines cross; one angle is 70°. What is the vertically opposite angle?

70° (vertically opposite angles are equal).

Quick recap

A linear pair adds up to 180°.
Vertically opposite angles are equal.

✓ Quick check

Vertically opposite angles are always ___ ?

Vertically opposite angles are equal.

Angles in a linear pair add up to ___ ?

A linear pair is supplementary, adding to 180°.

Parallel Lines, Transversals & Triangle Angle Sum

When a transversal crosses two parallel lines, 8 angles are formed with special relationships:

Angle Pair	Relationship	Examples
Corresponding	EQUAL	1=5, 2=6, 3=7, 4=8
Alternate Interior	EQUAL	3=6, 4=5
Alternate Exterior	EQUAL	1=8, 2=7
Consecutive Interior	Sum = 180 degrees	3+5=180, 4+6=180

Consecutive interior angles are also called co-interior or same-side interior angles.

PARALLEL LINES WITH TRANSVERSAL:

     1 | 2
    ---|---  L1
     3 | 4

     5 | 6
    ---|---  L2
     7 | 8

Corresponding (equal): 1=5, 2=6, 3=7, 4=8
Alternate interior (equal): 3=6, 4=5
Alternate exterior (equal): 1=8, 2=7
Consecutive interior (sum 180): 3+5=180, 4+6=180

Example 1: Two angles of a triangle are 60° and 70°. Find the third.

180 − 60 − 70 = 50°.

Example 2: When a transversal cuts parallel lines, what is true of corresponding angles?

They are equal.

Example 3: If angle 1 = 70 degrees, find angles 5, 3, and 6.

Angle 5 = 70 deg (corresponding). Angle 3 = 70 deg (vertical to angle 1). Angle 6 = 110 deg (consecutive interior: 70+110=180).

Example 4: Alternate interior angles are (3x+20) and (5x-40) degrees. Find x.

Alternate interior angles are equal: 3x+20 = 5x-40 => 60 = 2x => x = 30.

Quick recap

Corresponding and alternate interior angles are equal; co-interior angles are supplementary.
Triangle angle sum = 180°.
Corresponding angles are EQUAL
Alternate interior angles are EQUAL
Alternate exterior angles are EQUAL
Consecutive interior angles SUM to 180 degrees

✓ Quick check

Two angles of a triangle are 90° and 45°. What is the third angle?

180 − 90 − 45 = 45°.

Co-interior angles between parallel lines add up to ___ ?

Co-interior angles are supplementary, adding to 180°.