The Triangle & its Properties — Class 7 IMO

Medians, Altitudes & 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: How many medians does a triangle have?

3 medians.

Example 2: What is an altitude perpendicular to?

The opposite side.

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

Median: vertex to midpoint; altitude: perpendicular from a vertex.
Angle sum of a triangle = 180°.
Corresponding angles are EQUAL
Alternate interior angles are EQUAL
Alternate exterior angles are EQUAL
Consecutive interior angles SUM to 180 degrees

✓ Quick check

How many medians does a triangle have?

A triangle has 3 medians.

The angles of a triangle add up to ___ ?

The angle sum of a triangle is 180°.

Exterior Angle Property

The exterior angle of a triangle equals the sum of the two opposite (interior) angles. So if the two opposite interior angles are 60° and 70°, the exterior angle is 130°.

Example 1: Interior angles 60° and 70°. Find the exterior angle.

60 + 70 = 130°.

Example 2: An exterior angle is 110° and one opposite interior angle is 50°. Find the other.

110 − 50 = 60°.

Quick recap

Exterior angle = sum of the two opposite interior angles.

✓ Quick check

The exterior angle of a triangle equals the sum of the two ___ interior angles.

It equals the sum of the two opposite interior angles.

Interior angles 45° and 55° give an exterior angle of ___ ?

45 + 55 = 100°.

Pythagoras Theorem & Inequalities

In a right-angled triangle, the Pythagoras theorem states hypotenuse² = base² + height². So legs 3 and 4 give a hypotenuse of 5.

The triangle inequality says the sum of any two sides must be greater than the third side.

Example 1: A right triangle has legs 3 cm and 4 cm. Find the hypotenuse.

√(9 + 16) = √25 = 5 cm.

Example 2: Can sides 2, 3 and 6 form a triangle?

No, because 2 + 3 = 5 is not greater than 6.

Quick recap

hypotenuse² = base² + height² (right triangles).
Triangle inequality: any two sides sum to more than the third.

✓ Quick check

A right triangle has legs 5 cm and 12 cm. What is the hypotenuse?

√(25 + 144) = √169 = 13 cm.

Can sides 2 cm, 3 cm and 6 cm form a triangle?

2 + 3 = 5 is not greater than 6, so no.