Grade 10-12 Euclidean Geometry

Triangles, Midpoints & Proportion

This covers every single acceptable reason for triangles listed in the DBE 2015 Exam Guidelines, including all Grade 12 proportion and similarity rules.

1. Basic Triangle Angles

∠ sum in ∆

ext ∠ of ∆

Sums and Shortcuts

The interior angles add to 180°. The exterior angle is equal to the sum of the two interior opposite angles.

2. Isosceles Triangles

∠s opp equal sides

sides opp equal ∠s

The Hidden Radii Trick

If a triangle has two equal sides (like radii), the angles sitting opposite those sides are equal.

Converse: If you prove two angles are equal, the sides opposite them must be equal.

3. Theorem of Pythagoras

Pythagoras

Converse Pythagoras

Right-Angled Lengths

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Converse: If the square of the longest side equals the sum of the squares of the other two, the triangle is exactly 90° right-angled.

4. Congruency

SSS

SAS OR S∠S

AAS OR ∠∠S

RHS OR 90°HS

Proving Triangles are Identical

To prove two triangles are perfectly congruent (identical in shape and size), you must find three specific matching properties. These are the ONLY four acceptable reasons you can use.

5. The Midpoint Theorem

Midpt Theorem

line through midpt || to 2nd side

The Half-way Slice

If you connect the exact midpoint of one side of a triangle to the exact midpoint of another side, the line is parallel to the third side AND exactly half the length of the third side.

Converse: A line drawn from the midpoint of one side, parallel to another side, bisects the third side.

6. Triangles with Same Base and Height

same base; same height OR equal bases; equal height

The Area Rule

If two triangles sit on the same line (base) and are trapped between the same parallel lines, they have the exact same area.

7. Proportionality Theorem

line || one side of ∆ OR prop theorem; name || lines

line divides two sides of ∆ in prop

Grade 12 Proportion

Forward (Examinable Proof): A line drawn parallel to one side of a triangle divides the other two sides proportionally.

Converse: If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side.

→ View the formal proof for this theorem

8. Similarity Theorem

||| ∆s OR equiangular ∆s

Sides of ∆ in prop

Grade 12 Equiangular Triangles

Forward (Examinable Proof): If two triangles are equiangular, then their corresponding sides are in proportion (and they are similar).

Converse: If the corresponding sides of two triangles are proportional, then the triangles are equiangular (similar).

→ View the formal proof for this theorem