Grade 8-10 Euclidean Geometry

Lines & Quadrilaterals

This covers every single acceptable reason for lines and quadrilaterals listed in the DBE 2015 Exam Guidelines, including all the tricky converses.

1. Straight Lines & Points

∠s on a str line

adj ∠s supp

∠s round a pt OR ∠s in a rev

vert opp ∠s =

The Fundamentals

Adjacent angles on a straight line add up to 180°.
Converse: If adjacent angles add to 180°, the outer arms form a straight line (adj ∠s supp).
Angles around a point add up to 360°.
When two lines intersect, vertically opposite angles are equal.

2. Parallel Lines (FUN)

alt ∠s; AB || CD

corresp ∠s; AB || CD

co-int ∠s; AB || CD

The "FUN" Shapes & Their Converses

If two lines are parallel (F, U, N shapes), the respective angles are equal or supplementary. You MUST name the parallel lines in your reason!

The Converses (Proving lines are parallel):

If alternate angles are equal, the lines are parallel. alt ∠s =
If corresponding angles are equal, the lines are parallel. corresp ∠s =
If co-interior angles add to 180°, the lines are parallel. coint ∠s supp

3. Quadrilateral Basics

sum of ∠s in quad

The 360 Rule

The interior angles of ANY quadrilateral add up to 360°.

4. Properties of a Parallelogram

opp sides of ||m

opp ∠s of ||m

diag of ||m

diag bisect area of ||m

Proving a Parallelogram (The Converses)

To prove a shape is a parallelogram, use one of these converse reasons:

opp sides of quad are ||
opp sides of quad are =
opp ∠s of quad are =
diags of quad bisect each other
pair of opp sides = and ||

5. Special Quadrilaterals (Rhombus, Rectangle, Square, Kite)

diags of rhombus

sides of rhombus

diags of rect

sides of square

diags of kite

diag of kite

The Highly Tested Properties

Rhombus: All 4 sides are equal. The diagonals bisect each other at 90°, AND the diagonals bisect the interior angles.

Rectangle: The diagonals are equal in length.

Kite: The diagonals intersect at 90° (diags of kite), a diagonal of a kite bisects the other diagonal (diag of kite), and a diagonal bisects the opposite angles (diag of kite).