Everything Maths Grade 10

12 Euclidean geometry

Geometry (from the Greek “geo” = earth and “metria” = measure) arose as the field of knowledge dealing
with spatial relationships. Geometry can be split into Euclidean geometry and analytical geometry. Analytical
geometry deals with space and shape using algebra and a coordinate system. Euclidean geometry deals with
space and shape using a system of logical deductions.

12.1 Proofs and conjectures EMA7H

We will now apply what we have learnt about geometry and the properties of polygons (in particular triangles
and quadrilaterals) to prove some of these properties. We will also look at how we can prove a particular
quadrilateral is one of the special quadrilaterals.

VISIT:

This video shows how to prove that the the diagonals of a rhombus are perpendicular.

See video:2GQMatwww.everythingmaths.co.za

Worked example 1: Proving a quadrilateral is a parallelogram

QUESTION

In parallelogramABCD, the bisectors of the angles (AW,BX,CY andDZ) have been constructed. You

are also givenAB=CD,AD=BC,AB∥CD,AD∥BC,A^=C^,B^=D^. Prove thatM N OPis a

parallelogram.

A

D

B

C

X W

Y Z

M

P

O

N

1

2

2

1

1

2

1

2

1

2

(^12)
1
2
(^21)

SOLUTION

Step 1: Use properties of the parallelogramABCDto fill in on the diagram all equal sides and angles.

Step 2: Prove thatM^ 2 =O^ 2

In△CDZand△ABX,

DCZ^ =BAX^ (given)

D^ 1 =B^ 1 (given)

DC =AB (given)

)△CDZ △ABX (AAS)

)CZ =AX

andCZD^ =AXB^

404 12.1. Proofs and conjectures