Geometry Theorems

Information for Students and Teachers

Geometry Theorems

Geometry is a very formal part of the Mathematics course. Proofs need to be set out in a very particular way that ensures that all statements have a reason attached to them.

Most of the following pages were originally written by the mathematics staff at James Ruse Agricultural High School and we acknowledge that this is primarily their work (in particular Mr R Howlett).

The Mathematics Department at PLC Sydney made several elaborations and additions.

By looking at the following worked examples, it is hoped that students will be guided by the steps and justification needed to correctly write down a geometrical proof.

1

GEOMETRY THEOREMS AND PROOFS

Rationale:In order to gain full marks, students should present a solution in which there is a full equation showing the geometric property that is being used and a worded reason that again identifies the geometric property that is being used.

EXAMPLE:Find the value of x.

C

B

Ax

42

73

EQUATION REASON COMMENT(Angle sum of

)Desired level of proof to be reproduced by students:

full equation contains geometric property reason contains geometric property

General Notes:(1) the word “equals” may be replaced by the symbol “=” or words such as “is”(2) abbreviations such as “coint”, “alt”, “vert opp”, etc are to be used with caution. The Board of Studies advises “that commonly accepted abbreviations in geometrical proofs are accepted by markers, provided that the abbreviation left the marker in no doubt that the student knew the relevant theorem or property”.(3) the angle symbol (), the triangle symbol (), the parallel symbol (||), the perpendicular symbol (), etc are not to be used as substitutes for words unless used with labels such as PQR, ABC, AB||XY, PQST(4) If the diagram is not labelled then students may introduce their own labels or refer to the shape in general terms such as “angle sum of triangle is 180o” or “angle sum of straight angle is 180o”(5) is the same as (6) The ‘converse’ of a theorem or rule means the reverse of the rule or theorem, written in a back-to-front way. If a statement is true, its converse may be true or false.Example 1:Statement: If lines are parallel, then corresponding angles are equal. (True)Converse: If corresponding angles are equal, then the lines are parallel. (True)Example 2:Statement: If any two angles are right angles, then they are equal angles. (True)Converse: If any two angles are equal angles, then they are right angles. (False)

2

Revolution, Straight Angles, Adjacent angles, Vertically opposite anglesThe sum of angles about a point is 360o

Find the value of x. (angle sum at point P equals 360o)

OR (angles at a point equal

360o)

OR (angles in a revolution)

The sum of the angles in a right angle is 90o.

AB is perpendicular to BC. Find the value of x. (angle sum of right angle ABCequals 90o)

ORx + 36 = 90 (angles in a right angle equal 90o)

ORx + 36 = 90 ( is a right angle)

AB is perpendicular to BC. Find the value of x. (angle sum of right angle ABCequals 90o)

OR (angles in a right angle equal 90o)

3

D

CB

A

xo

36o

E

The sum of the angles in a straight angle is 180o.

FMJ is a straight line. Find the value of x. (angle sum of straight angle FMJ equals 180o)

OR (adjacent angles on a straight line

equal 180o)

OR ( is a straight angle)

FMJ is a straight line. Find the value of x.

J

I

H

G

F M

5046 4x

2x

(angle sum of straight angle FMJ equals 180o)

OR (angles on a straight line

equal 180o)

4

Three points are collinear if they form a straight angle

AKB is a straight line.Prove that the points P, K and Q are collinear.

Q

P

K

B

A

723x

2x

(angle sum of straight angle AKB equals 180o)

P, K and Q are collinear (PKQ is a straight angle which equals 180o)

Vertically opposite angles are equal.

AC and DE are straight lines. Find the value of y. (vertically opposite angles are equal)

5

Angles and Parallel LinesAlternate angles on parallel lines are equal.

All lines are straight. Find the value of x. (alternate angles are equal as AB||CD)

NOTE: it is important to name the lines, especially when there is more than one pair of parallel lines in the diagram.

OR (alternate angles on parallel lines are equal)

Corresponding angles on parallel lines are equal.

All lines are straight. Find the value of x. (corresponding angles are equal as AB||CD)

NOTE: it is important to name the lines, especially when there is more than one pair of parallel lines in the diagram.

OR (corresponding angles on parallel lines are

equal)

6

Co-interior angles on parallel lines are supplementary.

All lines are straight. Find the value of x. (co-interior angles are supplementary as AB||CD)

NOTE: it is important to name the lines, especially when there is more than one pair of parallel lines in the diagram.

OR (co-interior angles on parallel lines

are supplementary)

Two lines are parallel if a pair of alternate angles are equal

Prove that AB CD is alternate to (given)

, because a pair of alternate angles are equal

Two lines are parallel if a pair of corresponding angles are equal

Prove that AB CD is corresponding to EGB = GHD = 65o (given)

, because a pair of corresponding angles are equal

7

Two lines are parallel if a pair of cointerior angles are supplementary

Prove that PR KM is co-interior to RQL + QLM = 124o + 56o (given)

= 180o

, because a pair of co-interior angles are supplementary

8

Angles in PolygonsThe longest side is opposite the largest angle. The shortest side is opposite the smallest angle.

Longest side = BCLargest angle = z

Shortest side = ABSmallest angle = y

For a triangle to exist, the sum of the two smaller sides in a triangle must be greater than the longest side. This is known as the ‘Triangle Inequality.

AB + AC > BC

Conversely, the length of the longest side is always less than the sum of the other two sides.BC < AB + AC

The angle sum of a triangle is 180o.

Find the value of x. (angle sum of equals 180o)

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

Find the value of x. (exterior angle of equals sum of the two opposite interior angles)

The sum of the exterior angles of a triangle is 360o.

Find the value of x. (sum of exterior angles of equals 360o)

9

The angles opposite equal sides of a triangle are equal. (The converse is also true)

Find the value of x. (equal angles are opposite equal sides in )

ORAC = AB (given)

ABC is isosceles (base angles of isosceles are

equal)

NOTE: The ‘converse’ of a theorem or rule means the reverse of the rule or theorem, written in a back-to-front way. If a statement is true, its converse may be true or false.

The sides opposite equal angles of a triangle are equal (The converse is also true).

Find the value of x. (equal sides are opposite equal angles in )

ORABC = BAC (given)

ABC is isosceles(equal sides in isosceles triangle)

All angles at the vertices of an equilateral triangle are 60o.

is equilateral. EC and DB are angle bisectors and meet at P. Find the size of CPB.

ACB = 60o (all angles of an equilateral triangle are 60o)

similarly ABC = 60o

ECB = 30o (EC bisects ACB, given)similarly DBC = 30o

CPB + 60o = 180o (angle sum of equals 180o)

CPB = 120o

10

The angle sum of a quadrilateral is 360o.

Find the value of x. (angle sum of quadrilateral ABCD equals 360o)

The angle sum of a n-sided polygon is or (2n-4)right angles.

Find the value of x. Angle sum of a pentagon = (n-2) = (5-2) =

x +92+87+106+165 = 540 (angle sum of pentagon equals 540o)

x+450 = 540x = 90

The angle at each vertex of a regular n-sided polygon is

Find the size of each interior angle of a regular hexagon.

The sum of the exterior angles of a n-sided polygon is 360o.

Find the size of each interior angle of a regular decagon.

Sum of exterior angles = 360o

Exterior angles =

= 36o

Interior angles = 144o (angle sum of straight angle equals 180o)

11

Pythagoras’ TheoremPythagoras’ Theorem: In a right angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Find the value of x. (Pythagoras’ Theorem)

9=x (x>0 as it is a length)x=9 units

OR (3,4,5 Pythagorean Triad)

A triangle is right-angled if the square on the hypotenuse equals the sum of the squares on the other two sides (converse of Pythagoras’ Theorem)Prove that the triangle with sides 6, 10 and 8 is right-angled.

Therefore the triangle is right-angled, because it satisfies Pythagoras’ Theorem

More formally:Prove that ABC is right-angled. Prove

ABC is right-angled (Pythagoras’ theorem converse)

12

Congruent Triangles – The 4 Tests for Congruent Triangles‘ ’ means ‘is congruent to’Two triangles are congruent if three sides of one triangle are equal to three sides of the other triangle. (The SSS Test)

Given that AC = DB and AB = DC,prove that CAB BDC.

In CAB and BDCAC = BD = 8 (given)AB = CD = 12 (given)CB is common CAB BDC (SSS)

Two triangles are congruent if two sides of one triangle are equal to two sides of the other triangle and the angles included by these sides are equal. (The SAS Test)

Given that AC = BD and CAB = DBA,prove that CAB DBA.

In CAB and DBAAB is commonAC = BD (given)CAB = DBA (given)CAB DBA (SAS)

Two triangles are congruent if two angles of one triangle are equal to two angles of the other triangle and one pair of corresponding sides is equal. (The AAS Test)

Given that AD and BC are straight lines and AB = CD and EAB = ECD, prove that ABE  CDE.

In ABE and CDE.AB = CD (given)EAB = ECD (given)AEB = CED (vertically opposite angles are

equal)ABE CDE (AAS)

13

Two right-angled triangles are congruent if their hypotenuses are equal and a pair of sides is also equal. (The RHS Test)

Given that CD = AD, prove that ABD CBD. In ABD and CBDBCD = BAD = 90o

CD = AD (given)DB is commonABD CBD (RHS)

An example of a geometric proof that uses a congruency test:

Given that AB=CB and AD=CD, prove that BD bisects .

In ABD and CBDAB=CB (given)AD=CD (given)BD is commonABD CBD (SSS)

(corresponding angles in congruent triangles are equal)

bisects

NOTE: the word ‘matching’ can be used instead of ‘corresponding’.

Another example:

ABCD is a square. AE = CF. Prove that AF = CE Let AB = x Let AE = y

BE = x-y Similarly, DF = x-y

BE = DFIn and ,BE = DF (proved above)AD = CB (opposite sides of a square are equal)

(all angles in a square equal 90)

(SAS)AF = CE (corresponding sides in congruent

triangles are equal

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B

D

A C

A B

D C

E

F

Similar Triangles – The 4 Tests for Similar Triangles‘|||’ means ‘is similar to’There are NO abbreviations for the names of the similarity testsTwo triangles are similar if two angles of one triangle are equal to two angles of the other triangle NOTE: there is no need to mention the third angle.

Prove that ABC and DCA are similar. In ABC and DCAABC = DCA (given)BAC = CDA (given)ABC ||| DCA (equiangular)

Note: The abbreviations AA or AAA will not be accepted.

Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the angles included by these sides are equal.

Prove that BCA and ACD are similar. In BCA and ACDBCA = ACD (given)

BCA ||| ACD (sides adjacent to equal angles are in the same ratio)

ORIn BCA and ACDBCA = ACD (given)

BCA ||| ACD (sides adjacent to equal angles are in proportion)

15

ORIn BCA and ACDBCA = ACD (given)

BCA ||| ACD (two pairs of corresponding sides in the same ratio and included angles are equal)

NOTE: the word ‘matching’ can be used instead of ‘corresponding’.

Two triangles are similar if the ratios of the three pairs of sides are equal.

Prove that ABC and ACD are similar. In ABC and DCA

ABC ||| DCA (three pairs of sides in the same ratio)

ORIn ABC and DCA

ABC ||| DCA (three pairs of sides in proportion)

16

Two triangles are similar if the hypotenuse and a second side of a right-angled triangle are proportional to the hypotenuse and a second side of another right-angled triangle.

Prove that ABC and CDE are similar. In ABC and CDE

(right-angled triangles)ABC ||| DCA (hypotenuse and a second side

of a right-angled triangle are proportional to the hypotenuse and a second side of another right-angled triangle)

An example of a geometric proof that uses a similarity test:Example problem:Given that , find the value of x. In and

(corresponding angles are equal as )

(common) (equiangular)

(corresponding sides in similar triangles

are in the same ratio)

NOTE: (1) Either ‘in the same ratio’ or ‘are in proportion’ can be used in the reason.

(2) The word ‘matching’ can be used instead of ‘corresponding’.

17

A

B C

106

C

D E

53

Intercepts and ParallelsParallel lines preserve the ratio of intercepts on transversals. (The converse is not true)

Find the value of x. (parallel lines preserve the ratios of

intercepts on transversals) x = 24

OR

(when three or more parallel lines are cut

by two transversals, the ratio of intercepts is equal)

x = 24

An interval parallel to a side of a triangle divides the other sides in the same ratio. (The converse is true)

Find the value of x.(interval parallel to side of

divides other sides in same ratio)

x = 12

An interval joining the midpoints of the sides of a triangle is parallel to the third side and half its length.

E and F are midpoints of AB and AC.G and H are midpoints of FB and FC.Prove that EF = GH.

EF=½BC (interval joining midpoints of sides of is half the length 3rd side)

Similarly in , GH=½BC EF = GH

NOTE: It can also be proven that EF and GH are parallel

18

Quadrilateral PropertiesTrapeziumOne pair of sides of a trapezium are parallel

KiteTwo pairs of adjacent sides of a kite are equalOne diagonal of a kite bisects the other diagonalOne diagonal of a kite bisects the opposite anglesThe diagonals of a kite are perpendicularA kite has one axis of symmetry

ParallelogramThe opposite sides of a parallelogram are parallelThe opposite sides of a parallelogram are equalThe opposite angles of a parallelogram are equalThe diagonals of a parallelogram bisect each otherA parallelogram has point symmetry

RhombusThe opposite sides of a rhombus are parallelAll sides of a rhombus are equalThe opposite angles of a rhombus are equalThe diagonals of a rhombus bisect the opposite anglesThe diagonals of a rhombus bisect each otherThe diagonals of a rhombus are perpendicularA rhombus has two axes of symmetryA rhombus has point symmetry

RectangleThe opposite sides of a rectangle are parallelThe opposite sides of a rectangle are equalAll angles at the vertices of a rectangle are 90o

The diagonals of a rectangle are equalThe diagonals of a rectangle bisect each otherA rectangle has two axes of symmetryA rectangle has point symmetry

SquareOpposite sides of a square are parallelAll sides of a square are equalAll angles at the vertices of a square are 90o

The diagonals of a square are equalThe diagonals of a square bisect the opposite anglesThe diagonals of a square bisect each otherThe diagonals of a square are perpendicularA square has four axes of symmetryA square has point symmetry

19

Properties of Quadrilaterals:

kit

e

trap

eziu

m

par

alle

logr

am

rect

angl

e

rhom

bu

s

squ

are

How many axes of symmetry? 1 0 0 2 2 4

Does it have point symmetry? no no yes yes yes yes

How many pairs of opposite equal sides?

0 0 2 2 2 2

How many pairs of opposite parallel sides?

0 1 2 2 2 2

How many pairs of opposite equal angles?

1 0 2 2 2 2

Are the diagonals equal? no no no yes no yes

Do the diagonals bisect each other? no no yes yes yes yes

Do the diagonals meet at right angles?

yes no no no yes yes

Do the diagonals bisect each other at right angles?

no no no no yes yes

Do the diagonals bisect the angles through which they pass?

one does

no no no yes yes

Does it have 4 equal sides? no no no no yes yes

Does it have 4 right angles? no no no yes no yes

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The diagrams below show two different ways to represent the family of quadrilaterals. In the first diagram, the quadrilaterals become more and more special as you move down the branches.

quadrilateral

trapezium kite

parallelogram rhombus

rectangle

square

The second diagram shows, for example, that all rectangles are special parallelograms.

Source: About Maths

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Minimum conditions needed to prove a particular quadrilateral: Note: This list is not exhaustive.

A quadrilateral is a trapezium if: It has one pair of parallel sides

A quadrilateral is a kite if: Two pairs of adjacent sides are equalor The diagonals meet at right angles and one of them

is bisected by the other

A quadrilateral is a parallelogram if: both pairs of opposite sides are parallelor both pairs of opposite sides are equalor both pairs of opposite angles are equalor the diagonals bisect each other

(i.e. the diagonals have the same midpoint)or two sides are equal and parallel

A quadrilateral is a rhombus if: all sides are equalor diagonals bisect each other at right anglesor the diagonals bisect the angles at the verticesor a pair of adjacent sides are equal and opposite

angles are equal

A quadrilateral is a rectangle if: the diagonals are equal and they bisect each otheror it has three right anglesor it has two pairs of parallel sides and one right angleor it has two pairs of opposite sides equal and one

right angle

A quadrilateral is a square if: it has four equal sides and one right angleor the diagonals are equal, bisect each other and meet

at right angles

22

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Some useful tips for solving geometry problems under exam conditions

Copy the diagram, include all the “givens” and add extra information to it.Put your writing paper on top of the question booklet and trace the diagram (with pen), so that it looks almost identical to the original. As you try to solve the problem(s), add information to the diagram (with pencil). When you think you have answered the question on the diagram, re-trace your steps and write your solution.

Don’t start your solution by stating that the thing you are trying to prove is true.If the question says: “Prove that triangle ABP is congruent to triangle CAR”, don’t start your solution by writing .

You could start your solution with: Aim: To prove that

orRequired to prove

orIn

The last thing you write is:, as required

If the question has 3 parts and you need the answer for part (i) to do part (ii), but you can’t answer part (i), make up a reasonable estimate for the answer to part (i) and use it to do the other parts. Show all of your working and you might get full marks for parts (ii) and (iii).

Some useful terminology: If you have done a few steps and you want to repeat those steps, the word “similarly” is quite useful,

so that you don’t have to repeat all the steps. Remember the steps have to be exactly the same with exactly the same reasons.

If you proved something in part (ii) and want to use it in part (iii), you can write the phrase “from part (ii)” as a reason.

Be careful in using the word “given”. It means that the thing you have stated was actually written in the question. For example, if you are told that:

ABC is a straight line and BD is perpendicular to ACthen don’t write

(given)because that wasn’t “given”

You could write:(given)

24

“Angle Chasing”If you are trying to prove something like:

2 angles are equal one angle is twice the size of another two angles are supplementary or complementary

you can employ a method called “angle chasing”.

Eg Prove that

One way to do this is to start by writing: Let

Then, using , find other angles that might be, for example:

until you get to and so

25

Exercise: The Basic Tools of “Angle-Chasing”

Let . Write down an expression involving for (and give a reason) P C

A R

C

P R A

C R

A P

C P

R

PRCA is a rhombus P R

A C

PRAC is a rhombus R P

A C

Let . Write down an expression involving for (and give reasons) A C

R P

AC = CR A

P R C

C A

R P

Let . Write down an expression involving for (and give a reason) A

R C

C

R A

CART is a parallelogram C A

T R

AC = CR A

R C

AC = CR C

R A

AC = AR C

A R

26

y

yy

y

A

y

y

Here is a very good solution to a difficult geometry problem.

Example:

Let ABC be a triangle with AB = AC. D is a point on AC such that BC = BD. E is a point on AB such that BE = ED = AD. Find the size of the angle EAD.

Solution:

Let (given)

(angles opposite equal sides are equal in a triangle) (angle sum of a straight line)

(given) (angles opposite equal sides are equal in a triangle)

In

(angle sum of triangle)

(exterior angle equals sum of two interior opposite angles in triangle AED.

(adjacent angles)

27

A

B C

DE

A

B C

DE

x

x

180 x

A

B C

DE

x

x

2

x

2

x

(given)

(angles opposite equal sides are equal in a triangle)

(given)

(angles opposite equal sides are equal in a triangle)

In (angle sum of triangle)

28

A

B C

DE

x

x

2

x 3

2

x

3

2

x

3

2

x 3

2

x

A

B C

DE

x