Simple proofs for congruence in triangles

Congruent triangles are triangles that are identical in shape and size. In other words, all corresponding sides and angles are equal. 

We can confirm congruent shapes by seeing that if by using a combination of flips, slides or turns, we can reorient one shape to look like the other.  If you have to use enlargements to stretch or change the size of the shape than they are not congruent.  In this chapter, we are going to look at four different ways to prove triangles are congruent, as well as how to write these proofs.

 

Congruent Triangles

Two triangles are considered congruent if

• all three angles have the same value AND
• all three sides have the same value

Normally though we can identify congruent triangles with 3 pieces of information (not the whole 6).  This is because once we know 3 pieces we can use other geometrical angle rules or trigonometry side length rules to calculate the others.

 

SSS - all sides are equal

If all three sides on both triangles have the same corresponding lengths then the triangles are congruent.  We abbreviate this to SSS. 

Writing the proof:

In $\triangle ABC$△ABC and $\triangle FGE$△FGE:

$AB$AB	$=$=	$FG$FG (given)
$AC$AC	$=$=	$EF$EF (similarly)
$BC$BC	$=$=	$EG$EG (similarly)
$\therefore$∴ $\triangle ABC$△ABC	$\cong$≅	$\triangle FGE$△FGE (SSS)

 

ASA and AAS - two angles and one side are equal

If we know two angles and $1$1 corresponding side length, then the triangles are congruent.  We abbreviate this to 

• ASA is what we use when the side is between the two angles.  Formally we say two angles and the included side.  

• AAS is what we use when the side is not between the two angles, but one of the others.  Formally we say two angles and the non-included side. 

Some schools and countries consider ASA and AAS as different proofs for congruency, mathematically however they are the same because if you know two angles, you can work out the third using the angle sum of a triangle is equal to $180^\circ$180°. This can change the information we have from AAS to ASA, hence they are both equivalent proofs. 

 

SAS- two pairs of corresponding sides, and the pair of included angles are equal

If two sides and the included angle are equal on two triangles, then the triangles are congruent.  We abbreviate this to SAS.

The angle used must be the one between the two sides. 

Writing the proof:

In $\triangle MKP$△MKP and $\triangle AGZ$△AGZ:

$MK$MK	$=$=	$AG$AG (given)
$KP$KP	$=$=	$ZG$ZG (similarly)
$\angle MKP$∠MKP	$=$=	$\angle AGZ$∠AGZ (similarly)
$\therefore$∴ $\triangle MKP$△MKP	$\cong$≅	$\triangle AGZ$△AGZ (SAS)

 

RHS- right-angled triangles with equal hypotenuses and a pair of equal corresponding sides

If two right-angled triangles have equal hypotenuses and a pair of corresponding sides that are equal, then these triangles are congruent. We abbreviate this to RHS.

Writing the proof:

In $\triangle ABC$△ABC and $\triangle PRQ$△PRQ:

$\angle ABC$∠ABC	$=$=	$\angle PRQ$∠PRQ  $=$=  $90^\circ$90° (given)
$AC$AC	$=$=	$PQ$PQ (given)
$AB$AB	$=$=	$PR$PR (similarly)
$\therefore$∴ $\triangle ABC$△ABC	$\cong$≅	$\triangle PRQ$△PRQ

 

 

Examples

Question 1

a) Which pairs of triangles are congruent?

b) Give reasons for the congruence.

 

Question 2

Prove that $\triangle OAB$△OAB is congruent to $\triangle OCD$△OCD where $O$O is the centre of the circle.  Do this by stating three facts and then give the reason why these facts imply that the triangles are congruent. 

 

Question 3

Prove that $\triangle ABO$△ABO and $\triangle CDO$△CDO are congruent.