© T Madas

© T Madas

Two triangles are congruent if…

All 3 sides are equal SSS

2 sides and the contained angle are equal SAS

1 side and the 2 adjacent angles are equal ASA

© T Madas

© T Madas

Prove that the any point that lies on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment

A BM

C Let AB be a line segment and M its midpointLet C be a point on the perpendicular bisectorTwo right angled triangles are formedAM = MB MC is commonAMC = CMB = 90°

The two triangles have two sides and the contained angle of those sides, correspondingly equal (SAS)Therefore the triangles are congruentAC = CB

© T Madas

© T Madas

Given that a parallelogram has four equal sides, prove that its diagonals are perpendicular to each other.

A parallelogram with 4 equal sides is in general arhombus

A B

CDBDC = ABD

as alternate angles

© T Madas

Given that a parallelogram has four equal sides, prove that its diagonals are perpendicular to each other.

A parallelogram with 4 equal sides is in general arhombus

A B

CDBDC = ABD

as alternate angles

DCA = CABas alternate angles

© T Madas

Given that a parallelogram has four equal sides, prove that its diagonals are perpendicular to each other.

A parallelogram with 4 equal sides is in general arhombus

A B

CDBDC = ABD

as alternate angles

DCA = CABas alternate angles

DCA = CAB0

A S A

hence all their sides are equal

but all four sides of a rhombus are equal

thus all four triangles are congruent

So AOD = DOC = COB = AOB

Since all four add up to 360°, each must

be 90°

S S S

© T Madas

© T Madas

A

B C

D

E

F

G

In the diagram below ABCD and DEFG are squares.Prove that the triangles ADE and CDG are congruent.

© T Madas

A

B C

D

E

F

G

AD = DC

GD = DE

= +

In the diagram below ABCD and DEFG are squares.Prove that the triangles ADE and CDG are congruent.

© T Madas

A

B C

D

E

F

G

AD = DC

GD = DE

= +

= +

GDC =R GDAR 90+ °

ADE =R GDAR 90+ °

GDC =R ADER

In the diagram below ABCD and DEFG are squares.Prove that the triangles ADE and CDG are congruent.

© T Madas

A

B C

D

E

F

G

AD = DC

GD = DE

GDC =R ADERSAS

ADE and CDG are congruent because 2 sides and the contained angle of ADE are equal to 2 sides and the contained angle of ADE.

In the diagram below ABCD and DEFG are squares.Prove that the triangles ADE and CDG are congruent.

© T Madas

© T Madas

A

B

C D

O

In a circle, centre O, two chords AB and CD are marked, so that AB = CDProve that the chords are equidistant from the centre O

Need to prove OM = ON

If we prove that AOB and COD are congruent then their corresponding heights OM and ON will be equal

AB = CD AO = CO BO = DO

Triangle congruency SSS

OM = ON

M

N

(given)(circle radii)(circle radii)

© T Madas

© T Madas

In the diagram below ABD and BCE are equilateral triangles.Prove that the triangles ABE and DBC are congruent.

A

B

C

D

E

AB = DB [ABD is equilateral]

BC = BE [CBE is equilateral]

θ

60°

60°

DBC = ABE [both angles are 60° + θ ]

SAS

ABE and DBC are congruent because 2 sides and the contained angle of ABE are equal to 2 sides and the contained angle of DBC.

© T Madas