Teach GCSE Maths

Congruent Congruent TrianglesTriangles

Teach GCSE Maths

Congruent Congruent TrianglesTriangles

© Christine Crisp

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Congruent triangles form just one type of congruent shapes.

However, they are very useful so it’s worth looking at them in more detail.

D

E

F

A

B

C

Congruent triangles have 3 pairs of equal sides.

Sides AC and DF are a pair of equal sides

The other pairs of equal sides are AB and DE,and BC and

EF.

Which angle forms a pair with A?

Which with B? And which with C?

D

E

F

A

B

C

ANS: The pairs are A and D, B and E, C and F

There are also 3 pairs of equal angles.

Sides AC and DF are a pair of equal sides

Congruent triangles have 3 pairs of equal sides.

D

E

F

A

B

C

There are also 3 pairs of equal angles.

Sides AC and DF are a pair of equal sides

We use a variety of symbols to mark angles. The same symbol in 2 or more angles means

they are equal.

Congruent triangles have 3 pairs of equal sides.

Equal sides are always opposite equal angles.e.g. If we are told that these triangles are

congruentP Q

S

A

C

B

and QS =

CA ( opposite the 90º angle )BA ( opposite the angle

with the cross )So, PQ = CB ( 3rd sides of the triangles

)

PS =

X

Y

Z

A

B

C

7

cm7

cm

Equal sides are always opposite equal angles.

XY = CA ( opposite )

X = C ( opposite the 7 cm sides)

So, XZ = BC ( 3rd sides )

So, Y = A ( 3rd angles )

e.g. If these are congruent

SUMMARY

Equal sides are always opposite equal angles.

Congruent triangles have 3 pairs of equal sides and 3 pairs of equal angles.

X

Y

Z

A

B

C

Triangle XYZ and CAB are congruent.

Exercise1. If these triangles are congruent,

which sides are equal? Give the reasons.A

BC

XY

Z

2. If these triangles are congruent, which angles are equal? Give the reasons.

A

B C

Y Z

X

6 cm

8 cm

6 cm

8 cm

Ans:

AC = ZY ( opposite )

So, AB = ZX ( 3rd sides )

BC = XY ( opposite )

1. If these triangles are congruent, which sides are equal? Give the reasons.

A

BC

XY

Z

Exercise

2. If these triangles are congruent, which angles are equal? Give the reasons.

A

B C

Y Z

X

C = X ( opposite the 6 cm sides )

6 cm

8 cm

6 cm

8 cm

A = Z ( opposite the 8 cm sides )

B = Y ( 3rd side of triangles )

Ans:

Suppose we put 2 right angled congruent triangles together like this:

We now have an isosceles triangle.We can reverse this, so any isosceles triangle can be made into 2 right angled congruent triangles.

e.g.

Since the triangles are congruent, these lengths . . .are equal.

The line must be drawn from the vertex where the equal sides meet, vertex A.

The perpendicular from A bisects the base.

A

We say that