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SIMILAR TRIANGLES. Congruency. Two shapes are congruent if one of the shapes fits exactly on top of the other shape. These three triangles are all congruent. In congruent shapes: corresponding angles are equal corresponding lengths are equal.

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Congruency

Two shapes are congruent if one of the shapes fits exactly on top of the other shape.

These three triangles

are all congruent

  • In congruent shapes:

    • corresponding angles are equal

    • corresponding lengths are equal


To prove that two triangles are congruent you must show that they satisfy one

of the following four sets of conditions:

SSS: three sides are equal

SAS: two sides and the included angle are the same

ASA: two angles and the included side are the same

RHS: right-angled triangle with hypotenuse and one other side the same



Similar shapes they satisfy one

R

These two quadrilaterals are similar.

S

C

D

  • In similar shapes:

    • corresponding angles are equal

    • corresponding sides are in the same ratio

B

A

P

Q

To show that two triangles are similar it is sufficient

to show that just one of the above conditions is satisfied.



Examples they satisfy one

1

The triangles are similar. Find the values of x and y.

(All lengths are in cm.)

5.5

y

x

8

9

Using ratio of corresponding sides:

12


Examples they satisfy one

2

The triangles are similar. Find the values of x and y.

(All lengths are in cm.)

y

7

Turn one of the triangles so that you can see which are the corresponding sides.

x

12

15

8

7

Using ratio of corresponding sides:


Examples they satisfy one

3

Find the values of x and y.

(All lengths are in cm.)

x

6

Separate the two triangles.

9

2.5

2

Using ratio of corresponding sides:

y

x +2.5

8

y

x

6

9


6 they satisfy one

Examples

4

Find the values of x and y.

(All lengths are in cm.)

4

y

x

6

Turn the top triangle so that you can see which are the corresponding sides.

9

Using ratio of corresponding sides:

y

4

6

x

6

9


x they satisfy one

Examples

5

Find the values of x and y.

(All lengths are in cm.)

6

12

y

16

Turn the top triangle so that you can see which are the corresponding sides.

20

Using ratio of corresponding sides:

6

12

x

y

16

20


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