Sec 2 Maths Congruence and similarity worksheet with answers

A free Sec 2 Maths worksheet on congruence and similarity: 8 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 2 Maths worksheet: Congruence and similarity

Total: 30 marksSuggested time: 45 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

A lamp post is to be placed so that it is the same distance from the gates at

D

and

E

. On which line must it stand?

Answer[1]

In the diagram, the parts marked on triangles

ABC

and

PQR

are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Answer[2]

On the squared paper, shape

B

is an enlargement of shape

A

. Find the scale factor of the enlargement that maps shape

B

onto shape

A

Answer[2]

In the diagram, the bisector of

\angle ABC

has been constructed with compasses, and

Q

is a point on it.

QM

is perpendicular to

BA

and

QN

is perpendicular to

BC

. Find

x

Answer°[2]

In the diagram,

D

lies on

AB

and

E

lies on

AC

, with

DE

parallel to

BC

. Find

DE

Answercm[3]

Section B

(a)In the diagram, the parts marked on triangles

ABC

and

PQR

are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Answer (a)[2]

(b)In the diagram,

D

lies on

AB

and

E

lies on

AC

, with

DE

parallel to

BC

. Find

DE

, correct to 3 significant figures.

Answer (b)cm[3]

(a)

A

is the point

(4, 2)

and

B

is the point

(10, 2)

. The point

P

lies on the perpendicular bisector of

AB

and on the line

y = 7

. Find the

x

-coordinate of

P

Answer (a)[2]

(b)In the diagram, the parts marked on triangles

ABC

and

PQR

are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Answer (b)[2]

(c)In the diagram,

D

lies on

AB

and

E

lies on

AC

, with

DE

parallel to

BC

. Find

DE

Answer (c)cm[3]

(a)In triangles

ABC

and

PQR

\angle ABC = \angle PQR

and

\angle BCA = \angle QRP

. State one more condition that would make triangle

ABC

congruent to triangle

PQR

by the ASA test. Choose the condition.

Answer (a)[1]

(b)In the diagram, triangle

ABC

is similar to triangle

PRQ

. Find

x

Answer (b)cm[2]

(c)

A

is the point

(1, 5)

and

B

is the point

(9, 5)

. The point

P

lies on the perpendicular bisector of

AB

and on the line

y = 1

. Find the

x

-coordinate of

P

Answer (c)[2]

(d)In the diagram,

B

E

C

and

F

lie on a straight line, with

BE = CF

AB = DE

and

AC = DF

. Which statement explains why

AC

is parallel to

DF

Answer (d)[3]

END OF PAPER

Answers and worked solutions

Question 1

Answer: ⊥ bisector[1]

The same distance from two points,

D

and

E

.Points the same distance from

D

and

E

lie on the perpendicular bisector of

DE

Question 2

Answer: ASA[2]

BC = RP

(both

8

cm)

\angle CBA = \angle PRQ

(both

38^\circ

)

\angle ACB = \angle QPR

(both

59^\circ

)Two angles and the side between them are equal: ASA (or AAS).

Question 3

Answer: Scale factor

= \dfrac{2}{3}

(less than 1: the image is smaller)[2]

Bottom side of shape

B

3

units; of shape

A

2

unitsScale factor

= \dfrac{2}{3}

Scale factor

= \dfrac{2}{3}

(less than 1: the image is smaller)

Question 4

Answer:

x = 69

(∠ sum of △) °[2]

\angle QBN = \angle ABQ

\angle QBN = 21^\circ

\angle QNB = 90^\circ

x = 180^\circ - 90^\circ - 21^\circ

x = 69

(∠ sum of △)

Question 5

Answer:

DE = 6.75

cm[3]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

\mathstrut AB = 6 + 2

AB = 8

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\mathstrut \dfrac{DE}{9} = \dfrac{6}{8}

DE = \dfrac{6}{8} \times 9

DE = 6.75

Question 6

(a)Answer: AAS[2]

\angle BAC = \angle RQP

(both

57^\circ

)

\angle CBA = \angle PRQ

(both

44^\circ

)

BC = RP

(both

6

cm)Two angles and a side not between them are equal: AAS (or ASA).

(b)Answer:

\phantom{DE} = 5.82

cm (3 s.f.)[3]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

\mathstrut AB = 8 + 3

AB = 11

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\mathstrut \dfrac{DE}{8} = \dfrac{8}{11}

DE = \dfrac{8}{11} \times 8

DE = 5.8181\ldots

\phantom{DE} = 5.82

cm (3 s.f.)

Question 7

(a)Answer: The

x

-coordinate of

P

7

.[2]

\mathstrut x = \dfrac{4 + 10}{2}

x = 7

The

x

-coordinate of

P

7

(b)Answer: RHS[2]

\angle ACB = \angle PQR

(both

90^\circ

)

AB = PR

(both

8

cm)

BC = RQ

(both

5

cm)A right angle, the hypotenuse and one other side are equal: RHS.

(c)Answer:

DE = 6.4

cm[3]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

\mathstrut AB = 8 + 7

AB = 15

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\mathstrut \dfrac{DE}{12} = \dfrac{8}{15}

DE = \dfrac{8}{15} \times 12

DE = 6.4

Question 8

(a)Answer: R[1]

ASA needs two pairs of equal angles and the sides between them.With

BC = QR

, the triangles are congruent by ASA.

(b)Answer:

x = 10

cm[2]

CA

matches

QP

and

AB

matches

PR

\dfrac{AB}{PR} = \dfrac{CA}{QP}

\mathstrut \dfrac{x}{15} = \dfrac{7}{10.5}

x = \dfrac{7}{10.5} \times 15

x = 10

(c)Answer: The

x

-coordinate of

P

5

.[2]

\mathstrut x = \dfrac{1 + 9}{2}

x = 5

The

x

-coordinate of

P

5

(d)Answer: ok[3]

AB = DE

(given),

AC = DF

(given),

BC = BE + EC

BC = CF + EC

BC = EF

\triangle ABC \equiv \triangle DEF

(SSS)

\angle ACB = \angle DFE

They are equal corresponding angles, so

AC \parallel DF

Practice questions by skill

Sec 2 Maths Congruence and similarity worksheet with answers

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Congruence and similarity notes. Or practise it online with step-by-step help when you get one wrong.

Section B

(a)In the diagram, the parts marked on triangles

ABC

and

PQR

are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Answer (a)[2]

(b)In the diagram,

D

lies on

AB

and

E

lies on

AC

, with

DE

parallel to

BC

. Find

DE

, correct to 3 significant figures.

Answer (b)cm[3]

Answers and worked solutions

Question 1

Answer: ⊥ bisector[1]

The same distance from two points,

D

and

E

.Points the same distance from

D

and

E

lie on the perpendicular bisector of

DE

Question 2

Answer: ASA[2]

BC = RP

(both

8

cm)

\angle CBA = \angle PRQ

(both

38^\circ

)

\angle ACB = \angle QPR

(both

59^\circ

)Two angles and the side between them are equal: ASA (or AAS).

Question 3

Answer: Scale factor

= \dfrac{2}{3}

(less than 1: the image is smaller)[2]

Bottom side of shape

B

3

units; of shape

A

2

unitsScale factor

= \dfrac{2}{3}

Scale factor

= \dfrac{2}{3}

(less than 1: the image is smaller)

Question 4

Answer:

x = 69

(∠ sum of △) °[2]

\angle QBN = \angle ABQ

\angle QBN = 21^\circ

\angle QNB = 90^\circ

x = 180^\circ - 90^\circ - 21^\circ

x = 69

(∠ sum of △)

Question 5

Answer:

DE = 6.75

cm[3]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

\mathstrut AB = 6 + 2

AB = 8

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\mathstrut \dfrac{DE}{9} = \dfrac{6}{8}

DE = \dfrac{6}{8} \times 9

DE = 6.75

Question 6

(a)Answer: AAS[2]

\angle BAC = \angle RQP

(both

57^\circ

)

\angle CBA = \angle PRQ

(both

44^\circ

)

BC = RP

(both

6

cm)Two angles and a side not between them are equal: AAS (or ASA).

(b)Answer:

\phantom{DE} = 5.82

cm (3 s.f.)[3]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

\mathstrut AB = 8 + 3

AB = 11

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\mathstrut \dfrac{DE}{8} = \dfrac{8}{11}

DE = \dfrac{8}{11} \times 8

DE = 5.8181\ldots

\phantom{DE} = 5.82

cm (3 s.f.)

Question 7

(a)Answer: The

x

-coordinate of

P

7

.[2]

\mathstrut x = \dfrac{4 + 10}{2}

x = 7

The

x

-coordinate of

P

7

(b)Answer: RHS[2]

\angle ACB = \angle PQR

(both

90^\circ

)

AB = PR

(both

8

cm)

BC = RQ

(both

5

cm)A right angle, the hypotenuse and one other side are equal: RHS.

(c)Answer:

DE = 6.4

cm[3]

\angle ADE = \angle ABC

(corresponding angles)

\angle A

is common, so the triangles are similar (AA).

\mathstrut AB = 8 + 7

AB = 15

\dfrac{DE}{BC} = \dfrac{AD}{AB}

\mathstrut \dfrac{DE}{12} = \dfrac{8}{15}

DE = \dfrac{8}{15} \times 12

DE = 6.4

Question 8

(a)Answer: R[1]

ASA needs two pairs of equal angles and the sides between them.With

BC = QR

, the triangles are congruent by ASA.

(b)Answer:

x = 10

cm[2]

CA

matches

QP

and

AB

matches

PR

\dfrac{AB}{PR} = \dfrac{CA}{QP}

\mathstrut \dfrac{x}{15} = \dfrac{7}{10.5}

x = \dfrac{7}{10.5} \times 15

x = 10

(c)Answer: The

x

-coordinate of

P

5

.[2]

\mathstrut x = \dfrac{1 + 9}{2}

x = 5

The

x

-coordinate of

P

5

(d)Answer: ok[3]

AB = DE

(given),

AC = DF

(given),

BC = BE + EC

BC = CF + EC

BC = EF

\triangle ABC \equiv \triangle DEF

(SSS)

\angle ACB = \angle DFE

They are equal corresponding angles, so

AC \parallel DF