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worksheet

Congruent and similar triangle proofs

(For advanced students only)

Tests for congruent triangles: SSS, SAS, AAS, RHS1 Which test proves that:

a ∆ABC ≡ ∆ADC?

B

A

D

C

________

b ∆QRP ≡ ∆PNQ?

M

R N

QQN 5 PR

P

________

c ∆WZX ≡ ∆YZX?

Z

W

X

Y

________

d ∆PQO ≡ ∆SRO?QP

S R

O

________

2 AC and DB are diameters of a circle with centre O. Prove that ∆ABO ≡ ∆DCO.

D

A

C

BO

3 Prove ∆PSR ≡ ∆PQR and hence find the values of x and y.S Q

R

P

9 cm

2 cm 2 cm

xy8

958

508 508

4 a Prove that ∆KNM ≡ ∆KLM.

K M

N

L

b If NL 5 4 cm and KL 5 3 3 NM, then write the lengths of ML and KN.

__________________________________________

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5 a Prove that ∆DEH ≡ ∆FGH.D

E

H

F

G

b Name the side that has the same length as EH. _____

6 a Prove ∆OZY ≡ ∆OXY.

X

O

Y

Z

b List the three pairs of equal corresponding angles. ______________________________________

c What is special about ∠OYZ?

_________________________________________________________________________________

Tests for similar triangles: SSS, SAS, AA, RHS 7 Which test proves that:

a ∆RST ||| ∆WUV?

R S

T

3

5

1108

WV

U

9

15

1108

____________

b ∆MNP ||| ∆SQR?

M

P N

86

Q

S R

12

9

____________

c ∆HIJ ||| ∆LKM?H

I

J

1824

12

K

M

L6

4 8

____________

d ∆DEH ||| ∆FGH?

DE

FG

H

____________

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8 Prove that the triangles in each pair below are similar, then find x.

a

C

B

A

2.5

6.5

6

D

EF x

1026

x 5 ______

b M

NP

208258

4

x Q R

S

258

1358

20

x 5 ______c

GH

F

608

3

6

S

RT

608x

10

x 5 ______

d

Y

XW

V

U6

921

38.531.5

x

x 5 ______

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Answers1 a SSS b RHS c AAS d SAS2 SAS3 SAS; x 5 9 cm; y° 5 35°4 a AAS b ML 5 2 cm; KN 5 6 cm5 a AAS b GH6 a SSS b ∠Z 5 ∠X; ∠ZOY 5 ∠XOY; ∠ZYO 5 ∠XYO c It is a right angle (90°)7 a SAS b RHS c SSS d AA8 a RHS; x 5 24 b AA; x 5 8 c AA; x 5 5 d SAS; x 5 11

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