similarity and congruence - triangles

1. Which of these triangles are similar?

a.

b. Give a suitable reason for your choice.

2. Consider the attached figure.What is the value of x?

a.

b. Give a reason for your answer

3.

Consider the attached figure.

What is the value of x?

A B

C D

All sides are in the same ratio.A All the corresponding sides are equal.B

All matching angles are equal.C

We require more information.A 11B

44C 22D

All sides in similar shapes must have the same ratioA

Base angles in isosceles triangles must be equal.B

All matching angles in similar shapes must be equalC

a.

b. Give a reason for your answer


a. Which of these triangles are similar?

b. Give a suitable reason for their similarity.


We require more information.A 30B

60C 15D

Base angles in isosceles triangles must be equal.A

All matching angles in similar shapes must be equalB

All sides in similar shapes must have the same ratioC

A B

C D

All sides are in the same ratioA

All matching angles are equalB

2 angles are equal and one side is a multiple of the corresponding side of the other.C

a.

b. Give the reason for their similarity.

6. The pair of triangles in the diagram have side lengths as labelled.What is the minimal condition needed to identify if these are similar triangles?

7.

The triangles in the diagram have angles as marked. What is the minimal condition needed to identify if these are similar triangles?

A B

C D

All matching sides are in the same ratioA

All matching angles are equal.B

Two sides are in the same ratio and the included angles are equalC

two pairs of corresponding anglesA nothing more, this is sufficientB

one pair ofincluded corresponding anglesC a third pair of corresponding side lengthsD

nothing more, this is sufficientA two pairs of corresponding sidesB

two corresponding side lengthsC another pair of corresponding anglesD

8. Select the two triangles that are similar.

9. Given that all three angles in one triangles match all three angles in another triangle, can you be sure these triangles are similar?

10. Consider the two similar triangles.

a. DE corresponds to which side in LMN?

b. DCcorresponds to which side in LMN?

A B

C

NoA YesB

LNA MNB

MLC

MLA MNB

LNC

11. The pair of triangles in the diagram already have one pair of angles identified as being equal.What is the minimal condition needed to identify if these are similartriangles?

12. The two triangles in the diagram are similar.

Which angle inJIKis equal and corresponding to FEG?

13.

The two given triangles are similar.

a. Comparing the longer side length to the shorter side length, state the ratio of corresponding sides. Expressyour answer in the form a:b or as a fraction.

b. Find the value of each pronumeral:s=n=m=

two corresponding side lengthsA two more pairs of corresponding anglesB

one corresponding side lengthC another pair of corresponding anglesD

JIKA IJKB

JKIC

14. In congruent triangles, what is the ratio of corresponding sides?

15. The two given triangles are similar.

a. Determine the enlargementfactor.

b. Determine the value of the pronumeral.


a. Whatenlargement factor is applied to each side of the smaller triangle?

b. Solve forx.


a. Determine the enlargement factor, stating your answer in simplest fraction form.

b. Solve for the value ofa.

18. Consider the diagram below.

a. Why is ABC similar to AED?

b. Use the correct mathematical symbols to state that the two triangles are similar.

19.

There are two common sides and a common angle.A

Two pairs of angles are labelled as equal and there is a common angle.B

Corresponding sides are in the same ratio.C

In the diagram, JKMN.

a. Write a statement in which you identify the angle that is equal toLJK. Give a reason as to why they are equal.

b. Write a statement in which you identify the angle that is equal toLKJ. Give a reason as to why they are equal.

c. Use the correct mathematical symbols to state that the two triangles are similar.

20. In the diagram, JIKLIM.

a. Write a geometric statement in which you identify theangle that is equal to IKJ. Give a reason.

b. Write a geometric statement in which you identify theangle that is equal to IJK. Give a reason.

c. Complete the statement:ILIJ=JK

21.

Consider the diagram.

a. Why isBEparallel to CD?

b. Hence form a statement in which you identify the angle that is equal toBEA.

c. Why is ABEACD?

d. State the enlargement factor going from the smaller triangle to the larger triangle.

e. Find the value of the pronumeral.

ABE and ACD form a pair of supplementary cointerior angles with respect to BE and CD.A

ABE and ACD form a pair of equal corresponding angles with respect to BE and CD.B

ABE and ACD form a pair of equal alternate angles with respect to BE and CD.C

All three pairs of corresponding angles are equal.A

Two pairs of sides are in the same ratio and the included angle is equal.B

22. Prove that these two triangles are similar:

In ABC and DFE we have:

23. Match the sides and find the ratio between each pairs of sides. Use this to say whether triangles ABC and DFE are similar.

In ABC and DFE we have:

24.

Prove that ABC and EGF are similar.

In ABC and EGF we have:

25. Consider ABC and PQR.

a. Prove that ABC is similar to PQR:

b. Deduce the value of x.

c. Deduce the value ofy

26. LMN and LKJ are drawn such that JKMN.

Prove that LMN and LKJ are similar.

In the two triangles LMNand LKJ we have:

27. Prove the following triangles are similar.

a. In AOB and DOC we have:

b. Hence show that ABCD.

28. A large tree casts a shadow of 50m. At the same time, a 4m high stick, standing vertically, casts a shadow of 5m.

a. Show that ABE and DCE are similar.

b. If the height of the tree is hmetres, find h giving reasons.

29. In the diagram, QRST.

a. Show that PQR is similar to PST.

b. Find the scale factor of enlargement.

c. Deduce the value of .

30.

In the diagram, AB= 3, BC= 7.5 and BE= 1.

a. Prove that the ABE and ACD are similar.In the two triangles ABE and ACD we have:

b. Solve for the value of 3.5.

31. In the diagram, ABC is a right angledtriangle with the right angle at C. The midpoint of AB is M and MP is perpendicular to AC:

a. Prove that AMP is similar to ABC.

In the two triangles AMP and ABC we have:

b. What is the ratio of AP to AC?

32. Consider the figure:

a. Prove that is similar to .In the two triangles and we have:

b. Prove that is similar to .In the two triangles and we have:

c. Can we conclude that is similar to ?

NoA YesB

similarity and congruence - triangles

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