# Worksheet: Conditions for Similar Triangles

In this worksheet, we will practice identifying similar triangles.

**Q2: **

Given that and are parallel, are triangles and similar? If yes, why?

- A Yes, as the side lengths are equal.
- B no
- C Yes, as all the corresponding angles in each triangle have equal measures.

**Q3: **

The two triangles in the given figure have equal angles. Is this enough to prove that the two triangles are similar?

- A yes
- B no

**Q4: **

Two triangles are similar. What will be true of the measures of the corresponding angles in the two triangles?

- A Only two corresponding angles will be equal.
- B They will be different.
- C If the sides are equal, the angles will be equal.
- D They will be equal.
- EOnly two corresponding angles will be equal.

**Q5: **

Two quadrilaterals have corresponding angles that have equal measures. Can we be sure that the quadrilaterals are similar?

- A no
- B yes

**Q7: **

What does the AA criterion for triangles allow us to prove?

- A If a corresponding side and angle are equal in two triangles, then the two triangles are similar.
- B If the corresponding sides of two triangles are proportional, then the two triangles are similar.
- C If the corresponding sides of two triangles are equal, then the two triangles are congruent.
- D If two corresponding angles in two triangles have equal measures, then they must be similar.
- EIf, in the two triangles, one pair of corresponding sides are proportional and the included angles are equal, then the two triangles are similar.

**Q8: **

The figure shows two triangles: and .

Work out the size of angle .

- A
- B
- C
- D
- E

What does the criterion tell us about these two triangles?

- AAs both triangles share two angles of equal sizes, they must be similar.
- BAs both triangles share one angle of equal size, they must be similar.
- CAs both triangles share two sides of equal sizes, they must be similar.
- DAs both triangles share three angles of equal sizes, they must be similar.
- EAs both triangles share two angles and two sides of equal sizes, they must be similar.

**Q9: **

The figure shows two triangles: and .

Work out the size of angle .

- A
- B
- C
- D
- E

What does the criterion tell us about these two triangles?

- AAs both triangles only share one angle of equal sizes, they are not similar.
- BAs both triangles only share three angles of equal sizes, they are not similar.
- CAs both triangles only share one side of equal sizes, they are not similar.
- DAs both triangles only share two angles of equal sizes, they are not similar.
- EAs both triangles only share two sides of equal sizes, they are not similar.

**Q10: **

The figure shows two triangles: and .

Work out the size of angle .

- A
- B
- C
- D
- E

What does the criterion tell us about these two triangles?

- AAs both triangles share two angles of equal sizes, they must be similar.
- BAs both triangles share one angle of equal size, they must be similar.
- CAs both triangles share two sides of equal sizes, they must be similar.
- DAs both triangles share three angles of equal sizes, they must be similar.
- EAs both triangles share two angles and two sides of equal sizes, they must be similar.

**Q11: **

The figure shows three triangles: , , and .

Are triangles and similar?

- Ayes
- Bno

Justify your answer with one of the following reasons.

- A Triangle can first be rotated clockwise about onto , and then can be dilated from point by a scale factor of three onto ; hence, the triangles are similar.
- B No sequence of translations, reflections, rotations, or dilations exists that can map triangle onto triangle ; therefore, the two triangles cannot be similar.

**Q12: **

In the given figure, is constructed on the triangle parallel to .

What can we conclude about the size of angles and ?

- A
- B
- C
- D
- E

Using the criterion, what can we conclude about triangles and ?

- AThey are similar.
- BThey are equilateral triangles.
- CThey are neither similar nor congruent.
- DThey are congruent.
- EThey are isosceles triangles.

**Q13: **

The figure shows two triangles.

Are the two triangles similar?

- Ayes
- Bno

Why?

- AIf you calculate the measure of the third angle in one of the triangles, you can see that the triangles share two angles; therefore, by the criteria, the triangles are similar.
- BThe triangles do not share the same angles and, hence, are not similar.

**Q14: **

The figure shows a triangle where the line segment is parallel to

Which angle is equivalent to ? Give reasons.

- A , because the angles are alternate.
- B , because the angles are alternate.
- C , because the angles are corresponding.
- D , because the angles are corresponding.
- E , because the angles are corresponding.

Which angle is equivalent to ? Give reasons.

- A , because the angles are corresponding.
- B , because the angles are corresponding.
- C , because the angles are corresponding.
- D , because the angles are alternate.
- E , because the angles are alternate.

Hence, are triangles and similar? If yes, state why?

- AYes, they are similar by the AA criterion.
- BYes, they are similar by the SSS criterion.
- CYes, they are similar by the SAS criterion.
- DNo

**Q15: **

The figure shows two triangles and where the line segment is parallel to

Which angle is equivalent to ? Give reasons.

- A , because the angles are corresponding.
- B , because the angles are corresponding.
- C , because the angles are alternate.
- D , because the angles are alternate.
- E , because the angles are vertically opposite.

Which angle is equivalent to ? Give reasons.

- A , because the angles are alternate.
- B , because the angles are alternate.
- C , because the angles are corresponding.
- D , because the angles are corresponding.
- E , because the angles are vertically opposite.

Hence, are triangles and similar? If yes, state why?

- Ayes, they are similar by the AA criterion.
- Byes, they are similar by the SSS criterion.
- Cyes, they are similar by the SAS criterion.
- Dno

**Q16: **

Triangles and in the given figure are similar. What, if anything, must be true of the lines and ?

- A They are parallel.
- B They are perpendicular.

**Q17: **

Given the following four shapes, which two are similar?

- AShape 2 and Shape 4
- BShape 1 and Shape 2
- CShape 2 and Shape 3
- DShape 1 and Shape 3

**Q18: **

The figure shows two triangles.

Find the size of angle .

- A
- B
- C
- D
- E

Find the size of angle .

- A
- B
- C
- D
- E

The triangles, therefore, share the same angles and are *similar*. What is the fewest number of angles needed to
determine whether two triangles are similar?

- Athree
- Bone
- Ctwo

**Q19: **

The figure shows two triangles.

Find the size of angle .

- A
- B
- C
- D
- E

Find the size of angle .

- A
- B
- C
- D
- E

Are the two triangles similar?

- Ano
- Byes

What is the fewest number of angles needed to determine whether two triangles are similar?

- Atwo
- Bthree
- Cone

**Q20: **

Are the two triangles in the figure similar?

- Ayes
- Bno

**Q21: **

Are these two triangles similar?

- Ano
- Byes