Worksheet: Conditions for Similar Triangles

Practice

In this worksheet, we will practice identifying similar triangles.

Q1:

In the given figure, 𝐴 𝐵 and 𝐷 𝐸 are parallel. Using the 𝐴 𝐴 criterion, what can we say about triangles 𝐴 𝐵 𝐶 and 𝐷 𝐸 𝐶 ?

  • A They are neither similar nor congruent.
  • B They are congruent.
  • C They are equilateral triangles.
  • D They are similar.
  • EThey are isosceles triangles.

Q2:

Given that 𝐴 𝐵 and 𝐷 𝐶 are parallel, are triangles 𝐸 𝐶 𝐷 and 𝐸 𝐴 𝐵 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

Q6:

Triangles 𝐴 𝐵 𝐶 and 𝐴 ′ 𝐵 ′ 𝐶 ′ in the given figure are similar. Work out the value of 𝑥 .

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: 𝐴 𝐵 𝐶 , 𝐴 ′ 𝐵 ′ 𝐶 ′ , and 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ .

Are triangles 𝐴 𝐵 𝐶 and 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ similar?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • A Triangle 𝐴 𝐵 𝐶 can first be rotated 9 0 ∘ clockwise about 𝐷 onto 𝐴 ′ 𝐵 ′ 𝐶 ′ , and then 𝐴 ′ 𝐵 ′ 𝐶 ′ can be dilated from point 𝐷 by a scale factor of three onto 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ ; hence, the triangles are similar.
  • B No sequence of translations, reflections, rotations, or dilations exists that can map triangle 𝐴 𝐵 𝐶 onto triangle 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ ; therefore, the two triangles cannot be similar.

Q12:

In the given figure, 𝐷 𝐸 is constructed on the triangle 𝐴 𝐵 𝐶 parallel to 𝐵 𝐶 .

What can we conclude about the size of angles 𝐴 𝐷 𝐸 and 𝐴 𝐵 𝐶 ?

  • A 𝑚 ∠ 𝐴 𝐷 𝐸 + 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 8 0 ∘
  • B 𝑚 ∠ 𝐴 𝐷 𝐸 = 1 2 𝑚 ∠ 𝐴 𝐵 𝐶
  • C 𝑚 ∠ 𝐴 𝐷 𝐸 + 𝑚 ∠ 𝐴 𝐵 𝐶 = 9 0 ∘
  • D 𝑚 ∠ 𝐴 𝐷 𝐸 = 𝑚 ∠ 𝐴 𝐵 𝐶
  • E 𝑚 ∠ 𝐴 𝐷 𝐸 = 2 𝑚 ∠ 𝐴 𝐵 𝐶

Using the 𝐴 𝐴 criterion, what can we conclude about triangles 𝐴 𝐷 𝐸 and 𝐴 𝐵 𝐶 ?

  • 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 𝐴 𝐴 criteria, the triangles are similar.
  • BThe triangles do not share the same angles and, hence, are not similar.

Q14:

The figure shows a triangle 𝐴 𝐷 𝐸 where the line segment 𝐵 𝐶 is parallel to 𝐷 𝐸

Which angle is equivalent to ∠ 𝐴 𝐵 𝐶 ? Give reasons.

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

Which angle is equivalent to ∠ 𝐴 𝐶 𝐵 ? Give reasons.

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

Hence, are triangles 𝐴 𝐵 𝐶 and 𝐴 𝐷 𝐸 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 𝐴 𝐵 𝐶 and 𝐷 𝐶 𝐸 where the line segment 𝐴 𝐵 is parallel to 𝐷 𝐸

Which angle is equivalent to ∠ 𝐴 𝐵 𝐶 ? Give reasons.

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

Which angle is equivalent to ∠ 𝐵 𝐴 𝐶 ? Give reasons.

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

Hence, are triangles 𝐴 𝐵 𝐶 and 𝐶 𝐷 𝐸 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 𝐴 𝐷 𝐸 and 𝐴 𝐵 𝐶 in the given figure are similar. What, if anything, must be true of the lines ⃖ ⃗ 𝐷 𝐸 and ⃖ ⃗ 𝐵 𝐶 ?

  • 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

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