Worksheet: Conditions for Similar Triangles

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In this worksheet, we will practice identifying similar triangles using AA and SSS criteria.

Q1:

In the given figure, 𝐴𝐵 and 𝐷𝐸 are parallel. Using the 𝐴𝐴 criterion, what can we say about triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐶?

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

Q2:

Given that 𝐴𝐵 and 𝐷𝐶 are parallel, are triangles 𝐸𝐶𝐷 and 𝐸𝐴𝐵 similar? If yes, why?

  • Ano
  • BYes, as the side lengths are equal.
  • CYes, 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?

  • Ayes
  • Bno

Q4:

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

  • AThey will be different.
  • BThey will be equal.
  • COnly one corresponding angles will be equal.
  • DOnly two corresponding angles will be equal.
  • EIf the sides are equal, the angles will be equal.

Q5:

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

  • Ano
  • Byes

Q6:

Triangles 𝐴𝐵𝐶 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?

  • AIf two corresponding angles in two triangles have equal measures, then they must be similar.
  • BIf the corresponding sides of two triangles are proportional, then the two triangles are similar.
  • CIf, in the two triangles, one pair of corresponding sides are proportional and the included angles are equal, then the two triangles are similar.
  • DIf a corresponding side and angle are equal in two triangles, then the two triangles are similar.
  • EIf the corresponding sides of two triangles are equal, then the two triangles are congruent.

Q8:

The figure shows two triangles: 𝐴𝐵𝐶 and 𝐴𝐵𝐶.

Work out the measure of angle 𝐴𝐵𝐶.

What does the 𝐴𝐴 criterion tell us about these two triangles?

  • AAs both triangles share two angles of equal measures, they must be similar.
  • BAs both triangles share two angles and two sides of equal measures, they must be similar.
  • CAs both triangles share one angle of equal measure, they must be similar.
  • DAs both triangles share three angles of equal measures, they must be similar.
  • EAs both triangles share two sides of equal measures, they must be similar.

Q9:

The figure shows two triangles: 𝐸𝐹𝐺 and 𝐻𝐼𝐽.

Work out the measure of angle 𝐹𝐸𝐺.

What does the AA criterion tell us about these two triangles?

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

Q10:

The figure shows two triangles: 𝐴𝐵𝐶 and 𝐴𝐵𝐶.

Work out the measure of angle 𝐴𝐵𝐶.

What does the 𝐴𝐴 criterion tell us about these two triangles?

  • AAs both triangles share one angle of equal measure, they must be similar.
  • BAs both triangles share three angles of equal measures, they must be similar.
  • CAs both triangles share two angles and two sides of equal measures, they must be similar.
  • DAs both triangles share two sides of equal measures, they must be similar.
  • EAs both triangles share two angles of equal measures, they must be similar.

Q11:

The figure shows three triangles: 𝐴𝐵𝐶, 𝐴′𝐵′𝐶′, and 𝐴′′𝐵′′𝐶′′.

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

  • Ano
  • Byes

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐵𝐶 can first be rotated 90∘ 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.
  • BNo sequence of translations, reflections, rotations, or dilations exists that can map triangle 𝐴𝐵𝐶 onto triangle 𝐴′′𝐵′′𝐶′′; therefore, the two triangles cannot be similar.

Q12:

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

What can we conclude about the measure of angles 𝐴𝐷𝐸 and 𝐴𝐵𝐶?

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

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

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

Q13:

The figure shows two triangles.

Are the two triangles similar?

  • Ano
  • Byes

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

Q14:

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

Which angle is equivalent to ∠𝐴𝐵𝐶? Give reasons.

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

Which angle is equivalent to ∠𝐴𝐶𝐵? Give reasons.

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

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

  • AYes, they are similar by the SAS criterion.
  • BNo
  • CYes, they are similar by the AA criterion.
  • DYes, they are similar by the SSS criterion.

Q15:

The figure shows two triangles 𝐴𝐵𝐶 and 𝐷𝐶𝐸 where the line segment 𝐴𝐵 is parallel to 𝐷𝐸

Which angle is equivalent to ∠𝐴𝐵𝐶? Give reasons.

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

Which angle is equivalent to ∠𝐵𝐴𝐶? Give reasons.

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

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

  • Ano
  • Byes, they are similar by the SSS criterion.
  • Cyes, they are similar by the AA criterion.
  • Dyes, they are similar by the SAS criterion.

Q16:

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

  • AThey are parallel.
  • BThey 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 measure of angle 𝐴𝐵𝐶.

Find the measure of angle 𝐸𝐹𝐷.

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?

  • Aone
  • Btwo
  • Cthree

Q19:

The figure shows two triangles.

Find the measure of angle 𝐴𝐵𝐶.

Find the measure of angle 𝐸𝐹𝐷.

Are the two triangles similar?

  • Ano
  • Byes

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

  • Athree
  • Bone
  • Ctwo

Q20:

Are the two triangles in the figure similar?

  • Ayes
  • Bno

Q21:

Are these two triangles similar?

  • Ayes
  • Bno

Q22:

In the given figure, triangles 𝐶𝐷𝐸 and 𝐶𝐵𝐴 are similar. What must be true of 𝐷𝐸 and 𝐵𝐴?

  • A 𝐷 𝐸 ⟂ 𝐴 𝐵
  • B 𝐷 𝐸 = 1 3 𝐴 𝐵
  • C 𝐷 𝐸 ∥ 𝐴 𝐵
  • D 𝐷 𝐸 = 1 2 𝐴 𝐵
  • E 𝐷 𝐸 = 2 𝐴 𝐵

Q23:

The figure shows two triangles: 𝐴𝐵𝐶 and 𝐴𝐵𝐶.

Work out the measure of ∠𝐴𝐵𝐶.

What does the 𝐴𝐴 criterion tell us about these two triangles?

  • AAs both triangles share two angles of equal measures, they must be similar.
  • BAs both triangles share two angles and two sides of equal measures, they must be similar.
  • CAs both triangles share two sides of equal measures, they must be similar.
  • DAs both triangles share three angles of equal measures, they must be similar.
  • EAs both triangles share one angle of equal measure, then they must be similar.

Q24:

Which of the following triangles is similar to the one seen in the given figure?

  • A
  • B
  • C
  • D
  • E

Q25:

The figure shows three triangles: 𝐴𝐵𝐶, 𝐴𝐵𝐶, and 𝐴𝐵𝐶.

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

  • Ano
  • Byes

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐵𝐶 can first be rotated 90∘ 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.
  • BNo sequence of translations, reflections, rotations, or dilations exists that can map triangle 𝐴𝐵𝐶 onto triangle 𝐴′′𝐵′′𝐶′′; therefore, the two triangles cannot be similar.

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