Lesson Worksheet: Similarity of Triangles Mathematics • 8th Grade

In this worksheet, we will practice determining and proving whether two triangles are similar using equality of corresponding angles or proportionality of corresponding sides and practice using similarity to find unknown lengths and angles.

Q1:

In the given figure, 𝐴𝐡 and 𝐷𝐸 are parallel. Using the AA criterion, what can we say about triangles 𝐴𝐡𝐢 and 𝐷𝐸𝐢?

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

Q2:

Given that 𝐴𝐡 and 𝐷𝐢 are parallel, are triangles 𝐸𝐢𝐷 and 𝐸𝐴𝐡 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:

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.

Q6:

The figure shows two triangles: 𝐴𝐡𝐢 and 𝐴𝐡𝐢.

Work out the measure of angle 𝐴𝐡𝐢.

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.

Q7:

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.

Q8:

The figure shows two triangles: 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′.

Work out the measure of angle 𝐴𝐡𝐢.

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.

Q9:

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

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

  • Aπ‘šβˆ π΄π·πΈ=2π‘šβˆ π΄π΅πΆ
  • Bπ‘šβˆ π΄π·πΈ+π‘šβˆ π΄π΅πΆ=90∘
  • Cπ‘šβˆ π΄π·πΈ+π‘šβˆ π΄π΅πΆ=180∘
  • Dπ‘šβˆ π΄π·πΈ=12π‘šβˆ π΄π΅πΆ
  • Eπ‘šβˆ π΄π·πΈ=π‘šβˆ π΄π΅πΆ

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

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

Q10:

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.

Q11:

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

Which angle is equivalent to ∠𝐴𝐡𝐢? Give reasons.

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

Which angle is equivalent to ∠𝐴𝐢𝐡? Give reasons.

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

Hence, are triangles 𝐴𝐡𝐢 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.

Q12:

The figure shows two triangles 𝐴𝐡𝐢 and 𝐷𝐢𝐸 where the line segment 𝐴𝐡 is parallel to 𝐷𝐸.

Which angle is equivalent to ∠𝐴𝐡𝐢? Give reasons

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

Which angle is equivalent to ∠𝐡𝐴𝐢? Give reasons

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

Hence, are triangles 𝐴𝐡𝐢 and 𝐢𝐷𝐸 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.

Q13:

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

  • AThey are parallel.
  • BThey are perpendicular.

Q14:

The figure shows two triangles.

Find the measure of angle 𝐴𝐡𝐢.

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

Q15:

The figure shows two triangles.

Find the measure of angle 𝐴𝐡𝐢.

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

Q16:

Are the two triangles in the figure similar?

  • AYes
  • BNo

Q17:

Are these two triangles similar?

  • AYes
  • BNo

Q18:

In the given figure, triangles 𝐢𝐷𝐸 and 𝐢𝐡𝐴 are similar. What must be true of 𝐷𝐸 and 𝐡𝐴?

  • A𝐷𝐸=13𝐴𝐡
  • B𝐷𝐸=12𝐴𝐡
  • C𝐷𝐸βˆ₯𝐴𝐡
  • Dπ·πΈβŸ‚π΄π΅
  • E𝐷𝐸=2𝐴𝐡

Q19:

The figure shows two triangles: 𝐴𝐡𝐢 and 𝐴𝐡𝐢.

Work out the measure of ∠𝐴𝐡𝐢.

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.

Q20:

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

  • A
  • B
  • C
  • D
  • E

Q21:

In the two triangles shown, π‘šβˆ π·πΉπΈ=30∘ and π‘šβˆ π·πΈπΉ=42∘. What is π‘šβˆ π΄?

Q22:

Which two of these triangles are similar?

  • A(1), (2)
  • B(2), (3)
  • C(1), (4)
  • D(3), (4)

Q23:

Two triangles contain angles with measures of 43∘ and 93∘. Are the two triangles similar? If yes, why?

  • ANo
  • BYes, because they have different angles.
  • CYes, by the AA criterion

Q24:

Which of the following properties is enough to conclude that two triangles are similar?

  • AAll corresponding angles have the same ratio.
  • BThey have the same measure of angles.
  • CTwo corresponding sides have the same ratio.
  • DThey both contain a right angle.
  • EOne corresponding side and one corresponding angle are equal.

Q25:

Fill in the blank: β–³π΄π΅πΆβˆΌβ–³βˆΌβ–³.

  • A𝐷𝐴𝐢, 𝐷𝐡𝐴
  • B𝐷𝐢𝐴, 𝐷𝐴𝐡
  • C𝐴𝐷𝐢, 𝐴𝐷𝐡
  • D𝐢𝐴𝐷, 𝐴𝐡𝐷

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