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Worksheet: Similar Triangles

Q1:

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

  • 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:

In the given figure, triangles 𝐢 𝐷 𝐸 and 𝐢 𝐡 𝐴 are similar. What must be true of 𝐷 𝐸 and 𝐡 𝐴 ?

  • A 𝐷 𝐸 = 2 𝐴 𝐡
  • B 𝐷 𝐸 = 1 2 𝐴 𝐡
  • C 𝐷 𝐸 βŸ‚ 𝐴 𝐡
  • D 𝐷 𝐸 βˆ₯ 𝐴 𝐡
  • E 𝐷 𝐸 = 1 3 𝐴 𝐡

Q3:

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.

Q4:

Triangles 𝐴 𝐷 𝐸 and 𝐴 𝐡 𝐢 in the given figure are similar. What, if anything, must be true of the lines βƒ–     βƒ— 𝐷 𝐸 and βƒ–     βƒ— 𝐡 𝐢 ?

  • A They are parallel.
  • B They are perpendicular.

Q5:

The figure shows three triangles: 𝐴 𝐡 𝐢 , 𝐴 β€² 𝐡 β€² 𝐢 β€² , and 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² .

Are triangles 𝐴 𝐡 𝐢 and 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² similar?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • A Triangle 𝐴 𝐡 𝐢 can first be rotated 9 0 ∘ clockwise about 𝐷 onto 𝐴 β€² 𝐡 β€² 𝐢 β€² , and then 𝐴 β€² 𝐡 β€² 𝐢 β€² can be dilated from point 𝐷 by a scale factor of three onto 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² ; hence, the triangles are similar.
  • B No sequence of translations, reflections, rotations, or dilations exists that can map triangle 𝐴 𝐡 𝐢 onto triangle 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² ; therefore, the two triangles cannot be similar.

Q6:

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

Q7:

The figure shows two triangles.

Find the measure of angle 𝐴 𝐡 𝐢 .

Find the measure of angle 𝐸 𝐹 𝐷 .

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

Q8:

The figure shows two triangles 𝐴 𝐡 𝐢 and 𝐷 𝐢 𝐸 where the line segment 𝐴 𝐡 is parallel to 𝐷 𝐸

Which angle is equivalent to ∠ 𝐴 𝐡 𝐢 ? Give reasons.

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

Which angle is equivalent to ∠ 𝐡 𝐴 𝐢 ? Give reasons.

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

Hence, are triangles 𝐴 𝐡 𝐢 and 𝐢 𝐷 𝐸 similar? If yes, state why?

  • Ayes, they are similar by the 𝐴 𝐴 criterion.
  • Byes, they are similar by the 𝑆 𝑆 𝑆 criterion.
  • Cyes, they are similar by the 𝑆 𝐴 𝑆 criterion.
  • Dno

Q9:

The figure shows two triangles: 𝐴 𝐡 𝐢 and 𝐴 𝐡 𝐢 β€² β€² β€² .

Work out the measure of angle 𝐴 𝐡 𝐢 .

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

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

Q10:

In the given figure, 𝐷 𝐸 is constructed on the triangle 𝐴 𝐡 𝐢 parallel to 𝐡 𝐢 .

What can we conclude about the measure of angles 𝐴 𝐷 𝐸 and 𝐴 𝐡 𝐢 ?

  • A π‘š ∠ 𝐴 𝐷 𝐸 + π‘š ∠ 𝐴 𝐡 𝐢 = 1 8 0 ∘
  • B π‘š ∠ 𝐴 𝐷 𝐸 = 1 2 π‘š ∠ 𝐴 𝐡 𝐢
  • C π‘š ∠ 𝐴 𝐷 𝐸 + π‘š ∠ 𝐴 𝐡 𝐢 = 9 0 ∘
  • D π‘š ∠ 𝐴 𝐷 𝐸 = π‘š ∠ 𝐴 𝐡 𝐢
  • E π‘š ∠ 𝐴 𝐷 𝐸 = 2 π‘š ∠ 𝐴 𝐡 𝐢

Using the 𝐴 𝐴 criterion, what can we conclude about triangles 𝐴 𝐷 𝐸 and 𝐴 𝐡 𝐢 ?

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

Q11:

The figure shows two triangles: 𝐴 𝐡 𝐢 and 𝐴 𝐡 𝐢 β€² β€² β€² .

Work out the measure of angle 𝐴 𝐡 𝐢 .

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

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

Q12:

In the two triangles shown, and . What is ?

  • A
  • B
  • C
  • D

Q13:

The figure shows two triangles: 𝐸 𝐹 𝐺 and 𝐻 𝐼 𝐽 .

Work out the measure of angle 𝐹 𝐸 𝐺 .

What does the 𝐴 𝐴 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 three angles of equal measures, they are not similar.
  • CAs both triangles only share one side 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 two sides of equal measures, they are not similar.

Q14:

The figure shows two triangles.

Find the measure of angle 𝐴 𝐡 𝐢 .

Find the measure of angle 𝐸 𝐹 𝐷 .

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

Q15:

Are the two triangles in the figure similar?

  • Ayes
  • Bno

Q16:

Are these two triangles similar?

  • Ano
  • Byes

Q17:

The figure shows a triangle 𝐴 𝐷 𝐸 where the line segment 𝐡 𝐢 is parallel to 𝐷 𝐸

Which angle is equivalent to ∠ 𝐴 𝐡 𝐢 ? Give reasons.

  • A ∠ 𝐴 𝐢 𝐡 , 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 𝐢 𝐡 , because the angles are corresponding.

Which angle is equivalent to ∠ 𝐴 𝐢 𝐡 ? Give reasons.

  • A ∠ 𝐴 𝐸 𝐷 , because the angles are corresponding.
  • B ∠ 𝐴 𝐷 𝐸 , because the angles are corresponding.
  • C ∠ 𝐴 𝐡 𝐢 , 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 𝐡 𝐢 and 𝐴 𝐷 𝐸 similar? If yes, state why?

  • Ayes, they are similar by the 𝐴 𝐴 criterion.
  • Byes, they are similar by the 𝑆 𝑆 𝑆 criterion.
  • Cyes, they are similar by the 𝑆 𝐴 𝑆 criterion.
  • Dno

Q18:

Are the two triangles in the figure similar?

  • Ano
  • Byes

Q19:

What does the 𝐴 𝐴 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.

Q20:

The given figure shows a right triangle 𝐴 𝐡 𝐢 , where 𝐢 𝐷 is perpendicular to 𝐴 𝐡 .

Using similarity, express π‘Ž  in terms of 𝑐 and 𝑑 .

  • A π‘Ž = ( 𝑐 βˆ’ 𝑑 )  
  • B π‘Ž = 𝑐 βˆ’ 𝑑   
  • C π‘Ž = 𝑐 𝑑 
  • D π‘Ž = 𝑐 𝑑 
  • E π‘Ž = ( 𝑐 𝑑 )  

Using similarity, express 𝑏  in terms of 𝑐 and 𝑒 .

  • A 𝑏 = 𝑐 𝑒 
  • B 𝑏 = 𝑐 βˆ’ 𝑒   
  • C 𝑏 = 𝑐 𝑒 
  • D 𝑏 = ( 𝑐 βˆ’ 𝑒 )  
  • E 𝑏 = ( 𝑐 𝑒 )  

Express the sum of π‘Ž  and 𝑏  in terms of 𝑐 .

  • A π‘Ž + 𝑏 = 𝑐 2   
  • B π‘Ž + 𝑏 = 𝑐  
  • C π‘Ž + 𝑏 = 𝑐   
  • D π‘Ž + 𝑏 = 2 𝑐  
  • E π‘Ž + 𝑏 = 2 𝑐   

Q21:

Given that β–³ 𝐽 𝐾 𝐿 ∼ β–³ π‘Š π‘Œ 𝑍 , π‘š ∠ 𝐾 = ( 6 𝑦 βˆ’ 9 5 ) ∘ , and π‘š ∠ 𝐽 = ( 6 π‘₯ + 1 5 ) ∘ , find the values of π‘₯ and 𝑦 .

  • A π‘₯ = 4 6 , 𝑦 = 1 3 3
  • B π‘₯ = 1 1 , 𝑦 = 2 2 . 2
  • C π‘₯ = 4 6 , 𝑦 = 2 6
  • D π‘₯ = 1 1 , 𝑦 = 2 6
  • E π‘₯ = 7 . 7 , 𝑦 = 2 2 . 2