Worksheet: Triangle Congruence through Rigid Transformations

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In this worksheet, we will practice identifying congruent triangles using transformations.

Q1:

A triangle 𝐴 𝐵 𝐶 has been dilated from a center 𝑃 by a scale factor of 3 to triangle 𝐴 𝐵 𝐶 ′ ′ ′ .

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

  • Ayes
  • Bno

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

  • Ano
  • Byes

Q2:

Determine, by applying transformations, whether the two triangles seen in the given figure are congruent.

  • A They are congruent.
  • B They are not congruent.

Q3:

If there exists a combination of rotations, reflections, and translations that would map one shape to another, would the two shapes be congruent?

  • Ayes
  • Bno

Q4:

If triangle 𝐴 is mapped by a reflection in the line 𝑦 = 𝑥 to triangle 𝐴 ′ , would the two triangles be congruent?

  • A yes
  • B no

Q5:

If triangle 𝐵 is mapped by a 1 8 0 ∘ rotation about the origin to triangle 𝐵 ′ , would the two triangles be congruent?

  • A yes
  • B no

Q6:

A triangle 𝐴 𝐵 𝐶 is rotated by 1 8 0 ∘ about the origin to triangle 𝐴 𝐵 𝐶 ′ ′ ′ .

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

  • Ayes
  • Bno

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

  • Ayes
  • Bno

Q7:

The triangle 𝐴 𝐵 𝐶 has been transformed onto triangle 𝐴 𝐵 𝐶 which has then been transformed onto triangle 𝐴 𝐵 𝐶 as seen in the figure.

Describe the single transformation that would map 𝐴 𝐵 𝐶 onto 𝐴 ′ 𝐵 ′ 𝐶 ′ .

  • Aa translation two left and three up
  • Ba translation three right and two down
  • Ca translation two right and three up
  • Da translation two right and three down
  • Ea translation two left and three down

Describe the single transformation that would map 𝐴 ′ 𝐵 ′ 𝐶 ′ onto 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ .

  • Aa reflection in the line ⃖ ⃗ 𝐸 𝐹
  • Ba rotation of 9 0 ∘ clockwise about point 𝐸
  • Ca translation one right and four down
  • Da translation four right and one down
  • Ea rotation of 9 0 ∘ counterclockwise about point 𝐹

Hence, are triangles 𝐴 𝐵 𝐶 and 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ congruent?

  • Ano
  • Byes

Q8:

The triangle 𝐴 𝐵 𝐶 has been transformed onto triangle 𝐴 𝐵 𝐶 which has then been transformed onto triangle 𝐴 𝐵 𝐶 as seen in the figure.

Describe the single transformation that would map 𝐴 𝐵 𝐶 onto 𝐴 ′ 𝐵 ′ 𝐶 ′ .

  • Aa translation three left and two down
  • Ba translation two right and three up
  • Ca translation three left and two up
  • Da translation three right and two up
  • Ea translation three right and two down

Describe the single transformation that would map 𝐴 ′ 𝐵 ′ 𝐶 ′ onto 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ .

  • Aa reflection in the line ⃖ ⃗ 𝐸 𝐹
  • Ba translation four right
  • Ca rotation of 9 0 ∘ clockwise about 𝐹
  • Da rotation of 9 0 ∘ counterclockwise about 𝐸
  • Ea rotation of 9 0 ∘ counterclockwise about 𝐹

Hence, are triangles 𝐴 𝐵 𝐶 and 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ congruent?

  • Ano
  • Byes

Q9:

The figure shows triangles 𝐴 𝐵 𝐶 and 𝐷 𝐸 𝐹 .

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ATriangle 𝐴 𝐵 𝐶 can be rotated to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.
  • BTriangle 𝐴 𝐵 𝐶 can be reflected to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.
  • CNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴 𝐵 𝐶 onto triangle 𝐷 𝐸 𝐹 and, therefore, the two triangles cannot be congruent.
  • DWe can apply a two-stage transformation on triangle 𝐴 𝐵 𝐶 involving a reflection and then a rotation to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.

Q10:

The figure shows triangles 𝐴 𝐵 𝐶 and 𝐷 𝐸 𝐹 .

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ATriangle 𝐴 𝐵 𝐶 can be rotated to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.
  • BNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴 𝐵 𝐶 onto triangle 𝐷 𝐸 𝐹 and, therefore, the two triangles cannot be congruent.
  • CWe can apply a two-stage transformation on triangle 𝐴 𝐵 𝐶 involving a reflection and then a rotation to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.

Q11:

If triangle 𝑇 is mapped to triangle 𝑇 ′ by a reflection, translation, or rotation, which of the following statements will be true of the two triangles?

  • AThey are the same.
  • BThey will be similar.
  • CThey have exactly one side of the same length.
  • DThey will be congruent.

Q12:

Triangle 𝐴 𝐵 𝐶 has been reflected in the line 𝐿 to obtain triangle 𝐴 𝐵 𝐶 ′ ′ ′ as seen in the given figure.

Are the corresponding angles and sides of the two triangles equal?

  • Ayes
  • Bno

What is the length of 𝐵 𝐶 ?

What is the length of 𝐴 ′ 𝐵 ′ ?

What is the perimeter of triangle 𝐴 𝐵 𝐶 ?

Q13:

The triangle 𝐴 𝐵 𝐶 has been transformed onto triangle 𝐴 𝐵 𝐶 which has then been transformed onto triangle 𝐴 𝐵 𝐶 as seen in the figure.

Describe the single transformation that would map 𝐴 𝐵 𝐶 onto 𝐴 𝐵 𝐶 .

  • Aa 9 0 ∘ clockwise rotation about point 𝐷
  • Ba 1 8 0 ∘ rotation about point 𝐸
  • Ca 9 0 ∘ counterclockwise rotation about point 𝐷
  • Da 1 8 0 ∘ rotation about point 𝐷
  • Ea 9 0 ∘ clockwise rotation about point 𝐸

Describe the single transformation that would map 𝐴 𝐵 𝐶 onto 𝐴 𝐵 𝐶 .

  • Aa reflection in the line ⃖ ⃗ 𝐷 𝐸
  • Ba translation three left and three down
  • Ca 9 0 ∘ counterclockwise rotation about point 𝐸
  • Da 9 0 ∘ clockwise rotation about point 𝐷
  • Ea 9 0 ∘ clockwise rotation about point 𝐷

Hence, are triangles 𝐴 𝐵 𝐶 and 𝐴 𝐵 𝐶 congruent?

  • Ano
  • Byes

Q14:

The figure shows two triangles and .

Work out the size of angle .

  • A
  • B
  • C
  • D
  • E

Work out the size of angle .

  • A
  • B
  • C
  • D
  • E

What do you notice about the sizes of the angles in both shapes?

  • AThey are equal.
  • BThe sizes of the angles in triangle are double the sizes of the angles in triangle .
  • CThe sizes of the angles in triangle are half the sizes of the angles in triangle .
  • DThe sizes of the angles in both triangles depend on their lengths.

Are the two triangles similar?

  • Ano
  • Byes

Q15:

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?

  • Ano
  • Byes

Justify your answer with one of the following reasons.

  • A 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.
  • B Triangle 𝐴 𝐵 𝐶 can first be translated eight right and three down to 𝐴 ′ 𝐵 ′ 𝐶 ′ and then 𝐴 ′ 𝐵 ′ 𝐶 ′ can be reflected in the line ⃖ ⃗ 𝐸 𝐹 onto 𝐴 ′ ′ 𝐵 ′ ′ 𝐶 ′ ′ ; hence, the triangles are similar.

Q16:

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 translated eight right and two down to 𝐴 ′ 𝐵 ′ 𝐶 ′ and then 𝐴 ′ 𝐵 ′ 𝐶 ′ can be reflected in the line ⃖ ⃗ 𝐸 𝐹 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.

Q17:

A triangle 𝐴 𝐵 𝐶 has vertices at the points ( − 7 , 4 ) , ( − 4 , 3 ) , and ( − 1 , 3 ) . A triangle 𝐷 𝐸 𝐹 has vertices at the points ( 1 , − 1 ) , ( 4 , − 2 ) , and ( 7 , − 2 ) . By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

  • A They are congruent.
  • B They are not congruent.

Q18:

A triangle 𝐴 𝐵 𝐶 has vertices at the points ( 0 , 1 ) , ( 1 , 3 ) , and ( − 3 , 3 ) . A triangle 𝐷 𝐸 𝐹 has vertices at the points ( − 2 , − 2 ) , ( − 1 , − 4 ) , and ( − 5 , − 4 ) . By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

  • A They are congruent.
  • BThey are not congruent.

Q19:

A triangle 𝐴 𝐵 𝐶 has vertices at the points ( 0 , 1 ) , ( 1 , 2 ) , and ( 5 , 2 ) . A triangle 𝐷 𝐸 𝐹 has vertices at the points ( 0 , − 1 ) , ( 1 , − 2 ) , and ( 5 , − 1 ) . By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

  • A They are not congruent
  • B They are congruent.

Q20:

Triangle 𝐴 𝐵 𝐶 has been rotated to obtain triangle 𝐴 𝐵 𝐶 as seen in the given figure.

What is the length of 𝐵 𝐶 ?

What is the length of 𝐴 𝐶 ?

What type of triangle is 𝐴 𝐵 𝐶 ?

  • Ascalene
  • Bequilateral
  • Cisosceles

Q21:

In the given figure, △ 𝐷 𝐸 𝐶 is the image of △ 𝐴 𝐵 𝐶 by reflection in point 𝐶 . Find the length of 𝐷 𝐶 , rounding your result to the nearest hundredth.

Q22:

Triangle 𝐴 𝐵 𝐶 is right-angled at 𝐵 with 𝐴 𝐵 = 5 5 c m and 𝐵 𝐶 = 5 2 c m . Let 𝑋 be the image of 𝐵 after a translation through 78 cm in the direction of 𝐵 𝐴 . Let 𝑌 be the image of 𝐵 under a rotation centre 𝐴 through angle − 9 0 ∘ . Calculate the length 𝑋 𝑌 to the nearest hundredth.

Q23:

In the given figure, triangle 𝐴 𝐵 𝐶 has been reflected to triangle 𝐴 ′ 𝐵 ′ 𝐶 ′ . The perimeter of triangle 𝐴 𝐵 𝐶 is 10.5. What is the perimeter of triangle 𝐴 ′ 𝐵 ′ 𝐶 ′ ?

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