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Worksheet: Triangle Congruence through Rigid Transformations

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

Q1:

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

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

Q2:

The figure shows two triangles 𝐴 𝐡 𝐢 and 𝐷 𝐸 𝐹 .

Work out the measure of angle 𝐴 𝐢 𝐡 .

Work out the measure of angle 𝐷 𝐸 𝐹 .

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

  • AThey are equal.
  • BThe measures of the angles in triangle 𝐴 𝐡 𝐢 are double the measures of the angles in triangle 𝐷 𝐸 𝐹 .
  • CThe measures of the angles in triangle 𝐴 𝐡 𝐢 are half the measures of the angles in triangle 𝐷 𝐸 𝐹 .
  • DThe measures of the angles in both triangles depend on their lengths.

Are the two triangles similar?

  • Ano
  • Byes

Q3:

The triangle 𝐴 𝐡 𝐢 has been transformed onto triangle 𝐴 𝐡 𝐢    which has then been transformed onto triangle 𝐴 𝐡 𝐢       as seen in the figure.

Describe the single transformation that would map 𝐴 𝐡 𝐢 onto 𝐴 β€² 𝐡 β€² 𝐢 β€² .

  • 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 β€² 𝐡 β€² 𝐢 β€² onto 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² .

  • Aa reflection in the line βƒ–     βƒ— 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 𝐡 𝐢 and 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² congruent?

  • Ano
  • Byes

Q4:

Triangle 𝐴 𝐡 𝐢 has been rotated to obtain triangle 𝐴 𝐡 𝐢    as seen in the given figure.

What is the length of 𝐡 𝐢 ?

What is the length of 𝐴 𝐢 ?

What type of triangle is 𝐴 𝐡 𝐢 ?

  • Ascalene
  • Bequilateral
  • Cisosceles

Q5:

A triangle 𝐴 𝐡 𝐢 has coordinates at the points ( βˆ’ 7 , 4 ) , ( βˆ’ 4 , 3 ) , and ( βˆ’ 1 , 3 ) . A triangle 𝐷 𝐸 𝐹 has coordinates at the points ( 1 , βˆ’ 1 ) , ( 4 , βˆ’ 2 ) , and ( 7 , βˆ’ 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.

Q6:

If triangle 𝐡 is mapped by a 1 8 0 ∘ rotation about the origin to triangle 𝐡 β€² , would the two triangles be congruent?

  • A yes
  • B no

Q7:

A triangle 𝐴 𝐡 𝐢 has coordinates at the points ( 0 , 1 ) , ( 1 , 3 ) , and ( βˆ’ 3 , 3 ) . A triangle 𝐷 𝐸 𝐹 has coordinates at the points ( βˆ’ 2 , βˆ’ 2 ) , ( βˆ’ 1 , βˆ’ 4 ) , and ( βˆ’ 5 , βˆ’ 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.

Q8:

The figure shows triangles 𝐴 𝐡 𝐢 and 𝐷 𝐸 𝐹 .

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ATriangle 𝐴 𝐡 𝐢 can be rotated to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.
  • BTriangle 𝐴 𝐡 𝐢 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 𝐡 𝐢 onto triangle 𝐷 𝐸 𝐹 and, therefore, the two triangles cannot be congruent.
  • DWe can apply a two-stage transformation on triangle 𝐴 𝐡 𝐢 involving a reflection and then a rotation to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.

Q9:

The figure shows triangles 𝐴 𝐡 𝐢 and 𝐷 𝐸 𝐹 .

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ATriangle 𝐴 𝐡 𝐢 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 𝐡 𝐢 onto triangle 𝐷 𝐸 𝐹 and, therefore, the two triangles cannot be congruent.
  • CWe can apply a two-stage transformation on triangle 𝐴 𝐡 𝐢 involving a reflection and then a rotation to obtain triangle 𝐷 𝐸 𝐹 and, thus, the triangles are congruent.

Q10:

In the given figure, β–³ 𝐷 𝐸 𝐢 is the image of β–³ 𝐴 𝐡 𝐢 by reflection in point 𝐢 . Find the length of 𝐷 𝐢 , rounding your result to the nearest hundredth.

Q11:

Triangle 𝐴 𝐡 𝐢 has been rotated about point 𝐷 to obtain triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² as seen in the given figure. Work out the perimeter of triangle 𝐴 𝐡 𝐢 .

Q12:

A triangle 𝐴 𝐡 𝐢 has coordinates at the points ( 0 , 1 ) , ( 1 , 2 ) , and ( 5 , 2 ) . A triangle 𝐷 𝐸 𝐹 has coordinates at the points ( 0 , βˆ’ 1 ) , ( 1 , βˆ’ 2 ) , and ( 5 , βˆ’ 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.

Q13:

A triangle 𝐴 𝐡 𝐢 is rotated by 1 8 0 ∘ about the origin to triangle 𝐴 𝐡 𝐢 β€² β€² β€² .

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

  • Ayes
  • Bno

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

  • Ayes
  • Bno

Q14:

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

  • 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 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.
  • B Triangle 𝐴 𝐡 𝐢 can first be translated eight right and three down to 𝐴 β€² 𝐡 β€² 𝐢 β€² and then 𝐴 β€² 𝐡 β€² 𝐢 β€² can be reflected in the line βƒ–     βƒ— 𝐸 𝐹 onto 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² ; hence, the triangles are similar.

Q15:

The triangle 𝐴 𝐡 𝐢 has been transformed onto triangle 𝐴 𝐡 𝐢    which has then been transformed onto triangle 𝐴 𝐡 𝐢       as seen in the figure.

Describe the single transformation that would map 𝐴 𝐡 𝐢 onto 𝐴 β€² 𝐡 β€² 𝐢 β€² .

  • 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 β€² 𝐡 β€² 𝐢 β€² onto 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² .

  • Aa reflection in the line βƒ–     βƒ— 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 𝐡 𝐢 and 𝐴 β€² β€² 𝐡 β€² β€² 𝐢 β€² β€² congruent?

  • Ano
  • Byes

Q16:

Triangle 𝐴 𝐡 𝐢 has been reflected in the line 𝐿 to obtain triangle 𝐴 𝐡 𝐢 β€² β€² β€² as seen in the given figure.

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

  • Ayes
  • Bno

What is the length of 𝐡 𝐢 ?

What is the length of 𝐴 β€² 𝐡 β€² ?

What is the perimeter of triangle 𝐴 𝐡 𝐢 ?

Q17:

Triangle 𝐴 𝐡 𝐢 is right-angled at 𝐡 with 𝐴 𝐡 = 5 5 c m and 𝐡 𝐢 = 5 2 c m . Let 𝑋 be the image of 𝐡 after a translation through 78 cm in the direction of  𝐡 𝐴 . Let π‘Œ be the image of 𝐡 under a rotation centre 𝐴 through angle βˆ’ 9 0 ∘ . Calculate the length 𝑋 π‘Œ to the nearest hundredth.

Q18:

The triangle 𝐴 𝐡 𝐢 has been transformed onto triangle 𝐴 𝐡 𝐢    which has then been transformed onto triangle 𝐴 𝐡 𝐢       as seen in the figure.

Describe the single transformation that would map 𝐴 𝐡 𝐢 onto 𝐴 𝐡 𝐢    .

  • 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 𝐡 𝐢    onto 𝐴 𝐡 𝐢       .

  • Aa reflection in the line βƒ–     βƒ— 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 𝐡 𝐢 and 𝐴 𝐡 𝐢       congruent?

  • Ano
  • Byes

Q19:

A triangle 𝐴 𝐡 𝐢 has been dilated from a center 𝑃 by a scale factor of 3 to triangle 𝐴 𝐡 𝐢 β€² β€² β€² .

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

  • Ayes
  • Bno

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

  • Ano
  • Byes

Q20:

The figure shows three triangles: 𝐴 𝐡 𝐢 , 𝐴 𝐡 𝐢    , and 𝐴 𝐡 𝐢       .

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 translated eight right and two down to 𝐴 β€² 𝐡 β€² 𝐢 β€² and then 𝐴 β€² 𝐡 β€² 𝐢 β€² can be reflected in the line βƒ–     βƒ— 𝐸 𝐹 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.

Q21:

If triangle 𝐴 is mapped by a reflection in the line 𝑦 = π‘₯ to triangle 𝐴 β€² , would the two triangles be congruent?

  • A yes
  • B no

Q22:

If triangle 𝑇 is mapped to triangle 𝑇 β€² 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.

Q23:

In the given figure, triangle 𝐴 𝐡 𝐢 has been reflected to triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² . The perimeter of triangle 𝐴 𝐡 𝐢 is 10.5. What is the perimeter of triangle 𝐴 β€² 𝐡 β€² 𝐢 β€² ?

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