# Lesson Worksheet: Congruence of Polygons through Transformations Mathematics • 8th Grade

In this worksheet, we will practice identifying congruence of polygons after applying transformations.

Q1:

A triangle has been dilated from a center by a scale factor of 3 to triangle .

Are triangles and similar?

• ANo
• BYes

Are triangles and congruent?

• AYes
• BNo

Q2:

Determine, by applying transformations, whether the two triangles seen in the given figure are congruent. • AThey are not congruent.
• BThey are 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?

• ANo
• BYes

Q4:

If triangle is mapped by a reflection in the line to triangle , would the two triangles be congruent?

• AYes
• BNo

Q5:

If triangle is mapped by a rotation about the origin to triangle , would the two triangles be congruent?

• AYes
• BNo

Q6:

A triangle is rotated by about the origin to triangle .

Are triangles and similar?

• AYes
• BNo

Are triangles and congruent?

• AYes
• BNo

Q7:

The triangle has been transformed onto triangle , which has then been transformed onto triangle , as seen in the figure. Describe the single transformation that would map onto .

• AA translation two right and three up
• BA translation two left and three down
• CA translation three right and two down
• DA translation two left and three up
• EA translation two right and three down

Describe the single transformation that would map onto .

• AA translation four right and one down
• BA rotation of counterclockwise about point
• CA translation one right and four down
• DA rotation of clockwise about point
• EA reflection in the line

Hence, are triangles and congruent?

• AYes
• BNo

Q8:

The triangle has been transformed onto triangle which has then been transformed onto triangle as seen in the figure. Describe the single transformation that would map onto .

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

Describe the single transformation that would map onto .

• Aa rotation of counterclockwise about
• Ba translation four right
• Ca rotation of counterclockwise about
• Da rotation of clockwise about
• Ea reflection in the line

Hence, are triangles and congruent?

• ANo
• BYes

Q9:

The figure shows triangles and . Are the two triangles congruent?

• ANo
• BYes

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

Q10:

The figure shows triangles and . Are the two triangles congruent?

• AYes
• BNo

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

Q11:

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

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

Q12:

Triangle has been reflected in the line to obtain triangle as seen in the given figure. Are the corresponding angles and sides of the two triangles equal?

• ANo
• BYes

What is the length of ?

What is the length of ?

What is the perimeter of triangle ?

Q13:

The triangle has been transformed onto triangle which has then been transformed onto triangle as seen in the figure. Describe the single transformation that would map onto .

• BA counterclockwise rotation about point
• CA clockwise rotation about point
• DA clockwise rotation about point

Describe the single transformation that would map onto .

• AA translation three left and three down
• BA clockwise rotation about point
• CA reflection in the line
• DA counterclockwise rotation about point
• EA clockwise rotation about point

Hence, are triangles and congruent?

• AYes
• BNo

Q14:

The figure shows two triangles and . Work out the measure of angle .

Work out the measure of angle .

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

• AThe measures of the angles in both triangles depend on their lengths.
• BThe measures of the angles in triangle are half the measures of the angles in triangle .
• CThe measures of the angles in triangle are double the measures of the angles in triangle .
• DThey are equal.

Are the two triangles similar?

• ANo
• BYes

Q15:

The figure shows three triangles: , , and . Are triangles and similar?

• ANo
• BYes

• ATriangle can first be translated eight right and two down to and then can be reflected in the line onto ; hence, the triangles are similar.
• BNo sequence of translations, reflections, rotations, or dilations exists that can map triangle onto triangle ; therefore, the two triangles cannot be similar.

Q16:

The figure shows three triangles: , , and . Are triangles and similar?

• AYes
• BNo

• ANo sequence of translations, reflections, rotations, or dilations exists that can map triangle onto triangle ; therefore, the two triangles cannot be similar.
• BTriangle can first be translated eight right and two down to and then can be reflected in the line onto ; hence, the triangles are similar.

Q17:

A triangle has vertices at the points , , and . A triangle has vertices at the points , and . By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

• AThey are congruent.
• BThey are not congruent.

Q18:

A triangle has vertices at the points , , and . A triangle has vertices at the points , , and . By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

• AThey are not congruent.
• BThey are congruent.

Q19:

A triangle has vertices at the points , , and . A triangle has vertices at the points , , and . By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

• AThey are congruent.
• BThey are not congruent

Q20:

Triangle has been rotated to obtain triangle as seen in the given figure. What is the length of ?

What is the length of ?

What type of triangle is ?

• Aequilateral
• Bscalene
• Cisosceles

Q21:

In the given figure, is the image of by reflection in point . Find the length of , rounding your result to the nearest hundredth. Q22:

Triangle is right-angled at with and . Let be the image of after a translation through 78 cm in the direction of . Let be the image of under a rotation center through angle . Calculate the length to the nearest hundredth.

Q23:

In the given figure, triangle has been reflected to triangle . The perimeter of triangle is 10.5. What is the perimeter of triangle ? Q24:

Triangle has been rotated about point to obtain triangle as seen in the given figure. Work out the perimeter of triangle . Q25:

A quadrilateral has been dilated by a scale factor of 2 to . Are and congruent?

• ANo
• BYes