Worksheet: Congruence of Triangles: SSS, SAS, and RHS

In this worksheet, we will practice proving that two triangles are congruent using either the side-side-side (SSS), the side-angle-side (SAS), or the right angle-hypotenuse-side (RHS) criterion.

Q1:

Determine whether the triangles in the given figure are congruent, and, if they are, state which of the congruence criteria proves this.

  • Acongruent, ASA
  • Bcongruent, SAS
  • Ccongruent, SSS
  • Dnot congruent

Q2:

Two triangles share two sides and a contained angle. Would the two triangles be congruent?

  • Ayes
  • Bno

Q3:

Determine whether the triangles in the given figure are congruent by applying SSS, SAS, or ASA. If they are congruent, state which of the congruence criteria proves this.

  • Ano
  • Byes, SAS
  • Cyes, ASA
  • Dyes, SSS

Q4:

Can you use SAS to prove the triangles in the given figure are congruent? Please state your reason.

  • ANo, because the angle must be contained between the two sides.
  • BYes, because there are two pairs of corresponding sides equal in length and one pair of equal angles.

Q5:

Which congruence criteria can be used to prove that the two triangles in the given figure are congruent?

  • ASSS
  • BASA
  • CSAS

Q6:

In the given quadrilateral, 𝐴𝐹 and 𝐵𝐹 have the same length and 𝐸𝐹 and 𝐶𝐹 have the same length.

Which angle has the same measure as 𝐴𝐹𝐸?

  • A 𝐹 𝐴 𝐸
  • B 𝐴 𝐸 𝐹
  • C 𝐸 𝐹 𝐶
  • D 𝐵 𝐹 𝐶
  • E 𝐴 𝐹 𝐵

Hence, are the triangles 𝐴𝐹𝐸 and 𝐵𝐹𝐶 congruent? If yes, state which congruence criteria proves this.

  • ANo
  • BYes, ASA
  • CYes, SAS
  • DYes, SSS

Q7:

State whether the figures are congruent or not congruent.

  • Acongruent
  • Bnot congruent

Q8:

Which of the following statements will be true for two triangles that are congruent?

  • AThere will always exist a combination of translations, reflections, and rotations that could be used to map one triangle onto the other.
  • BThere will be only reflection that could be used to map one triangle onto the other.
  • CThere will always exist a combination of translations, reflections, rotations, and dilations that could be used to map one triangle onto the other.
  • DThere will be only rotation that could be used to map one triangle onto the other.
  • EThere will be only translation that could be used to map one triangle onto the other.

Q9:

In the given figure, triangle 𝐴𝐵𝐶 and triangle 𝐵𝐶𝐷 have two equal sides and share one equal angle. Are triangles 𝐴𝐵𝐶 and 𝐵𝐶𝐷 congruent?

  • Ayes
  • Bno

Q10:

Determine whether the triangles in the given figure are congruent, and, if they are, state which of the congruence criteria proves this.

  • ACongruent, ASA
  • BCongruent, SSS
  • CCongruent, SAS
  • DNot congruent

Q11:

Which congruence criteria can be used to prove that the two triangles in the given figure are congruent?

  • A 𝐴 𝑆 𝐴
  • B 𝑆 𝑆 𝑆
  • C 𝑆 𝐴 𝑆

Q12:

Given that 𝐸 is the midpoint of 𝐴𝐶 in the given figure, without referencing angles, which congruence criteria could you use to prove triangles 𝐴𝐵𝐸 and 𝐶𝐵𝐸 are congruent?

  • ASSS
  • BASA
  • CSAS

Q13:

The figure shows triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ANo sequence of translations, reflections, or rotations exists that can map triangle 𝐴𝐵𝐶 onto triangle 𝐷𝐸𝐹 and, therefore, the two triangles cannot be congruent.
  • BTriangle 𝐴𝐵𝐶 can be translated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • CTriangle 𝐴𝐵𝐶 can be rotated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • DTriangle 𝐴𝐵𝐶 can be reflected onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.

Q14:

The figure shows triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐵𝐶 can be rotated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • BTriangle 𝐴𝐵𝐶 can be translated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • CNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴𝐵𝐶 onto triangle 𝐷𝐸𝐹 and, therefore, the two triangles cannot be congruent.
  • DTriangle 𝐴𝐵𝐶 can be reflected onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.

Q15:

The figure shows triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • Ano
  • Byes

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐵𝐶 can be rotated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • BNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴𝐵𝐶 onto triangle 𝐷𝐸𝐹 and, therefore, the two triangles cannot be congruent.
  • CTriangle 𝐴𝐵𝐶 can be reflected onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • DTriangle 𝐴𝐵𝐶 can be translated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.

Q16:

From the following figure, what can we conclude about a possible Side-Side-Angle (SSA) congruence criterion?

  • ASSA is a criterion that works sometimes.
  • BThere is nothing we can conclude.
  • CSSA is a valid congruence criterion.
  • DSSA is not a valid congruence criterion.

Q17:

The figure shows triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐵𝐶 can be reflected onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • BNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴𝐵𝐶 onto triangle 𝐷𝐸𝐹 and, therefore, the two triangles cannot be congruent.
  • CTriangle 𝐴𝐵𝐶 can be rotated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • DTriangle 𝐴𝐵𝐶 can be translated onto triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.

Q18:

The figure shows triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • Ayes
  • Bno

Justify your answer with one of the following reasons.

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

Q19:

The two triangles in the given figure have two sides and a contained angle that are equal. Triangle 𝐴𝐵𝐶 could be mapped to triangle 𝐷𝐸𝐹 by a reflection in the line 𝐺𝐻. Are the two triangles congruent?

  • Ayes
  • Bno

Q20:

Draw a triangle 𝑋𝑌𝑍 which is right angled at 𝑌 and has 𝑋𝑌=𝑌𝑍=4. Bisect 𝑋𝑍 at 𝐿 and draw 𝑌𝐿. Find 𝑚𝑋𝐿𝑌.

Q21:

Complete the sentence: These figures are .

  • Aneither similar nor congruent
  • Bsimilar but not congruent
  • Ccongruent

Q22:

The two triangles in the given figure have equal sides. Are the two triangles congruent?

  • Ayes
  • Bno

Q23:

The diagonal of the rectangle divides its surface into two triangles.

  • Adifferent
  • Bcongruent

Q24:

State whether the figures are congruent or not congruent.

  • Acongruent
  • Bnot congruent

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