Worksheet: Congruence of Triangles: ASA and AAS

In this worksheet, we will practice proving that two triangles are congruent using either the angle-side-angle (ASA) or the angle-angle-side (AAS) criterion and determining whether angle-side-side is a valid criterion for triangle congruence or not.

Q1:

Determine the lengths of 𝐴𝑁 and 𝐵𝑁.

  • A 𝐴 𝑁 = 3 8 c m , 𝐵 𝑁 = 2 2 c m
  • B 𝐴 𝑁 = 3 8 c m , 𝐵 𝑁 = 2 1 c m
  • C 𝐴 𝑁 = 3 7 c m , 𝐵 𝑁 = 2 1 c m
  • D 𝐴 𝑁 = 3 7 c m , 𝐵 𝑁 = 2 2 c m

Q2:

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

  • AASA
  • BSSS
  • CSAS

Q3:

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

  • ASSS
  • BASA
  • CSAS

Q4:

Find the lengths of 𝐶𝐵 and 𝐴𝐷.

  • A 𝐶 𝐵 = 2 0 c m , 𝐴 𝐷 = 2 0 c m
  • B 𝐶 𝐵 = 1 2 c m , 𝐴 𝐷 = 2 0 c m
  • C 𝐶 𝐵 = 2 0 c m , 𝐴 𝐷 = 1 2 c m
  • D 𝐶 𝐵 = 1 2 c m , 𝐴 𝐷 = 1 2 c m

Q5:

The two triangles in the given figure share the measures of two angles and the length of one side. Does a rigid transformation exist that would map triangle 𝐴𝐵𝐶 to triangle 𝐷𝐸𝐹, and, hence, are the two triangles congruent?

  • Ayes
  • Bno

Q6:

Two triangles have two corresponding angles and one corresponding side that are equal. Are the two triangles congruent?

  • Ano
  • Byes

Q7:

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

  • Acongruent, ASA
  • Bnot congruent
  • Ccongruent, SAS
  • Dcongruent, SSS

Q8:

Two triangles contain angles of equal measures.

Can we prove that the triangles are similar?

  • Ano
  • Byes

Can we prove that the two triangles are congruent?

  • Ano
  • Byes

Q9:

In the given figure, a reflection in the line 𝐴𝐵 would map triangle 𝐴𝐵𝐶 to triangle 𝐴𝐵𝐷. Does it follow that the two triangles congruent?

  • Ayes
  • Bno

Q10:

Find the length of 𝐸𝐶.

Q11:

Are two triangles congruent only when the angles of one triangle are equal to the angles in the other?

  • AYes
  • BNo

Q12:

Triangle 𝐴𝐵𝐶 has been rotated to obtain triangle 𝐴𝐵𝐶 as seen in the given figure.

What is the measure of 𝐴𝐵𝐶?

What is the length of 𝐴𝐶?

What is the measure of 𝐴𝐶𝐵?

What type of triangle is 𝐴𝐵𝐶?

  • Ascalene
  • Bequilateral
  • Cisosceles

Q13:

Which is the correct congruence from this figure?

  • A 𝐵 𝐷 𝐴 𝐶 𝐷 𝐴
  • B 𝐴 𝐵 𝐷 𝐶 𝐷 𝐴
  • C 𝐴 𝐷 𝐵 𝐶 𝐷 𝐴
  • D 𝐴 𝐵 𝐷 𝐴 𝐷 𝐶

Q14:

In the given figure, 𝑆𝑃=3𝑥7 and 𝑆𝑀=2𝑥2. Find 𝑆𝑃.

Q15:

The triangle in the given figure has been constructed in the following way: 𝑀 is the midpoint of 𝐴𝐵, 𝑀𝑃 is the line parallel to 𝐵𝐶, and 𝑀𝑄 is the line parallel to 𝐴𝐶.

What do we know about the measures of angles 𝐴𝑀𝑃 and 𝐴𝐵𝐶?

  • AAngle 𝐴𝐵𝐶 is bigger than angle 𝐴𝑀𝑃.
  • BAngle 𝐴𝑀𝑃 is bigger than angle 𝐴𝐵𝐶.
  • CThey are equal.

What do we know about the measures of angles 𝐴𝑃𝑀 and 𝑀𝑄𝐵?

  • AAngle 𝑀𝑄𝐵 is bigger than angle 𝐴𝑃𝑀.
  • BThey are equal.
  • CAngle 𝐴𝑃𝑀 is bigger than angle 𝑀𝑄𝐵.

What do we know about the lengths of 𝐴𝑀 and 𝐵𝑀?

  • A 𝐵 𝑀 is longer than 𝐴𝑀.
  • B 𝐴 𝑀 is longer than 𝐵𝑀.
  • CThey are equal.

Are triangles 𝐴𝑀𝑃 and 𝐵𝑀𝑄 congruent? If yes, state by which congruence criteria.

  • AYes, AAA
  • BYes, SAS
  • CYes, ASA
  • DNo
  • EYes, SSS

Therefore, what can be said of the lengths 𝐴𝑃, 𝑀𝑄, 𝑀𝑃, and 𝐵𝑄?

  • A 𝐴 𝑃 = 𝑀 𝑄 and 𝑀𝑃=𝐵𝑄
  • B 𝐴 𝑃 = 𝑀 𝑃 and 𝑀𝑄=𝐵𝑄
  • C 𝐴 𝑃 𝑀 𝑄 and 𝑀𝑃=𝐵𝑄
  • D 𝐴 𝑃 = 𝐵 𝑄 and 𝑀𝑃=𝑀𝑄
  • E 𝐴 𝑃 = 𝑀 𝑄 and 𝑀𝑃𝐵𝑄

Since 𝑀𝑃𝐶𝑄 is a parallelogram, what can be said of the points 𝑃 and 𝑄?

  • AWe cannot conclude anything.
  • B 𝑃 and 𝑄 are midpoints, because 𝐴𝑃=𝑃𝐶 and 𝐵𝑄=𝑄𝐶.
  • C 𝑄 is closer to 𝐵 than 𝐶 because 𝐴𝑃 is less than 𝑃𝐶.
  • D 𝑃 is closer to 𝐴 than 𝐶 because 𝐴𝑀 is less than 𝑀𝑃.

Q16:

In the given figure, 𝐴𝐵 is the perpendicular bisector of 𝐶𝐷. By definition, 𝐶𝐸 is equal to 𝐸𝐷, 𝐴𝐸 is a common side to both angles, and 𝐴𝐸𝐶 and 𝐴𝐸𝐷 are both right angles.

Which congruence criterion could be used to prove that triangle 𝐴𝐸𝐶 and triangle 𝐴𝐸𝐷 are congruent?

  • ASSS
  • BASA
  • CSAS

As triangle 𝐴𝐸𝐶 and triangle 𝐴𝐸𝐷 are congruent, determine what will be true of the line segments 𝐴𝐶 and 𝐴𝐷, wherever 𝐴 may lie on the line.

  • AThey will be congruent.
  • BThey will be parallel.
  • CThey will be perpendicular.

Q17:

In the figure, 𝐴𝐵𝐷𝐸.

Which congruence criterion could be used to prove the two triangles are congruent?

  • AThe SAS rule will prove congruence.
  • BThe AAS rule will prove congruence.
  • CThe AAA rule will prove congruence.
  • DThe ASA rule will prove congruence.
  • EThere is not enough information to prove congruence.

Q18:

The shape 𝐴𝐵𝐹𝐶 in the given figure is a parallelogram.

What can be said of the lengths of 𝐴𝐶 and 𝐵𝐹?

  • AThey are equal.
  • BThey are different.

Which angle has the same measure as 𝐴𝐶𝐸?

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

Which angle has the same measure as 𝐶𝐴𝐸?

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

Given the information gained from the previous sections and the ASA congruence criterion, are triangles 𝐴𝐶𝐸 and 𝐹𝐸𝐵 congruent?

  • AYes
  • BNo

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