Lesson Worksheet: Congruence of Triangles: ASA and AAS Mathematics • 8th Grade

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 𝐡𝑁.

  • A𝐴𝑁=38cm, 𝐡𝑁=22cm
  • B𝐴𝑁=38cm, 𝐡𝑁=21cm
  • C𝐴𝑁=37cm, 𝐡𝑁=21cm
  • D𝐴𝑁=37cm, 𝐡𝑁=22cm

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 𝐢𝐡 and 𝐴𝐷.

  • A𝐢𝐡=20cm, 𝐴𝐷=20cm
  • B𝐢𝐡=12cm, 𝐴𝐷=20cm
  • C𝐢𝐡=20cm, 𝐴𝐷=12cm
  • D𝐢𝐡=12cm, 𝐴𝐷=12cm

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 𝐴𝐡𝐢 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 ⃖⃗𝐴𝐡 would map triangle 𝐴𝐡𝐢 to triangle 𝐴𝐡𝐷. Does it follow that the two triangles congruent?

  • Ayes
  • Bno

Q10:

Find the length of 𝐸𝐢.

Q11:

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

  • AYes
  • BNo

Q12:

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

What is the measure of ∠𝐴𝐡𝐢?

What is the length of 𝐴𝐢?

What is the measure of ∠𝐴𝐢𝐡?

What type of triangle is 𝐴𝐡𝐢?

  • Ascalene
  • Bequilateral
  • Cisosceles

Q13:

Which is the correct congruence from this figure?

  • A△𝐡𝐷𝐴≅△𝐢𝐷𝐴
  • B△𝐴𝐡𝐷≅△𝐢𝐷𝐴
  • C△𝐴𝐷𝐡≅△𝐢𝐷𝐴
  • D△𝐴𝐡𝐷≅△𝐴𝐷𝐢

Q14:

In the given figure, 𝑆𝑃=3π‘₯βˆ’7 and 𝑆𝑀=2π‘₯βˆ’2. Find 𝑆𝑃.

Q15:

The triangle in the given figure has been constructed in the following way: 𝑀 is the midpoint of 𝐴𝐡, 𝑀𝑃 is the line parallel to 𝐡𝐢, and ⃖⃗𝑀𝑄 is the line parallel to 𝐴𝐢.

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

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

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

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

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

  • A𝐡𝑀 is longer than 𝐴𝑀.
  • B𝐴𝑀 is longer than 𝐡𝑀.
  • CThey are equal.

Are triangles 𝐴𝑀𝑃 and 𝐡𝑀𝑄 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 𝐡𝑄?

  • A𝐴𝑃=𝑀𝑄 and 𝑀𝑃=𝐡𝑄
  • B𝐴𝑃=𝑀𝑃 and 𝑀𝑄=𝐡𝑄
  • C𝐴𝑃≠𝑀𝑄 and 𝑀𝑃=𝐡𝑄
  • D𝐴𝑃=𝐡𝑄 and 𝑀𝑃=𝑀𝑄
  • E𝐴𝑃=𝑀𝑄 and 𝑀𝑃≠𝐡𝑄

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

  • AWe cannot conclude anything.
  • B𝑃 and 𝑄 are midpoints, because 𝐴𝑃=𝑃𝐢 and 𝐡𝑄=𝑄𝐢.
  • C𝑄 is closer to 𝐡 than 𝐢 because 𝐴𝑃 is less than 𝑃𝐢.
  • D𝑃 is closer to 𝐴 than 𝐢 because 𝐴𝑀 is less than 𝑀𝑃.

Q16:

In the given figure, ⃖⃗𝐴𝐡 is the perpendicular bisector of 𝐢𝐷. By definition, 𝐢𝐸 is equal to 𝐸𝐷, 𝐴𝐸 is a common side to both angles, and ∠𝐴𝐸𝐢 and ∠𝐴𝐸𝐷 are both right angles.

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

  • ASSS
  • BASA
  • CSAS

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

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

Q17:

The shape 𝐴𝐡𝐹𝐢 in the given figure is a parallelogram.

What can be said of the lengths of 𝐴𝐢 and 𝐡𝐹?

  • AThey are equal.
  • BThey are different.

Which angle has the same measure as ∠𝐴𝐢𝐸?

  • A∠𝐴𝐸𝐢
  • B∠𝐢𝐸𝐹
  • C∠𝐸𝐢𝐹
  • D∠𝐢𝐴𝐸
  • E∠𝐸𝐡𝐹

Which angle has the same measure as ∠𝐢𝐴𝐸?

  • A∠𝐸𝐡𝐹
  • B∠𝐴𝐸𝐢
  • C∠𝐴𝐸𝐡
  • D∠𝐸𝐹𝐡
  • E∠𝐴𝐢𝐸

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

  • AYes
  • BNo

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