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4. Congruence & Triangles
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United States of AmericaFL
Grade 912

4.03 ASA and AAS congruence criteria

Lesson
Practice
Lesson

Concept summary

We have seen two congruency tests, SSS and SAS, but there are five in total. If we are given two congruent corresponding angles and one congruent corresponding side, then we will be proving the triangles congruent by Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS) congruency depending on the position of the given side.

Included side

The side between two angles of a polygon is the included side of those two angles.

Triangle DEF. Angle D is marked with one mark, angle E is marked with 2 marks, and side DE is highlighted
Angle-Side-Angle congruency theorem

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Angle-Angle-Side congruency theorem

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Triangle A B C and triangle X Y Z are drawn such that Angle C A B and angle Y X Z are marked congruent, as well as side A B and side X Y, and also angle A B C and angle X Y Z.
Angle-Side-Angle (ASA)
Vertical triangles M N R and Q N P are drawn such that segment M P and Segment Q R are straight lines intersecting at N. Angle R M N and angle Q P N are marked congruent as well as angle M N R and angle Q N P. Segment R N and segment Q N are also marked congruent.
Angle-Angle-Side (AAS)

When proving triangles congruent, it can be difficult to distinguish between angle-side-angle and angle-angle-side congruence. That's usually due to the third angles theorem:

Third angles theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Two triangles. Two angles of one triangle are congruent to the two angles of the other triangle.

Because of this theorem, any triangles that can be proven by angle-side-angle congruence can also be proven by angle-angle-side congruence and vice versa without any additional givens.

Worked examples

Example 1

Identify a corresponding side pair that, if congruent, would make these triangles congruent by:

Triangle A B C and triangle D E F are drawn such that Angle B A C and Angle E D F are marked congruent as well as Angle A B C and angle D E F.
a

Angle-side-angle congruence.

Approach

To use angle-side-angle congruence the included side must be congruent. We are given \angle{A}\cong\angle{D} and \angle{B}\cong \angle{E} so we need to identify the side included by \angle{A} and \angle{B} and the side included by \angle{D} and \angle{E}.

Solution

\overline{AB} \cong \overline{DE}

b

Angle-angle-side congruence.

Approach

To use angle-angle-side congruence any side, except the included side, can be congruent. We previously identified the included sides as \overline{AB} and \overline{DE}. What are the other corresponding sides in the diagram?

Solution

\overline{BC} \cong \overline{EF} or \overline{AC} \cong \overline{DF}

Example 2

In the following diagram, \overline{AD} and \overline{BC} are both straight line segments.

Vertical triangles A B X and C D X are drawn such that segment A D and segment C B are straight line intersecting at point X. Segment A B and segment C D are marked parallel with each other. Segment B X and segment C X are marked congruent.
a

Identify the theorem that justifies these triangles are congruent.

Approach

Consider all of the information given in the diagram. We are given that \overline{BX}\cong \overline{CX} and that \overline{AB} \parallel \overline{CD}. We also know that \angle{AXB}\cong \angle{DXC}. Since \overline{AB} \parallel \overline{CD}, we can conclude that \angle{ABX}\cong \angle{DCX}.

Label the additional information on the diagram to decide the applicable congruency theorem.

Vertical triangles A B X and C D X are drawn such that segment A D and segment C B are straight lines intersecting at point X. Segment A B and segment C D are marked parallel with each other. Segment B X and segment C X are marked congruent. Angle A B X and angle D C X are marked congruent as well as angle C X D and angle B X A. .

Solution

Angle-Side-Angle (ASA) congruency theorem.

Reflection

The parallel lines in the diagram would also allow us to conclude that \angle{BAX}\cong \angle{CDX} which would make the triangles congruent by angle-angle-side instead.

b

List the statements and reasons that establish \triangle{ABX}\cong \triangle{DCX}.

Approach

We determined in part (a) that the triangles are congruent by angle-side-angle and identified the additional information to justify it. Now we just need to list the specific statements and reasons that justify the angle, the side, and the angle for the theorem.

Solution

\angle{AXB}\cong \angle{DXC} by the vertical angles theorem.

\overline{BX} \cong \overline{CX} was given.

\angle{ABX}\cong \angle{DCX} by the alternate interior angles theorem.

Outcomes

MA.912.GR.1.2

Prove triangle congruence or similarity using Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, Angle-Angle and Hypotenuse-Leg.

MA.912.GR.1.6

Solve mathematical and real-world problems involving congruence or similarity in two-dimensional figures.

MA.912.GR.2.7

Justify the criteria for triangle congruence using the definition of congruence in terms of rigid transformations.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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