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10.04 ASA and AAS congruence criteria

Introduction

We will continue to learn about new criteria that we can use to justify congruence in triangles in this lesson, using rigid transformations and what we know about relationships in triangles.

ASA and AAS congruence criteria

Exploration

We are given two triangles with 2 pairs of congruent angles, and the corresponding sides between those angles are congruent.

  1. Sketch two triangles to fit the description
  2. Label the triangles GHI and LMN, so that \angle G \cong \angle L, \angle H \cong \angle M, and \overline{GH} \cong \overline{LM}
  3. Formulate a plan for proving that there is a sequence of rigid transformation that will map \triangle GHI to \triangle LMN and explain how you know one or more vertices will align at each step

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 or angle-angle-side 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 D E F. Angle D is marked with one mark, angle E is marked with 2 marks, and side D E is highlighted.
Angle-Side-Angle (ASA) 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 (AAS) 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.
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.
AAS

When proving triangles congruent, it can be difficult to distinguish between ASA and AAS congruence. That's usually due to a result of a corollary to the triangle sum 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 ASA congruence can also be proven by AAS congruence and vice versa without any additional givens.

Examples

Example 1

Use rigid transformations to prove the ASA 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.

Worked Solution
Create a strategy

We can set up a figure so we can do the proof in terms of a particular diagram. Since we can't use the ASA congruency theorem to prove itself, we must use another strategy. In this case, we need to identify a series of rigid transformations that will map \triangle ABC to \triangle DEF. Our rigid transformations are:

  • Translation
  • Rotation
  • Reflection
Apply the idea
Two triangles: A B C and D E F. B C and E F are congruent. Angles C and F are congruent, as well as angles B and E.

Given:

  • \angle{ABC}\cong \angle{DEF}
  • \overline{BC}\cong \overline{EF}
  • \angle{ACB}\cong \overline{DFE}

Prove: \triangle ABC \cong \triangle DEF

We need to consider that the two triangles could be any orientation.

  1. There is a rigid motion that will map \overline{BC} onto \overline{EF} because \overline{BC}\cong \overline{EF}. First, we can rotate and reflect the triangle to get \overline{BC}\parallel \overline{EF} so that and \angle B and \angle E are oriented in the same direction. Call this triangle \triangle A'B'C'. If A' and D are on the same side, reflect over \overline{EF} until the figure is as shown.

    Three triangles: A B C, A prime B prime C prime, and D E F. B prime C prime and E F are overlapping and congruent. C B is congruent to B prime C prime and E F. Angles C, C prime and F are congruent, as well as angles B, B prime and E.
  2. Since these transformations are all rigid motions, we have that: \angle{ABC}\cong \angle{A'B'C'} \text{ and }\angle{ACB}\cong \angle{A'C'B'}
  3. Using the given information and the transitive property of congruence, we have that:\angle{DEF}\cong \angle{A'B'C'} \text{ and }\angle{DFE}\cong \angle{A'C'B'}
  4. Point A' can be mapped on to point D by a reflection across \overline{EF}.

    We can justify that point A' coincides with point D after the reflection as follows:

    If the reflection of \triangle{A'FE} is \triangle{A''FE}, points E and F will remain fixed during the reflection. \overrightarrow{EA''} will coincide with \overrightarrow{ED}, and \overrightarrow{FA''} will coincide with \overrightarrow{FD} because reflections preserve angles, and rays coming from the same point at the same angle will coincide. We have that\overrightarrow{EA''} and \overrightarrow{FA''} can only intersect at point A'', and \overrightarrow{ED} and \overrightarrow{FD} can only intersect at point D. Since the rays coincide, their intersection points, A'' and D, will also coincide.

  5. We have now shown that:
    • A maps to D using rigid motions
    • B maps to E using rigid motions
    • C maps to F using rigid motions
    So, we have that \triangle ABC can be mapped onto \triangle DEF using rigid motions, so: \triangle ABC \cong \triangle DEF

Example 2

Prove that the two triangles are congruent.

Two triangles: left triangle has three congruent sides of length 8, right triangle has base angles both measuring 60 degrees and opposite one of the 60 degree angle is a side of length 8.
Worked Solution
Create a strategy

Notice that one triangle is equilateral, while the other is equiangular. The corollary to the base angles theorem and its converse tells us that both triangles will be equilateral and equiangular.

Apply the idea

Both triangles are equiangular, which means that the two triangles share three common angle measures. Since they both also have a corresponding side of length 8, we can use the AAS test to justify that they are congruent.

Reflect and check

We could also show that the triangles are congruent by stating that both triangles are equilateral, which means that both must have three sides of length 8. We can then use the SSS test to justify that they are congruent.

Example 3

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

Triangles A B X sharing vertex X with triangle D C X. A B is congruent and parallel to C D.

Prove that \triangle{ABX}\cong\triangle{DCX}.

Worked Solution
Create a strategy

We want to find as much information as we can in order to satisfy one of the congruence tests.

Since \overline{AD} and \overline{BC} are straight line segments, we can find vertically opposite angles, and since \overline{AB}\parallel\overline{DC} we can find alternate angles on parallel lines.

Triangles A B X sharing vertex X with triangle D C X. A B is congruent and parallel to C D. Angles A X B and D X C are congruent, as well as A B X and D C X, and B A X and C D X.
Apply the idea
To prove: \triangle{ABX}\cong\triangle{DCX}
StatementsReasons
1.\overline{AD} and \overline{BC} are straight line segmentsGiven
2.\angle{AXB} and \angle{DXC} are vertically opposite anglesOpposite angles between straight line segments
3.\angle{AXB}\cong\angle{DXC}Vertically opposite angles are congruent
4.\angle{ABX} and \angle{DCX} are alternate interior angles\overline{BC} is a transversal of \overline{AB} and \overline{DC}
5.\overline{AB}\parallel\overline{DC}Given
6.\angle{ABX}\cong\angle{DCX}Alternate interior angles are congruent
7.\overline{AB}\cong\overline{DC}Given
8.\triangle{ABX}\cong\triangle{DCX}AAS congruence
Reflect and check

We could also use ASA to prove that \triangle{ABX} \cong \triangle{DCX}, ignoring vertical angles in a proof like the one that follows:

To prove: \triangle{ABX}\cong\triangle{DCX}
StatementsReasons
1.\overline{AD} and \overline{BC} are straight line segmentsGiven
2.\angle{ABX} and \angle{DCX} are alternate interior angles\overline{BC} is a transversal of \overline{AB} and \overline{DC}
3.\overline{AB}\parallel\overline{DC}Given
4.\angle{ABX}\cong\angle{DCX}Alternate interior angles are congruent
5.\angle{BAX}\cong\angle{CDX}Alternate interior angles are congruent
6.\overline{AB}\cong\overline{DC}Given
7.\triangle{ABX}\cong\triangle{DCX}ASA congruence

Example 4

Find the value of x that makes the triangles congruent.

Two triangles D E F and X Y Z. Angle D measures 23 degrees, angle E is 64 degrees. Side D F is of length 13.5 feet. Angle X measures 23 degrees; angle Y, 64 degrees; side X Y is of length 4 x minus 1 feet.
Worked Solution
Create a strategy

The triangles are congruent by ASA, so we can identify corresponding parts and create an equation to solve for x.

Based on the diagram, we see that\angle E \cong \angle Y and \angle D \cong \angle X, so the included side for the first triangle is \overline{DE} and the included side for the second triangle is \overline{XY}. By the definition of congruence, if \overline{DE}\cong \overline{XY}, then the two segments will be equal in length. Therefore, we need to find x so that 15=4x-1.

Apply the idea
\displaystyle 15\displaystyle =\displaystyle 4x-1DE=XY
\displaystyle 16\displaystyle =\displaystyle 4xAdd 1 to both sides
\displaystyle 4\displaystyle =\displaystyle xDivide by 4 on both sides
Idea summary

To show that two triangles are congruent, it is sufficient to demonstrate the following:

  • Angle-side-angle, or ASA: 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, or AAS: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent

Outcomes

G.CO.B.7

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.CO.B.8

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G.CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

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