Topics
4. Congruence & Triangles
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United States of AmericaFL
Grade 912

4.02 SAS and SSS congruence criteria

Lesson
Practice
Lesson

Concept summary

There are five theorems for triangle congruency. If we are given two congruent corresponding sides then we will be proving the triangles congruent by Side-Side-Side (SSS) or Side-Angle-Side (SAS) congruency.

Side-Side-Side congruency theorem

If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Included angle

The angle between two sides of a polygon is known as the included angle of those two sides.

Triangle ABC. Angle B is marked and sides AB and BC are highlighted
Side-Angle-Side congruency theorem

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

Triangle A B C and triangle D E F are drawn such that segment A B and segment D E are marked congruent, as well as segment A C and segment D F, and also segment B C and segment E F
Side-Side-Side (SSS)
Triangle A B C and triangle D E F are drawn such that segment A B and segment D E are marked congruent, as well as segment A C and segment D F, Angle B A C and angle E D F are also marked congruent.
Side-Angle-Side (SAS)

Sometimes, congruent parts are not given to us directly and instead have to be concluded from the diagram. For example, we know from previous topics that vertical angles are congruent by the vertical angles theorem and that any segment is congruent to itself by the reflexive property of segments. We can use these facts when proving triangles congruent.

Isosceles triangle J K N is drawn such that segment J K and segment K N are marked congruent with one tick mark. A vertical segment K A intersects segment J N at A and divides it into two congruent segments J A and A N with two tick marks. Segment K A has 3 tick marks.
\overline{AK}\cong \overline{AK} by Reflexive property of segments
Vertical triangles A B C and E D C with common vertex C are drawn such that A E and B D are straight line segments intersecting at C. Segment A C and segment C E are marked congruent as well as segment B C and segment C D.
\angle{ACB}\cong \angle{DCE} by Vertical angle theorem

Worked examples

Example 1

Identify the additional information needed to prove these triangles congruent by Side-Angle-Side (SAS) congruence.

Triangle R E M and triangle A B C are drawn such that segment R E and segment A B are marked congruent as well as segment R M and segment A C.

Approach

From the diagram we know that \overline{ER}\cong \overline{BA} and \overline{RM}\cong \overline{AC}. If we want these triangles to be congruent by SAS we will need to identify the corresponding angles that complete the congruency theorem.

Solution

\angle{R}\cong \angle{A}

Reflection

Be sure that the angle identified is in between the given congruent sides.

Example 2

This two-column proof shows that \triangle{DEH}\cong \triangle{FEG} as seen in the diagram, but it is incomplete.

Given: E is the midpoint of \overline{DF}

Triangle D E H and triangle F E G are drawn such that D F is a straight line segment and E is a common point that lies on D F. Segment D H and segment F G are marked congruent as well as segment E H and E G.
To prove: \triangle{DEH}\cong \triangle{FEG}
StatementsReasons
1.E is the midpoint of \overline{DF}Given
2.\overline{DH}\cong \overline{FG}Given
3.\overline{EH}\cong \overline{EG}Given
4.⬚⬚
5.\triangle{DEH}\cong \triangle{FEG}⬚

Approach

The best way to approach a proof is to label the given information and any information that can be concluded based on the given information. In this example, we can label \overline{DE} \cong \overline{EF} because E is a midpoint. Take a look at the labeled diagram:

Triangle D E H and triangle F E G are drawn such that D F is a straight line segment and E is a common point that lies on D F. Segment D H and segment F G are marked congruent as well as segment E H and E G, and also D E and E F.

We can see that these triangles have all three corresponding sides labeled as congruent so the triangles will be congruent be Side-Side-Side (SSS) congruence.

Solution

To prove: \triangle{DEH}\cong \triangle{FEG}
StatementsReasons
1.E is the midpoint of \overline{DF}Given
2.\overline{DH}\cong \overline{FG}Given
3.\overline{EH}\cong \overline{EG}Given
4.\overline{DE}\cong \overline{EF}Definition of midpoint
5.\triangle{DEH}\cong \triangle{FEG}Side-Side-Side (SSS) congruence

Reflection

For the most part, the order of the statements in a proof is up to us. Just make sure that any statements based on a given piece of information (such as the definition of midpoint in this proof) come after that given statement.

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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