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7.02 SAS and SSS congruence criteria

Lesson

Concept summary

There are five theorems for triangle congruency. If we are given two congruent corresponding sides then we will be explaining why the triangles are 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. We can use these facts when justifying why two triangles are 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} as any segment is congruent to itself
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 show these triangles are 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

Considering the diagram as well as the following information:

  • E is the midpoint of \overline{DF}

Explain why \triangle{DEH}\cong \triangle{FEG}.

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.

Approach

The best way to show that two triangles are congruent is to label any given information or information that can be inferred on the diagram. 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 then deduce by which theorem the triangles are congruent.

Solution

From the diagram we know that \overline{HD} \cong \overline{FG} and \overline{EH} \cong \overline{GE}. Since E is the midpoint of \overline{DF}, we know that \overline{DE} \cong \overline{EF}. So, we can conclude that \triangle{DEH}\cong \triangle{FEG} by Side-Side-Side congruence (SSS).

Reflection

It's important to remember to justify the statements we make when showing two triangles are congruent, such as using the definition of a midpoint to show why \overline{DE} \cong \overline{EF}, in order for our explanation to be valid.

Outcomes

M2.G.CO.B.6

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.

M2.G.CO.B.7

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

M2.G.SRT.B.3

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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