Topics
10. Congruence and Triangles
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United States of America
Grades 9-12

10.03 SSS and SAS congruence criteria

Lesson
Practice

Introduction

We will use rigid transformations in this lesson to convince ourselves of congruency between triangles given specific parts of triangles. We will extend this concept to learn about new criteria that we can use to justify congruence in triangles.

SSS congruence criteria

Exploration

Use the points to change the side lengths and try to construct valid triangles.

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  1. Can you make triangles that aren't congruent? How do you know?

We can prove the congruence between two triangles when we are given all three sides on each triangle.

Side-Side-Side (SSS) congruency theorem

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

In the diagram shown, \triangle ABC \cong \triangle DEF by SSS congruency.

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.

Sometimes, congruent parts are not given to us directly and instead have to be concluded from the diagram.

For example, we know that any segment is congruent to itself by the reflexive property of segments. We can use this fact when proving triangles congruent.

In the diagram shown, \overline{AK}\cong \overline{AK} by the reflexive property of segments.

Examples

Example 1

Consider the following initial step in the proof of the SSS congruency theorem, which states if the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Given:

  • \overline{AB}\cong \overline{DE}
  • \overline{AC}\cong \overline{DF}
  • \overline{BC}\cong \overline{EF}

Prove: \triangle ABC \cong \triangle DEF

Triangles A B C and D E F with three pairs of congruent sides: A B and D E, A C and D F, B C and E F.

Step 1.

There is a rigid motion that will map \overline{BC} onto \overline{EF} because \overline{BC}\cong \overline{EF}.

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. A B and D E are congruent, as well as A C and D F.
a

Give an example of a sequence of rigid transformations that would map \triangle ABC onto \triangle A'B'C' as shown in Step 1.

Worked Solution
Apply the idea

A reflection across a vertical line of symmetry, followed by a translation along a vector from A to A' will map \triangle ABC onto \triangle A'B'C'.

Reflect and check

There exists more than one sequence of rigid transformations that could map \triangle ABC onto \triangle A'B'C', such as a rotation, followed by a reflection, and then a translation.

b

Complete the proof of SSS congruency theorem using a rigid transformation mapping.

Worked Solution
Create a strategy

We can use the converse of perpendicular bisector theorem, which states if a point is equidistant from the end points of a line segment, then it is on the perpendicular bisector of that line segment.

Apply the idea

Step 2.

Since \overline{DE}\cong \overline{A'B'} \text{ and }\overline{DF}\cong \overline{A'C'} we have that \overline{EF} must be the perpendicular bisector of \overline{A'D} using the converse of the perpendicular bisector theorem.

Two triangles: A prime B prime C prime and D E F. B prime C prime and E F are overlapping and congruent. A prime B prime and D E are congruent, as well as A prime C prime and D F. A dashed line is drawn from A prime to D, intersecting B prime C prime and E F at point G. The dashed line is also perpendicular to B prime C prime and E F.

Step 3.

Since \overline{EF} is the perpendicular bisector of \overline{A'D}, we have that: \overline{A'G}\cong\overline{GD}. And this means that point A' can be mapped on to point D by a reflection across \overline{EF}.

Step 4.

Since \triangle ABC maps to \triangle DEF using a sequence of rigid transformations, \triangle ABC \cong \triangle DEF.

Example 2

Find the value of x.

Worked Solution
Create a strategy

We know by SSS congruency that \triangle{LKM} \cong \triangle{RQS} so by CPCTC, \overline{KM} \cong \overline{QS}, so by the definition of congruence, KM = QS. We will use this information to write and solve an equation to find the unknown variable.

Apply the idea
\displaystyle KM\displaystyle =\displaystyle QSDefinition of congruent segments
\displaystyle \dfrac{2}{3}x - 4\displaystyle =\displaystyle \dfrac{3}{5}x-2Substitution
\displaystyle \dfrac{2}{3}x - 2\displaystyle =\displaystyle \dfrac{3} {5}xAdd 2 to both sides
\displaystyle -2\displaystyle =\displaystyle -\dfrac{1}{15}xSubtract \dfrac{2}{3}x from both sides
\displaystyle 30\displaystyle =\displaystyle xMultiply both sides by -\dfrac{15}{1}
Idea summary

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

  • Side-side-side, or SSS: The two triangles have three pairs of congruent sides

SAS congruence criteria

We can prove the congruence between two triangles when we are given two sides on each triangle with their included angle.

Side-Angle-Side (SAS) 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

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
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.

In the diagram shown, \triangle ABC \cong \triangle DEF by SAS congruency.

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.

We can use the fact that vertical angles are congruent by the vertical angles theorem to help us prove triangles congruent.

In the diagram shown, \angle{ACB}\cong \angle{DCE} by Vertical angle theorem.

Examples

Example 3

Consider the triangles shown:

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.
a

Identify the additional information needed to prove these triangles congruent by SAS congruence.

Worked Solution
Create a strategy

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.

Apply the idea

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

Reflect and check

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

b

Suppose that the triangles are congruent by SAS and that \angle R = 57 \degree and \angle A = \left(\dfrac{3x + 90}{2} \right) \degree. Solve for x.

Worked Solution
Create a strategy

If the triangles are congruent by SAS, we know that \angle R \cong \angle A and therefore m \angle R = m \angle A by the definition of congruence. We will use this to solve for x.

Apply the idea
\displaystyle m \angle R\displaystyle =\displaystyle m \angle ADefinition of congruent angles
\displaystyle 57\displaystyle =\displaystyle \dfrac{3x + 90}{2}Substitution
\displaystyle 114\displaystyle =\displaystyle 3x+90Multiply both sides by 2
\displaystyle 24\displaystyle =\displaystyle 3xSubtract 90 from both sides
\displaystyle 8\displaystyle =\displaystyle xDivide both sides by 3

Example 4

Using the truss bridge shown, the steel beam that makes up the base of the bridge is divided into 4 segments of equal length. The beams that appear horizontal and vertical in the diagram are perpendicular to one another.

Identify two triangles from the braces of the bridge that are congruent by naming their vertices and stating the correspondence.

Worked Solution
Apply the idea

\triangle ABC \cong \triangle EBC because \overline{BC} is the same in both triangles. We are given that \overline{AC} = \overline{CE}. \overline{BC} is a perpendicular bisector of \overline{AE} since it is vertical to the base of the bridge, so m \angle BCA = m \angle BCE = 90 \degree. Since \angle BCA is the shared angle of \overline{BC} and \overline{AC}, and \angle BCE is the shared angle of \overline{BC} and \overline{CE}, the triangles are congruent by SAS.

Reflect and check

We could use the same justification for \triangle {EFG} and \triangle {HFG}.

Idea summary

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

  • Side-angle-side, or SAS: The two triangles have two pairs of congruent sides, and the angles between these sides are also 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.

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