What if your parents were remodeling their kitchen so that measurements between the sink, refrigerator, and oven were as close to an equilateral triangle as possible? The measurements are in the picture at the left, below. Your neighbor’s kitchen has the measurements on the right, below. Are the two triangles congruent?

### SSS Triangle Congruence

Consider the question: If I have three lengths, 3 in, 4 in, and 5 in, can I construct more than one triangle with these measurements? In other words, can I construct two different triangles with these same three lengths?

#### Investigation: Constructing a Triangle Given Three Sides

Tools Needed: compass, pencil, ruler, and paper

- Draw the longest side (5 in) horizontally, halfway down the page.
*The drawings in this investigation are to scale.* - Take the compass and, using the ruler, widen the compass to measure 4 in, the next side.
- Using the measurement from Step 2, place the pointer of the compass on the left endpoint of the side drawn in Step 1. Draw an arc mark above the line segment.
- Repeat Step 2 with the last measurement, 3 in. Then, place the pointer of the compass on the right endpoint of the side drawn in Step 1. Draw an arc mark above the line segment. Make sure it intersects the arc mark drawn in Step 3.
- Draw lines from each endpoint to the arc intersections. These lines will be the other two sides of the triangle.

Can you draw another triangle, with these measurements that looks different? The answer is NO. Only one triangle can be created from any given three lengths.

**Side-Side-Side (SSS) Triangle Congruence Postulate:** If three sides in one triangle are congruent to three sides in another triangle, then the triangles are congruent.

Now, we only need to show that all three sides in a triangle are congruent to the three sides in another triangle. This is a postulate so we accept it as true without proof. Think of the SSS Postulate as a shortcut. You no longer have to show 3 sets of angles are congruent and 3 sets of sides are congruent in order to say that the two triangles are congruent.

In the coordinate plane, the easiest way to show two triangles are congruent is to find the lengths of the 3 sides in each triangle. Finding the measure of an angle in the coordinate plane can be a little tricky, so we will avoid it in this text. Therefore, you will only need to apply SSS in the coordinate plane. To find the lengths of the sides, you will need to use the distance formula, \begin{align*}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\end{align*}.

#### Writing a Triangle Congruence Statement

Write a triangle congruence statement based on the diagram below:

From the tic marks, we know \begin{align*}\overline{AB} \cong \overline{LM}, \overline{AC} \cong \overline{LK}, \overline{BC} \cong \overline{MK}\end{align*}. Using the SSS Postulate we know the two triangles are congruent. Lining up the corresponding sides, we have \begin{align*}\triangle ABC \cong \triangle LMK\end{align*}.

Don’t forget ORDER MATTERS when writing triangle congruence statements. Here, we lined up the sides with one tic mark, then the sides with two tic marks, and finally the sides with three tic marks.

#### Writing a Two-Column Proof

Write a two-column proof to show that the two triangles are congruent.

Given: \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}

\begin{align*}C\end{align*} is the midpoint of \begin{align*}\overline{AE}\end{align*} and \begin{align*}\overline{DB}\end{align*}.

Prove: \begin{align*}\triangle ACB \cong \triangle ECD\end{align*}

Statement |
Reason |
---|---|

1. \begin{align*}\overline{AB} \cong \overline{DE}\end{align*} \begin{align*}C\end{align*} is the midpoint of \begin{align*}\overline{AE}\end{align*} and \begin{align*}\overline{DB}\end{align*} |
Given |

2. \begin{align*}\overline{AC} \cong \overline{CE}, \overline{BC} \cong \overline{CD}\end{align*} | Definition of a midpoint |

3. \begin{align*}\triangle ACB \cong \triangle ECD\end{align*} | SSS Postulate |

Make sure that you clearly state the three sets of congruent sides BEFORE stating that the triangles are congruent.

**Prove Move:** Feel free to mark the picture with the information you are given as well as information that you can infer (vertical angles, information from parallel lines, midpoints, angle bisectors, right angles).

#### Calculating the Distance of Line Segments

Find the distances of all the line segments from both triangles to see if the two triangles are congruent.

Begin with \begin{align*}\triangle ABC\end{align*} and its sides.

\begin{align*}AB &= \sqrt{(-6-(-2))^2+(5-10)^2}\\ &= \sqrt{(-4)^2+(-5)^2}\\ &= \sqrt{16+25}\\ &= \sqrt{41}\end{align*}

\begin{align*}BC &= \sqrt{(-2-(-3))^2+(10-3)^2}\\ &= \sqrt{(1)^2+(7)^2}\\ &= \sqrt{1+49}\\ &= \sqrt{50}=5\sqrt{2}\end{align*}

\begin{align*}AC &= \sqrt{(-6-(-3))^2+(5-3)^2}\\ &= \sqrt{(-3)^2+(2)^2}\\ &= \sqrt{9+4}\\ &= \sqrt{13}\end{align*}

Now, find the distances of all the sides in \begin{align*}\triangle DEF\end{align*}.

\begin{align*}DE &= \sqrt{(1-5)^2+(-3-2)^2}\\ &= \sqrt{(-4)^2+(-5)^2}\\ &= \sqrt{16+25}\\ &= \sqrt{41}\end{align*}

\begin{align*}EF &= \sqrt{(5-4)^2+(2-(-5))^2}\\ &= \sqrt{(1)^2+(7)^2}\\ &= \sqrt{1+49}\\ &= \sqrt{50}=5\sqrt{2}\end{align*}

\begin{align*}DF &= \sqrt{(1-4)^2+(-3-(-5))^2}\\ &= \sqrt{(-3)^2+(2)^2}\\ &= \sqrt{9+4}\\ &= \sqrt{13}\end{align*}

We see that \begin{align*}AB = DE, BC = EF\end{align*}, and \begin{align*}AC = DF\end{align*}. Recall that if two lengths are equal, then they are also congruent. Therefore, \begin{align*}\overline{AB} \cong \overline{DE}, \overline{BC} \cong \overline{EF}\end{align*}, and \begin{align*}\overline{AC} \cong \overline{DF}\end{align*}. Because the corresponding sides are congruent, we can say that \begin{align*}\triangle ABC \cong \triangle DEF\end{align*} by SSS.

#### Earlier Problem Revisited

From what we have learned in this section, the two triangles are not congruent because the distance from the fridge to the stove in your house is 4 feet and in your neighbor’s it is 4.5 ft. The SSS Postulate tells us that all three sides have to be congruent.

### Examples

#### Example 1

Determine if the two triangles are congruent.

Start with \begin{align*}\triangle ABC\end{align*}.

\begin{align*}AB & = \sqrt{(-2-(-8))^2+(-2-(-6))^2}\\ & = \sqrt{(6)^2+(4)^2}\\ & = \sqrt{36+16}\\ & = \sqrt{52} = 2\sqrt{13}\end{align*}

\begin{align*}BC & = \sqrt{(-8-(-6))^2+(-6-(-9))^2}\\ & = \sqrt{(-2)^2+(3)^2}\\ & = \sqrt{4+9}\\ & = \sqrt{13}\end{align*}

\begin{align*}AC & = \sqrt{(-2-(-6))^2+(-2-(-9))^2}\\ & = \sqrt{(4)^2+(7)^2}\\ & = \sqrt{16+49}\\ & = \sqrt{65}\end{align*}

Now find the sides of \begin{align*}\triangle DEF\end{align*}.

\begin{align*}DE & = \sqrt{(3-6)^2 + (9-4)^2}\\ & = \sqrt{(-3)^2 + (5)^2}\\ & = \sqrt{9+25}\\ & = \sqrt{34}\end{align*}

\begin{align*}EF & = \sqrt{(6-10)^2+(4-7)^2}\\ & = \sqrt{(-4)^2+(-3)^2}\\ & = \sqrt{16+9}\\ & = \sqrt{25} = 5\end{align*}

\begin{align*}DF & = \sqrt{(3-10)^2+(9-7)^2}\\ & = \sqrt{(-7)^2+(2)^2}\\ & = \sqrt{49+4}\\ & = \sqrt{53}\end{align*}

No sides have equal measures, so the triangles are not congruent.

#### Example 2

Fill in the blanks in the proof below.

Given: \begin{align*}\overline{AB} \cong \overline{DC}, \ \overline{AC} \cong \overline{DB}\end{align*}

Prove: \begin{align*}\triangle ABC \cong \triangle DCB\end{align*}

Statement |
Reason |
---|---|

1. | 1. |

2. | 2. Reflexive PoC |

3. \begin{align*}\triangle ABC \cong \triangle DCB\end{align*} | 3. |

Statement |
Reason |
---|---|

1. \begin{align*}\overline{AB} \cong \overline{DC}, \ \overline{AC} \cong \overline{DB}\end{align*} | 1. Given |

2. \begin{align*}\overline{BC} \cong \overline{CB}\end{align*} | 2. Reflexive PoC |

3. \begin{align*}\triangle ABC \cong \triangle DCB\end{align*} | 3. SSS Postulate |

#### Example 3

Is the pair of triangles congruent? If so, write the congruence statement and why.

The triangles are congruent because they have three pairs of sides congruent. \begin{align*} \triangle DEF \cong \triangle IGH\end{align*}.

### Review

Are the pairs of triangles congruent? If so, write the congruence statement and why.

State the additional piece of information needed to show that each pair of triangles is congruent.

- Use SSS
- Use SSS

Fill in the blanks in the proofs below.

- Given: \begin{align*}B\end{align*} is the midpoint of \begin{align*}\overline{DC}\end{align*} \begin{align*}\overline{AD} \cong \overline{AC}\end{align*} Prove: \begin{align*}\triangle ABD \cong \triangle ABC\end{align*}

Statement |
Reason |
---|---|

1. | 1. |

2. | 2. Definition of a Midpoint |

3. | 3. Reflexive PoC |

4. \begin{align*}\triangle ABD \cong \triangle ABC\end{align*} | 4. |

Find the lengths of the sides of each triangle to see if the two triangles are congruent.

- \begin{align*}\triangle ABC: \ A(-1, 5), \ B(-4, 2), \ C(2, -2)\end{align*} and \begin{align*}\triangle DEF: \ D(7, -5), \ E(4, 2), \ F(8, -9)\end{align*}
- \begin{align*}\triangle ABC: \ A(-8, -3), \ B(-2, -4), \ C(-5, -9)\end{align*} and \begin{align*}\triangle DEF: \ D(-7, 2), \ E(-1, 3), \ F(-4, 8)\end{align*}
- \begin{align*}\triangle ABC: \ A(0, 5), \ B(3, 2), \ C(1, 4)\end{align*} and \begin{align*}\triangle DEF: \ D(1, 2), \ E(4, 4), \ F(7, 1)\end{align*}
- \begin{align*}\triangle ABC: \ A(1, 7), \ B(2, 2), \ C(4, 6)\end{align*} and \begin{align*}\triangle DEF: \ D(4, 10), \ E(5, 5), \ F(7, 9)\end{align*}
- Draw an example to show why SS is not enough to prove that two triangles are congruent.
- If you know that two triangles are similar, how many pairs of corresponding sides do you need to know are congruent in order to know that the triangles are congruent?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.6.