### SAS Triangle Congruence

An **included angle** is when an angle is between two given sides of a triangle (or polygon). In the picture below, the markings indicate that \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*} are the given sides, so \begin{align*}\angle B\end{align*} would be the included angle.

Consider the question: If I have two sides of length 2 in and 5 in and the angle between them is \begin{align*}45^\circ\end{align*}, can I construct only one triangle?

#### Investigation: Constructing a Triangle Given Two Sides and Included Angle

Tools Needed: protractor, pencil, ruler, and paper

- Draw the longest side (5 in) horizontally, halfway down the page.
*The drawings in this investigation are to scale.* - At the left endpoint of your line segment, use the protractor to measure a \begin{align*}45^\circ\end{align*} angle. Mark this measurement.
- Connect your mark from Step 2 with the left endpoint. Make your line 2 in long, the length of the second side.
- Connect the two endpoints by drawing the third side.

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

**Side-Angle-Side (SAS) Triangle Congruence Postulate:** If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent.

The markings in the picture are enough to say that \begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}.

#### Information Necessary to Prove Congruency

What additional piece of information would you need to prove that these two triangles are congruent using the SAS Postulate?

a) \begin{align*}\angle ABC \cong \angle LKM\end{align*}

b) \begin{align*}\overline{AB} \cong \overline{LK}\end{align*}

c) \begin{align*}\overline{BC} \cong \overline{KM}\end{align*}

d) \begin{align*}\angle BAC \cong \angle KLM\end{align*}

For the SAS Postulate, you need two sides and the included angle in both triangles. So, you need the side on the other side of the angle. In \begin{align*}\triangle ABC\end{align*}, that is \begin{align*}\overline{BC}\end{align*} and in \begin{align*}\triangle LKM\end{align*} that is \begin{align*}\overline{KM}\end{align*}. The correct answer is c.

#### Writing a Two-Column Proof

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

Given: \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*}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*}\angle ACB \cong \angle DCE\end{align*} | Vertical Angles Postulate |

4. \begin{align*}\triangle ACB \cong \triangle ECD\end{align*} | SAS Postulate |

#### Determining if Two Triangles are Congruent

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

While the triangles have two pairs of sides and one pair of angles that are congruent, the angle is not in the same place in both triangles. The first triangle fits with SAS, but the second triangle is SSA. There is not enough information for us to know whether or not these triangles are congruent.

### Examples

#### Example 1

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

The pair of triangles is congruent by the SAS postulate. \begin{align*} \triangle CAB \cong \triangle QRS\end{align*}.

#### Example 2

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

We know that one pair of sides and one pair of angles are congruent from the diagram. In order to know that the triangles are congruent by SAS we need to know that the pair of sides on the other side of the angle are congruent. So, we need to know that \begin{align*}\overline{EF} \cong \overline{BA}\end{align*}.

#### Example 3

Fill in the blanks in the proof below.

Given:

\begin{align*}\overline{AB} \cong \overline{DC}, \ \overline{BE} \cong \overline{CE}\end{align*}

Prove: \begin{align*}\triangle ABE \cong \triangle ACE\end{align*}

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

1. | 1. |

2. \begin{align*}\angle{AEB} \cong \angle{DEC}\end{align*} | 2. |

3. \begin{align*}\triangle ABE \cong \triangle ACE\end{align*} | 3. |

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

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

2. \begin{align*}\angle{AEB} \cong \angle{DEC}\end{align*} | 2. Vertical Angle Theorem |

3. \begin{align*}\triangle ABE \cong \triangle ACE\end{align*} | 3. SAS postulate |

### 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 SAS
- Use SAS
- Use SAS

Complete the proofs below.

- Given: \begin{align*}B\end{align*} is a midpoint of \begin{align*}\overline{DC}\end{align*} \begin{align*}\overline{AB} \perp \overline{DC}\end{align*} Prove: \begin{align*}\triangle ABD \cong \triangle ABC\end{align*}
- Given: \begin{align*}\overline{AB}\end{align*} is an angle bisector of \begin{align*}\angle{DAC}\end{align*} \begin{align*}\overline{AD} \cong \overline{AC}\end{align*} Prove: \begin{align*}\triangle ABD \cong \triangle ABC\end{align*}
- Given: \begin{align*}B\end{align*} is the midpoint of \begin{align*}\overline{DE}\end{align*} and \begin{align*}\overline{AC}\end{align*} \begin{align*}\angle{ABE}\end{align*} is a right angle Prove: \begin{align*}\triangle ABE \cong \triangle CBD\end{align*}
- Given: \begin{align*}\overline{DB}\end{align*} is the angle bisector of \begin{align*}\angle{ADC}\end{align*} \begin{align*}\overline{AD} \cong \overline{DC}\end{align*} Prove: \begin{align*}\triangle ABD \cong \triangle CBD\end{align*}

For each pair of triangles, write what needs to be congruent in order for the triangles to be congruent by SAS. Then, write the congruence statement for the triangles.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.7.