What if your parents changed their minds at the last second about their kitchen layout? Now, they have decided they to have the distance between the sink and the fridge be 3 ft, the angle at the sink \begin{align*}71^\circ\end{align*} and the angle at the fridge is \begin{align*}50^\circ\end{align*}. You used your protractor to measure the angle at the stove and sink at your neighbor’s house. Are the kitchen triangles congruent now?

### ASA and AAS Triangle Congruence

Consider the question: If I have two angles that are \begin{align*}45^\circ\end{align*} and \begin{align*}60^\circ\end{align*} and the side between them is 5 in, can I construct only one triangle? We will investigate it here.

#### Investigation: Constructing a Triangle Given Two Angles and Included Side

Tools Needed: protractor, pencil, ruler, and paper

- Draw the 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 the \begin{align*}45^\circ\end{align*} angle. Mark this measurement and draw a ray from the left endpoint through the \begin{align*}45^\circ\end{align*} mark.
- At the right endpoint of your line segment, use the protractor to measure the \begin{align*}60^\circ\end{align*} angle. Mark this measurement and draw a ray from the left endpoint through the \begin{align*}60^\circ\end{align*} mark. Extend this ray so that it crosses through the ray from Step 2.
- Erase the extra parts of the rays from Steps 2 and 3 to leave only 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 two angle measures and the INCLUDED side.

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

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

A variation on ASA is AAS, which is Angle-Angle-Side. Recall that for ASA you need two angles and the side between them. But, if you know two pairs of angles are congruent, then the third pair will also be congruent by the Third Angle Theorem. Therefore, you can prove a triangle is congruent whenever you have any two angles and a side.

Be careful to note the placement of the side for ASA and AAS. As shown in the pictures above, the side is ** between** the two angles for ASA and it is not for AAS.

**Angle-Angle-Side (AAS or SAA) Triangle Congruence Theorem:** If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent.

**Proof of AAS Theorem:**

Given: \begin{align*}\angle A \cong \angle Y, \angle B \cong \angle Z, \overline{AC} \cong \overline{XY}\end{align*}

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

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

1. \begin{align*}\angle A \cong \angle Y, \angle B \cong \angle Z, \overline{AC} \cong \overline{XY}\end{align*} | Given |

2. \begin{align*}\angle C \cong \angle X\end{align*} | Third Angle Theorem |

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

#### Using the ASA Postulate

What information would you need to prove that these two triangles are congruent using the ASA Postulate?

a) \begin{align*}\overline{AB} \cong \overline{UT}\end{align*}

b) \begin{align*}\overline{AC} \cong \overline{UV}\end{align*}

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

d) \begin{align*}\angle B \cong \angle T\end{align*}

For ASA, we need the side between the two given angles, which is \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{UV}\end{align*}. The answer is b.

#### Writing a Two-Column Proof

Write a two-column proof.

Given: \begin{align*}\angle C \cong \angle E, \overline{AC} \cong \overline{AE}\end{align*}

Prove: \begin{align*}\triangle ACF \cong \triangle AEB\end{align*}

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

1. \begin{align*}\angle C \cong \angle E, \overline{AC} \cong \overline{AE}\end{align*} | Given |

2. \begin{align*}\angle A \cong \angle A\end{align*} | Reflexive PoC |

3. \begin{align*}\triangle ACF \cong \triangle AEB\end{align*} | ASA |

#### Information Necessary to Prove Congruency

What information do you need to prove that these two triangles are congruent using:

a) ASA?

For ASA, we need the angles on the other side of \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{QR}\end{align*}. Therefore, we would need \begin{align*}\angle F \cong \angle Q\end{align*}.

b) AAS?

For AAS, we would need the angle on the other side of \begin{align*}\angle E\end{align*} and \begin{align*}\angle R\end{align*}. \begin{align*}\angle G \cong \angle P\end{align*}.

#### Kitchen Problem Revisited

Even though we do not know all of the angle measures in the two triangles, we can find the missing angles by using the Third Angle Theorem. In your parents’ kitchen, the missing angle is \begin{align*}39^\circ\end{align*}. The missing angle in your neighbor’s kitchen is \begin{align*}50^\circ\end{align*}. From this, we can conclude that the two kitchens are now congruent, either by ASA or AAS.

### Examples

#### Example 1

Can you prove that the following triangles are congruent? Why or why not?

Even though \begin{align*}\overline{KL} \cong \overline{ST}\end{align*}, they are not corresponding. Look at the angles around \begin{align*}\overline{KL}, \angle K\end{align*} and \begin{align*}\angle L\end{align*}. \begin{align*}\angle K\end{align*} has **one** arc and \begin{align*}\angle L\end{align*} is unmarked. The angles around \begin{align*}\overline{ST}\end{align*} are \begin{align*}\angle S\end{align*} and \begin{align*}\angle T\end{align*}. \begin{align*}\angle S\end{align*} has **two** arcs and \begin{align*}\angle T\end{align*} is unmarked. In order to use AAS, \begin{align*}\angle S\end{align*} needs to be congruent to \begin{align*}\angle K\end{align*}. They are not congruent because the arcs marks are different. Therefore, we cannot conclude that these two triangles are congruent.

#### Example 2

Write a 2-column proof.

Given: \begin{align*}\overline{BD}\end{align*} is an angle bisector of \begin{align*}\angle CDA, \angle C \cong \angle A\end{align*}

Prove: \begin{align*}\triangle CBD \cong \angle ABD\end{align*}

Here is the proof:

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

1. \begin{align*}\overline{BD}\end{align*} is an angle bisector of \begin{align*}\angle CDA, \angle C \cong \angle A\end{align*} | Given |

2. \begin{align*}\angle CDB \cong \angle ADB\end{align*} | Definition of an Angle Bisector |

3. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*} | Reflexive PoC |

3. \begin{align*}\triangle CBD \cong \triangle ABD\end{align*} | AAS |

#### Example 3

Write a two-column proof.

Given: \begin{align*}\overline{AB} \ || \ \overline{ED}, \angle C \cong \angle F, \overline{AB} \cong \overline{ED}\end{align*}

Prove: \begin{align*}\overline{AF} \cong \overline{CD}\end{align*}

First, prove that the triangles are congruent. Once you have proved they are congruent, you need one more step to show that the corresponding pair of sides must be congruent. Remember that CPCTC stands for *corresponding parts of congruent triangles are congruent*.

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

1. \begin{align*}\overline{AB} \ || \ \overline{ED}, \angle C \cong \angle F, \overline{AB} \cong \overline{ED}\end{align*} | Given |

2. \begin{align*}\angle ABE \cong \angle DEB\end{align*} | Alternate Interior Angles Theorem |

3. \begin{align*}\triangle ABF \cong \triangle DEC\end{align*} | ASA |

4. \begin{align*}\overline{AF} \cong \overline{CD}\end{align*} | CPCTC |

### Review

For questions 1-3, determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

For questions 4-8, use the picture and the given information below.

Given: \begin{align*}\overline{DB} \perp \overline{AC}, \ \overline{DB}\end{align*} is the angle bisector of \begin{align*}\angle CDA\end{align*}

- From \begin{align*}\overline{DB} \perp \overline{AC}\end{align*}, which angles are congruent and why?
- Because \begin{align*}\overline{DB}\end{align*} is the angle bisector of \begin{align*}\angle CDA\end{align*}, what two angles are congruent?
- From looking at the picture, what additional piece of information are you given? Is this enough to prove the two triangles are congruent?
- Write a two-column proof to prove \begin{align*}\triangle CDB \cong \triangle ADB\end{align*}, using #4-6.
- What would be your reason for \begin{align*}\angle C \cong \angle A\end{align*}?

For questions 9-13, use the picture and the given information.

Given: \begin{align*}\overline{LP}||\overline{NO}, \ \overline{LP} \cong \overline{NO}\end{align*}

- From \begin{align*}\overline{LP}||\overline{NO}\end{align*}, which angles are congruent and why?
- From looking at the picture, what additional piece of information can you conclude?
- Write a two-column proof to prove \begin{align*}\triangle LMP \cong \triangle OMN\end{align*}.
- What would be your reason for \begin{align*}\overline{LM} \cong \overline{MO}\end{align*}?
- Fill in the blanks for the proof below. Use the given from above. Prove: \begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{PN}\end{align*}.

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

1. \begin{align*}\overline{LP} || \overline{NO}, \ \overline{LP} \cong \overline{NO}\end{align*} | 1. Given |

2. | 2. Alternate Interior Angles |

3. | 3. ASA |

4. \begin{align*}\overline{LM} \cong \overline{MO}\end{align*} | 4. |

5. \begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{PN}\end{align*}. | 5. |

Determine the additional piece of information needed to show the two triangles are congruent by the given postulate.

- AAS
- ASA
- ASA
- AAS

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.8.