### Hypotenuse Leg Triangle Congruence

Recall that a right triangle has exactly one right angle. The two sides adjacent to the right angle are called legs and the side opposite the right angle is called the hypotenuse.

The Pythagorean Theorem says, for any ** right** triangle, \begin{align*}(leg)^2+(leg)^2=(hypotenuse)^2\end{align*}. What this means is that if you are given two sides of a right triangle, you can always find the third. Therefore, if you know that two sides of a

**triangle are congruent to two sides of another**

*right***triangle, you can conclude that third sides are also congruent.**

*right***HL Triangle Congruence Theorem:** If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right 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*}.

Notice that this theorem is only used with a hypotenuse and a leg. If you know that the two legs of a right triangle are congruent to two legs of another triangle, the two triangles would be congruent by SAS, because the right angle would be between them.

#### Proving Congruency using the HL Theorem

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

For HL, you need the hypotenuses to be congruent. So, \begin{align*}\overline{AC} \cong \overline{MN}\end{align*}.

#### Determining of Two Triangles are Congruent

Determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

We know the two triangles are right triangles. The have one pair of legs that is congruent and their hypotenuses are congruent. This means that \begin{align*}\triangle ABC \cong \triangle RQP\end{align*} by HL.

#### Information Necessary to use the HL Theorem

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

We already know one pair of legs is congruent and that they are right triangles. The additional piece of information we need is that the two hypotenuses are congruent, \begin{align*}\overline{UT} \cong \overline{FG} \end{align*}.

### Examples

#### Example 1

Determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

All we know is that two pairs of sides are congruent. Since we do not know if these are right triangles, we cannot use HL. We do not know if these triangles are congruent.

#### Example 2

Fill in the blanks in the proof below.

Given:

\begin{align*}\overline{SV} \perp \overline{WU}\end{align*}

\begin{align*}T\end{align*} is the midpoint of \begin{align*}\overline{SV}\end{align*} and \begin{align*}\overline{WU}\end{align*}

Prove: \begin{align*}\overline{WS} \cong \overline{UV}\end{align*}

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

1. | 1. |

2. \begin{align*}\angle STW\end{align*} and \begin{align*}\angle UTV\end{align*} are right angles | 2. |

3. | 3. |

4. \begin{align*}\overline{ST} \cong \overline{TV}, \ \overline{WT} \cong \overline{TU}\end{align*} | 4. |

5. \begin{align*}\triangle STW \cong \triangle UTV\end{align*} | 5. |

6. \begin{align*}\overline{WS} \cong \overline{UV}\end{align*} | 6. |

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

1. \begin{align*}\overline{SV} \perp \overline{WU}\end{align*} | 1. Given |

2. \begin{align*}\angle STW\end{align*} and \begin{align*}\angle UTV\end{align*} are right angles | 2. Definition of perpendicular lines. |

3. \begin{align*}T\end{align*} is the midpoint of \begin{align*}\overline{SV}\end{align*} and \begin{align*}\overline{WU}\end{align*} | 3. Given |

4. \begin{align*}\overline{ST} \cong \overline{TV}, \ \overline{WT} \cong \overline{TU}\end{align*} | 4. Definition of midpoint |

5. \begin{align*}\triangle STW \cong \triangle UTV\end{align*} | 5. SAS |

6. \begin{align*}\overline{WS} \cong \overline{UV}\end{align*} | 6. CPCTC |

Note that even though these were right triangles, we did not use the HL congruence shortcut because we were not originally given that the two hypotenuses were congruent. The SAS congruence shortcut was quicker in this case.

#### Example 3

If two right triangles have congruent hypotenuses and one pair of non-right angles that are congruent, are the two right triangles definitively congruent?

Yes, by the AAS Congruence shortcut. One pair of congruent angles is the right angles, and another pair is given. The congruent pair of sides are the hypotenuses that are congruent. Note that just like in #2, even though the triangles are right triangles, it is possible to use a congruence shortcut other than HL to prove the triangles are congruent.

### Review

Using the HL Theorem, what information do you need to prove the two triangles are congruent?

The triangles are formed by two parallel lines cut by a perpendicular transversal. \begin{align*}C\end{align*} is the midpoint of \begin{align*}\overline{AD}\end{align*}. Complete the proof to show the two triangles are congruent.

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

1. \begin{align*}\angle ACB\end{align*} and \begin{align*}\angle DCE\end{align*} are right angles. | (4.) |

2. (5.) | Definition of midpoint |

3. (6.) | Given |

4. \begin{align*}\Delta ACD \cong \Delta DCE\end{align*} | (7.) |

Based on the following details, are the two right triangles definitively congruent?

- The hypotenuses of two right triangles are congruent.
- Both sets of legs in the two right triangles are congruent.
- One set of legs are congruent in the two right triangles.
- The hypotenuses and one pair of legs are congruent in the two right triangles.
- One of the non right angles of the two right triangles is congruent.
- All of the angles of the two right triangles are congruent.
- All of the sides of the two right triangles are congruent.
- Both triangles have one leg that is twice the length of the other.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.9.