Max constructs a triangle in *Geogebra*. He tells Alicia that his triangle has a \begin{align*}42^\circ\end{align*}

### Applications for Congruent Triangles

Two triangles are **congruent** if and only if corresponding pairs of sides and corresponding pairs are congruent. While one way to show that two triangles are congruent is to verify that all side and angle pairs are congruent, there are five “shortcuts”. The following list summarizes the different criteria that can be used to show triangle congruence.

**AAS (Angle-Angle-Side):**If two triangles have two pairs of congruent angles and a non-common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.**ASA (Angle-Side-Angle):**If two triangles have two pairs of congruent angles and the common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.**SAS (Side-Angle-Side):**If two triangles have two pairs of congruent sides and the included angle in one triangle is congruent to the included angle in the other triangle, then the triangles are congruent.**SSS (Side-Side-Side):**If two triangles have three pairs of congruent sides, then the triangles are congruent.If two*[FOR RIGHT TRIANGLES]*HL (Hypotenuse-Leg):triangles have one pair of legs congruent and hypotenuses congruent, then the triangles are congruent.*right*

If two triangles don't satisfy at least one of the criteria above, you cannot be confident that they are congruent.

#### Recognizing Perpendicular Bisectors

\begin{align*}\overline{BC}\end{align*}

If \begin{align*}\overline{BC}\end{align*}**reflexive property**).

The triangles are congruent by SAS. *Note that even though these are right triangles, you would not use HL to show triangle congruence in this case since you are not given that the hypotenuses are congruent.*

#### Measuring Angles

Using the information from the previous problem, if \begin{align*}m\angle A=50^\circ\end{align*}

\begin{align*}m\angle D=50^\circ\end{align*}

#### Congruent Triangles

Does one diagonal of a rectangle divide the rectangle into congruent triangles?

Recall that a rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are congruent.

There is more than enough information to show that \begin{align*}\triangle EFG \cong \triangle GHE\end{align*}

**Method #1:**The triangles have three pairs of congruent sides, so they are congruent by SSS.**Method #2:**The triangles have two pairs of congruent sides and congruent included angles, so they are congruent by SAS.**Method #3:**The triangles are right triangles with congruent hypotenuses and a pair of congruent legs, so they are congruent by HL.

**Examples**

**Example 1**

Earlier, you were asked will Alicia be able to construct a triangle that must be congruent to Max's triangle.

Max constructs a triangle in Geogebra. He tells Alicia that his triangle has a \begin{align*}42^\circ\end{align*}

For each pair of triangles, tell whether the given information is enough to show that the triangles are congruent. If the triangles are congruent, state the criterion that you used to determine the congruence and write a congruency statement. *Note that the figures are not necessarily drawn to scale!*

#### Example 2

Notice that besides the one pair of congruent sides and the one pair of congruent angles, \begin{align*}\overline{AC}\cong \overline{CA}\end{align*}

#### Example 3

The congruent sides are not corresponding in the same way that the congruent angles are corresponding. The given information for \begin{align*}\triangle ACB\end{align*}

#### Example 4

\begin{align*}G\end{align*}

Because \begin{align*}G\end{align*}

### Review

1. List the five criteria for triangle congruence and draw a picture that demonstrates each.

2. Given two triangles, do you always need at least three pieces of information about each triangle in order to be able to state that the triangles are congruent?

For each pair of triangles, tell whether the given information is enough to show that the triangles are congruent. If the triangles are congruent, state the criterion that you used to determine the congruence and write a congruency statement.

3.

4.

5.

6.

7.

8.

For 9-11, state whether the given information about a hidden triangle would be enough for you to construct a triangle that must be congruent to the hidden triangle. Explain your answer.

9. \begin{align*}\triangle ABC\end{align*}

10. \begin{align*}\triangle ABC\end{align*}

11. \begin{align*}\triangle ABC\end{align*}

12. Recall that a square is a quadrilateral with four right angles and four congruent sides. Show and explain why a diagonal of a square divides the square into two congruent triangles.

13. Show and explain using a different criterion for triangle congruence why a diagonal of a square divides the square into two congruent triangles.

14. Recall that a kite is a quadrilateral with two pairs of adjacent, congruent sides. Will one of the diagonals of a kite divide the kite into two congruent triangles? Show and explain your answer.

15. In the picture below, \begin{align*}G\end{align*}

16. Explain why AAA is not a criterion for triangle congruence.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.5.