# Congruent Triangles

## Sides and angles have the same size.

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Congruent Triangles

What if you had a quilt whose pattern was geometric and made up of several congruent figures? In order for these patterns to come together, the quilter rotates and flips each block (in this case, a large triangle, smaller triangle, and a smaller square) to get new patterns and arrangements.

How many different sets of colored congruent triangles are there? How many triangles are in each set? How do you know these triangles are congruent?

### Congruent Triangles

Recall that two figures are congruent if and only if they have exactly the same size and shape. If two triangles are congruent, they will have exactly the same three sides and exactly the same three angles. In other words, two triangles are congruent if you can turn, flip, and/or slide one so it fits exactly on the other.

\begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*} are congruent because

\begin{align*}& \overline{AB} \cong \overline{DE} \qquad \qquad \angle A \cong \angle D\\ & \overline{BC} \cong \overline{EF} \quad \text{and} \quad \ \angle B \cong \angle E\\ & \overline{AC} \cong \overline{DF} \quad \qquad \ \angle \ C \cong \angle F\end{align*}

Notice that when two triangles are congruent their three pairs of corresponding angles and their three pairs of corresponding sides are congruent.

When referring to corresponding congruent parts of congruent triangles, you can use the phrase Corresponding Parts of Congruent Triangles are Congruent, or its abbreviation CPCTC.

##### Properties of Congruence Review

Recall the Properties of Congruence:

Reflexive Property of Congruence: Any shape is congruent to itself.

\begin{align*}\overline{AB} \cong \overline{AB}\end{align*} or \begin{align*}\triangle ABC \cong \triangle ABC\end{align*}

Symmetric Property of Congruence: If two shapes are congruent, the statement can be written with either shape on either side of the \begin{align*}\cong\end{align*} sign.

\begin{align*}\angle EFG \cong \angle XYZ\end{align*} and \begin{align*}\angle XYZ \cong \angle EFG\end{align*} or \begin{align*}\triangle ABC \cong \triangle DEF\end{align*} and \begin{align*}\triangle DEF \cong \triangle ABC\end{align*}

Transitive Property of Congruence: If two shapes are congruent and one of those is congruent to a third, the first and third shapes are also congruent.

\begin{align*}\triangle ABC \cong \triangle DEF\end{align*} and \begin{align*}\triangle DEF \cong \triangle GHI\end{align*}, then \begin{align*}\triangle ABC \cong \triangle GHI\end{align*}

These three properties will be very important when you begin to prove that two triangles are congruent.

#### Determining if Two Triangles are Congruent

Are the two triangles below congruent?

To determine if the triangles are congruent, each pair of corresponding sides and angles must be congruent.

Start with the sides and match up sides with the same number of tic marks. Using the tic marks: \begin{align*}\overline{BC} \cong \overline{MN}, \overline{AB} \cong \overline{LM}, \overline{AC} \cong \overline{LN}\end{align*}.

Next match the angles with the same markings; \begin{align*}\angle A \cong \angle L, \angle B \cong \angle M\end{align*}, and \begin{align*}\angle C \cong \angle N\end{align*}. Because all six parts are congruent, the two triangles are congruent.

#### Determining Congruency by the Reflexive Property

In order to say that \begin{align*}\triangle ABD \cong \triangle ABC\end{align*}, you must determine that the three corresponding angles and sides are congruent. Which pair of sides is congruent by the Reflexive Property?

The side \begin{align*}\overline{AB}\end{align*} is shared by both triangles. So, in a geometric proof, \begin{align*}\overline{AB} \cong \overline{AB}\end{align*} by the Reflexive Property of Congruence.

#### Understanding Congruency

If all three pairs of angles for two given triangles are congruent does that mean that the triangles are congruent?

Without knowing anything about the lengths of the sides you cannot tell whether or not two triangles are congruent. The two triangles described above might be congruent, but we would need more information to know for sure.

#### Quilt Problem Revisited

There are 16 “\begin{align*}A\end{align*}” triangles and they are all congruent. There are 16 “\begin{align*}B\end{align*}” triangles and they are all congruent. The quilt pattern is made from dividing up the square into smaller squares. The “\begin{align*}A\end{align*}” triangles are all \begin{align*}\frac{1}{32}\end{align*} of the overall square and the “\begin{align*}B\end{align*}” triangles are each \begin{align*}\frac{1}{128}\end{align*} of the large square. Both the “\begin{align*}A\end{align*}” and “\begin{align*}B\end{align*}” triangles are right triangles.

### Examples

#### Example 1

Determine if the triangles are congruent using the definition of congruent triangles.

We can see from the markings that \begin{align*}\angle B \cong \angle C\end{align*}, \begin{align*}\angle A \cong \angle D\end{align*}, and \begin{align*}\angle AEB \cong \angle DEC\end{align*} because they are vertical angles. Also, we know that \begin{align*}\overline{BA} \cong \overline {CD}\end{align*}, \begin{align*}\overline{EA} \cong \overline {ED}\end{align*}, and \begin{align*}\overline{BE} \cong \overline {CE}\end{align*}. Because three pairs of sides and three pairs of angles are all congruent and they are corresponding parts, this means that the two triangles are congruent.

#### Example 2

Determine if the triangles are congruent using the definition of congruent triangles.

While there are congruent corresponding parts, there are only two pairs of congruent sides, the marked ones and the shared side. Without knowing whether or not the third pair of sides is congruent we cannot say if the triangles are congruent using the definition of congruent triangles. Note, this does not mean that the triangles are not congruent, it just means that we need more information in order to say they are congruent using the definition of congruent triangles (congruent triangles have three pairs of congruent angles and three pairs of congruent sides).

#### Example 3

Determine if the triangles are congruent using the definition of congruent triangles.

We can see from the markings that \begin{align*}\angle G \cong \angle L\end{align*}, \begin{align*}\angle F \cong \angle K\end{align*}, and therefore \begin{align*}\angle H \cong \angle M\end{align*} by the Third Angle Theorem. Also, we know that \begin{align*}\overline{MK} \cong \overline {FH}\end{align*}, \begin{align*}\overline{GF} \cong \overline {LK}\end{align*}, and \begin{align*}\overline{GH} \cong \overline {LM}\end{align*}. Because three pairs of sides and three pairs of angles are all congruent and they are corresponding parts, this means that the two triangles are congruent.

### Review

The following illustrations show two parallel lines cut by a transversal. Are the triangles formed definitively congruent?

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

1. Both triangles are right triangles in which one angle measures \begin{align*}55^{\circ}\end{align*}. All of their corresponding sides are congruent.
2. Both triangles are equiangular triangles.
3. Both triangles are equilateral triangles. All sides are 5 inches in length.
4. Both triangles are obtuse triangles in which one angle measures \begin{align*}35^{\circ}\end{align*}. Two of their corresponding sides are congruent.
5. Both triangles are obtuse triangles in which two of their angles measure \begin{align*}40^{\circ}\end{align*} and \begin{align*}20^{\circ}\end{align*}. All of their corresponding sides are congruent.
6. Both triangles are isosceles triangles in which one angle measures \begin{align*}15^{\circ}\end{align*}.
7. Both triangles are isosceles triangles with two equal angles of \begin{align*}55^{\circ}\end{align*}. All corresponding sides are congruent.
8. Both triangles are acute triangles in which two of their angles measure \begin{align*}40^{\circ}\end{align*} and \begin{align*}80^{\circ}\end{align*}. All of their corresponding sides are congruent.
9. Both triangles are acute triangles in which one angle measures \begin{align*}60^{\circ}\end{align*}. Two of their corresponding sides are congruent.
10. Both triangles are equilateral triangles.

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