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4.3: Corresponding Parts (CPCTC) and Identifying Minimal Conditions

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Define congruence in triangles.
• Create accurate congruence statements.
• Understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining angles will also be congruent.

Defining Congruence in Triangles

Two figures are congruent if they have exactly the same size and shape. Another way of saying this is that the two figures can be perfectly aligned when one is placed on top of the other—but you may need to rotate or reflect (flip) the figures to make them line up.

When figures have exactly the same size and shape, they are ___________.

When that alignment is done, the angles that are matched are called corresponding angles, and the sides that are matched are called corresponding sides.

In congruent figures, the angles that match up are called ___________ angles and the matching sides are called ___________ sides.

Though the two triangles above may not look the same at first, when you rotate and flip triangle DEF on the right, the sides with the same number of tic marks line up!

In the diagram above,

• Sides \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{DE}\end{align*} each have one tic mark, indicating that they have the same length. Since they have the same length and are in matching positions in the triangle, they are corresponding sides.
• Sides \begin{align*}\overline{BA}\end{align*} and \begin{align*}\overline{DF}\end{align*} each have two tic marks, showing that they are also congruent and thus, corresponding sides.
• Finally, as you can see, \begin{align*}\overline{BC} \cong \overline{EF}\end{align*} because they each have three tic marks.

Each of these pairs corresponds because they are congruent to each other.

When two triangles are congruent, the three pairs of corresponding angles are also congruent. Notice the tic marks in the triangles below.

In congruent triangles, all three pairs of corresponding angles are ____________.

We use arcs inside the angle to show congruence in angles just as tic marks show congruence in sides. You can see that 1, 2, or 3 arcs inside each angle show which angles are congruent and corresponding. From the markings in the angles we can see:

\begin{align*}\angle A \cong \angle D, \angle B \cong \angle F, \qquad\end{align*} and \begin{align*}\angle C \cong \angle E.\end{align*}

Which angle is congruent to \begin{align*}\angle F\end{align*}? ___________________

Which angle is congruent to \begin{align*}\angle E\end{align*}? ___________________

Which angle is congruent to \begin{align*}\angle D\end{align*}? ___________________

Which side is congruent to \begin{align*}\overline{AC}\end{align*}? ___________________

Which side is congruent to \begin{align*}\overline{DF}\end{align*}? ___________________

Which side is congruent to \begin{align*}\overline{BC}\end{align*}? ___________________

A term used to describe sides and angles is part. Since a triangle has three sides and three angles, we say that it has six parts.

Angles and sides are also called ___________________________ of a triangle.

By definition, if two triangles are congruent, then you know that all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent. We can therefore say that the corresponding parts (sides and angles) of congruent triangles are congruent. This is often called CPCTC.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

1. Give 2 examples of a part of a triangle.

2. Fill in the blanks: If two triangles are congruent, this means that all of its corresponding _________ and _________ are congruent.

3. What do the letters CPCTC stand for?

Example 1

Are the two triangles below congruent?

Begin by examining the sides:

• \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{RI}\end{align*} both have one tic mark, so they are congruent.
• \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{TI}\end{align*} both have two tic marks, so they are congruent as well.
• \begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{RT}\end{align*} have three tic marks each.

So each pair of sides is congruent.

Next you must check each angle:

• \begin{align*}\angle I\end{align*} and \begin{align*}\angle A\end{align*} both have one arc, so they are congruent.
• \begin{align*}\angle T \cong \angle B\end{align*} because they each have two arcs.
• Finally, \begin{align*}\angle R \cong \angle C\end{align*} because they have three arcs.

We can check that each angle in the first triangle matches with its corresponding angle in the second triangle by examining the sides. \begin{align*}\angle B\end{align*} corresponds with \begin{align*}\angle T\end{align*} because they are formed by the sides with two and three tic marks. Since all pairs of corresponding sides and angles are congruent in these two triangles, we conclude that YES, the two triangles are congruent.

Creating Congruence Statements

We have already been using the congruence sign \begin{align*}\cong\end{align*} when talking about congruent sides and congruent angles.

When writing congruence statements involving angles or triangles, use other symbols:

• The symbol \begin{align*}\overline{BC}\end{align*} means “segment \begin{align*}BC\end{align*}
• The symbol \begin{align*}\angle B\end{align*} means “angle \begin{align*}B\end{align*}
• Similarly, the symbol \begin{align*}\Delta ABC\end{align*} means “triangle \begin{align*}ABC\end{align*}

In words, the symbol \begin{align*}\cong\end{align*} means _________________________________.

In words, the symbol \begin{align*}\Delta LMN\end{align*} means _________________________________.

In words, the symbol \begin{align*}\angle N\end{align*} means _________________________________.

In words, the symbol \begin{align*}\overline{LM}\end{align*} means _________________________________.

When you are creating a congruence statement of two triangles, the order of the letters is very important. Corresponding parts must be written in order. This means that the angle at the first letter of the first triangle corresponds with the angle at the first letter of the second triangle, the angles at the second letter correspond, and the angles at the third letter correspond. If the angles are not matched up between the triangles, the parts will not correspond.

When writing congruence statements, __________________________________ parts must be written in order.

You can use either congruence statements or triangle pictures to tell if corresponding parts are in order.

For instance, in the congruence statement \begin{align*}\Delta XYZ \cong \Delta LMN\end{align*}, the letters that match up tell you which angles in the triangles are congruent:

You can see that angles \begin{align*}X\end{align*} and \begin{align*}L\end{align*} match up, so \begin{align*}\angle X \cong \angle L\end{align*} angles \begin{align*}Y\end{align*} and \begin{align*}M\end{align*} match up, so \begin{align*}\angle Y \cong \angle M\end{align*} and angles \begin{align*}Z\end{align*} and \begin{align*}N\end{align*} match up, so \begin{align*}\angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \cong \angle \ \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Likewise, in the picture of the triangles below, you can match up the marked angles (or sides) to see what parts correspond:

If you are writing a congruence statement, you could NOT say that \begin{align*}\Delta BCD \cong \Delta PQR\end{align*} because the order of the letters does not match up to corresponding congruent angles.

If you look at \begin{align*}\angle B\end{align*}, it does not correspond to \begin{align*}\angle P\end{align*}. \begin{align*}\angle B\end{align*} corresponds to \begin{align*}\angle Q\end{align*} instead (indicated by the two arcs in the angles).

\begin{align*}\angle C\end{align*} corresponds to \begin{align*}\angle\end{align*} ______ (three arcs), and \begin{align*}\angle D\end{align*} corresponds to \begin{align*}\angle\end{align*} ______ (one arc).

Remember, you must compose the congruent statement so that the vertices are lined up for congruence, which is noted by the number of arcs inside the angles. The statement below is correct:

\begin{align*}\Delta BCD \cong \Delta QPR\end{align*}

Use the congruence statement \begin{align*}\Delta ABC \cong \Delta FGH\end{align*} to name all congruent angles in the two triangles:

\begin{align*}\angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;} & \cong \angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;} & \cong \angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;} & \cong \angle \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Example 2

Compose a congruence statement for the two triangles below.

To write an accurate congruence statement, you must be able to identify the corresponding pairs in the triangles above. Notice that:

• \begin{align*}\angle R\end{align*} and \begin{align*}\angle F\end{align*} each have one arc mark.
• Similarly, \begin{align*}\angle S\end{align*} and \begin{align*}\angle E\end{align*} each have two arcs, and
• \begin{align*}\angle T\end{align*} and \begin{align*}\angle D\end{align*} have three arcs.

Additionally, you can see from the tic marks on each side that:

• \begin{align*}RS = FE\end{align*} (or \begin{align*}\overline{RS} \cong \overline{FE})\end{align*},
• \begin{align*}ST = ED\end{align*}, and
• \begin{align*}RT = FD\end{align*}.

So, the two triangles are congruent, and to make the most accurate statement, this should be expressed by matching corresponding vertices. You can spell the first triangle in alphabetical order, for example, and then align the second triangle so its angles match with the angles in the first one:

\begin{align*}\Delta RST \cong \Delta FED\end{align*}

Notice in example 2 that you don’t need to write the angles in alphabetical order, as long as the corresponding parts match.

There are six ways to name any triangle by its vertices. You can start at any of the three vertices and then name the triangle’s other vertices by progressing clockwise or counter-clockwise around the diagram. This process would give six different possible names for a triangle. For the diagram in Example 2, we could also express the congruence statement as follows:

\begin{align*}\Delta DEF \cong \Delta TSR\end{align*}

Observe that this time we named the triangle on the left of the diagram first. The order does not matter. Both of these congruence statements are accurate because corresponding sides and angles are aligned within the statement.

1. In the diagram below, the two triangles are congruent. Create a congruence statement (using the geometry symbols \begin{align*}\Delta\end{align*} and \begin{align*}\cong\end{align*}) for the diagram. Remember to be careful with corresponding angles and sides!

2. For the same diagram above, create another congruence statement that is also true. Make sure your angles match!

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

3. For the same diagram above, create a third congruence statement that is also true. How many more true congruence statement can you write?

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

The Third Angle Theorem

Previously, you studied the Triangle Sum Theorem, which states that the sum of the measures of the interior angles in a triangle will always be equal to \begin{align*}180^\circ\end{align*}. This information is useful when showing congruence.

The Triangle Sum Theorem says that the measures of all three angles inside a triangle add up to ____________.

As you practiced, if you know the measures of two angles within a triangle, there is only one possible measurement of the third angle. Thus, if you can prove two corresponding angle pairs congruent, the third pair is also guaranteed to be congruent.

Third Angle Theorem

If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles are also congruent.

This means that once you know two congruent angle pairs, then the last angle pair is also _______________________________.

Example 3

Identify whether or not the missing angles in the triangles below are congruent.

One triangle looks bigger than the other. Does that mean all of its angles are bigger? To identify whether or not the third angles are congruent, you must first find their measures.

Start with the triangle on the left. Since you know two of the angles in the triangle, you can use the triangle sum theorem to find the missing angle. In \begin{align*}\Delta WVX\end{align*} we know:

\begin{align*}m\angle W + m\angle V + m\angle X & = 180^\circ\\ 80^\circ + 35^\circ + m\angle X & = 180^\circ\\ 115^\circ + m\angle X & = 180^\circ\\ m\angle X & = 65^\circ\end{align*}

The missing angle of the triangle on the left measures \begin{align*}65^\circ\end{align*}. Repeat this process for the triangle on the right:

\begin{align*}m\angle C + m\angle A + m\angle T & = 180^\circ\\ 80^\circ + 35^\circ + m\angle T & = 180^\circ\\ 115^\circ + m\angle T & = 180^\circ\\ m\angle T & = 65^\circ\end{align*}

Since the measure of both angles is \begin{align*}65^\circ\end{align*}, \begin{align*}\angle X \cong \angle T\end{align*}.

1. Fill in the blanks:

The Third Angle ____________ says that if two angles in one triangle are ____________ to two angles in another triangle, then the third pair of angles are also ____________.

2. In Example 3 (on the previous page), create three true congruence statements for the two triangles.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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