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4.2: Congruent Figures

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Define congruent triangles and use congruence statements.
  • Understand the Third Angle Theorem.

Review Queue

What part of each pair of triangles are congruent? Write out each congruence statement for the marked congruent sides and angles.

  1. Determine the measure of \begin{align*}x\end{align*}x.
    1. What is the measure of each angle?
    2. What type of triangle is this?

Know What? Quilt patterns are very geometrical. The pattern to the right is 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

Two figures are congruent if they have exactly the same size and shape.

Congruent Triangles: Two triangles are congruent if the three corresponding angles and sides are congruent.

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

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

When referring to corresponding congruent parts of congruent triangles it is called Corresponding Parts of Congruent Triangles are Congruent, or CPCTC.

Example 1: Are the two triangles below congruent?

Solution: To determine if the triangles are congruent, match up sides with the same number of tic marks: \begin{align*}\overline{BC}\cong \overline{MN}, \ \overline{AB}\cong \overline{LM}, \ \overline{AC} \cong \overline{LN}\end{align*}BC¯¯¯¯¯¯¯¯MN¯¯¯¯¯¯¯¯¯¯, AB¯¯¯¯¯¯¯¯LM¯¯¯¯¯¯¯¯¯, AC¯¯¯¯¯¯¯¯LN¯¯¯¯¯¯¯¯.

Next match up the angles with the same markings:

\begin{align*}\angle{A} \cong \angle{L}, \ \angle{B} \cong \angle{M}\end{align*}AL, BM, and \begin{align*}\angle{C} \cong \angle{N}\end{align*}CN.

Lastly, we need to make sure these are corresponding parts. To do this, check to see if the congruent angles are opposite congruent sides. Here, \begin{align*}\angle{A}\end{align*}A is opposite \begin{align*}\overline{BC}\end{align*}BC¯¯¯¯¯¯¯¯ and \begin{align*}\angle{L}\end{align*}L is opposite \begin{align*}\overline{MN}\end{align*}MN¯¯¯¯¯¯¯¯¯¯. Because \begin{align*}\angle{A} \cong \angle{L}\end{align*}AL and \begin{align*}\overline{BC} \cong \overline{MN}\end{align*}BC¯¯¯¯¯¯¯¯MN¯¯¯¯¯¯¯¯¯¯, they are corresponding. Doing this check for the other sides and angles, we see that everything matches up and the two triangles are congruent.

Creating Congruence Statements

In Example 1, we determined that \begin{align*}\triangle ABC\end{align*}ABC and \begin{align*}\triangle LMN\end{align*}LMN are congruent. When stating that two triangles are congruent, the corresponding parts must be written in the same order. Using Example 1, we would have:

Notice that the congruent sides also line up within the congruence statement.

\begin{align*}\overline{AB} \cong \overline{LM}, \ \overline{BC} \cong \overline{MN}, \ \overline{AC} \cong \overline{LN}\end{align*}AB¯¯¯¯¯¯¯¯LM¯¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯MN¯¯¯¯¯¯¯¯¯¯, AC¯¯¯¯¯¯¯¯LN¯¯¯¯¯¯¯¯

We can also write this congruence statement five other ways, as long as the congruent angles match up. For example, we can also write \begin{align*}\triangle ABC \cong \triangle LMN\end{align*} as:

\begin{align*}&\triangle ACB \cong \triangle LNM && \triangle BCA \cong \triangle MNL && \triangle BAC \cong \triangle MLN\\ &\triangle CBA \cong \triangle NML && \triangle CAB \cong \triangle NLM && \end{align*}

Example 2: Write a congruence statement for the two triangles below.

Solution: Line up the corresponding angles in the triangles:

\begin{align*}\angle{R} \cong \angle{F}, \ \angle{S} \cong \angle{E}\end{align*}, and \begin{align*}\angle{T} \cong \angle{D}\end{align*}.

\begin{align*}\triangle RST \cong \angle{FED}\end{align*}

Example 3: If \begin{align*}\triangle CAT \cong \triangle DOG\end{align*}, what else do you know?

Solution: From this congruence statement, we know three pairs of angles and three pairs of sides are congruent.

Third Angle Theorem

Example 4: Find \begin{align*}m\angle{C}\end{align*} and \begin{align*}m\angle{J}\end{align*}.

Solution: The sum of the angles in a triangle is \begin{align*}180^\circ\end{align*}.

\begin{align*}\triangle ABC: \ 35^\circ + 88^\circ + m\angle{C} & = 180^\circ\\ m\angle{C} & = 57^\circ\\ \triangle HIJ: \ 35^\circ + 88^\circ + m\angle{J} & = 180^\circ\\ m\angle{J} & = 57^\circ\end{align*}

Notice we were given \begin{align*}m\angle{A} = m\angle{H}\end{align*} and \begin{align*}m\angle{B} = m\angle{I}\end{align*} and we found out \begin{align*}m\angle{C} = m\angle{J}\end{align*}. This can be generalized into the Third Angle Theorem.

Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent.

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

Example 5: Determine the measure of the missing angles.

Solution: From the Third Angle Theorem, we know \begin{align*}\angle{C} \cong \angle{F}\end{align*}.

\begin{align*}m\angle{A}+m\angle{B}+m\angle{C}&=180^\circ\\ m\angle{D}+m\angle{B}+m\angle{C}&=180^\circ\\ 42^\circ + 83^\circ+m\angle{C}&=180^\circ\\ m\angle{C}&=55^\circ=m\angle{F}\end{align*}

Congruence Properties Recall the Properties of Congruence from Chapter 2. They will be very useful in the upcoming sections.

\begin{align*}&\text{Reflexive Property of Congruence:} && \overline{AB} \cong \overline{AB} \ \text{or} \ \triangle ABC \cong \triangle ABC\\ &\text{Symmetric Property of Congruence:} && \angle{EFG} \cong \angle{XYZ} \ \text{and} \ \angle{XYZ} \cong \angle{EFG}\\ &&& \triangle ABC \cong \triangle DEF \ \text{and} \ \triangle DEF \cong \triangle ABC\\ &\text{Transitive Property of Congruence:} && \triangle ABC \cong \triangle DEF \ \text{and} \ \triangle DEF \cong \triangle GHI \ \text{then} \\ &&&\triangle ABC \cong \triangle GHI\end{align*}

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

Example 6: In order to say that \begin{align*}\triangle ABD \cong \triangle ABC\end{align*}, you must show the three corresponding angles and sides are congruent. Which pair of sides is congruent by the Reflexive Property?

Solution: The side \begin{align*}\overline{AB}\end{align*} is shared by both triangles. In a geometric proof, \begin{align*}\overline{AB} \cong \overline{AB}\end{align*} by the Reflexive Property.

Know What? Revisited The 16 “\begin{align*}A\end{align*}” triangles are congruent. The 16 “\begin{align*}B\end{align*}” triangles are also congruent. The quilt pattern is made from dividing up the entire square into smaller squares. Both the “\begin{align*}A\end{align*}” and “\begin{align*}B\end{align*}” triangles are right triangles.

Review Questions

  • Questions 1 and 2 are similar to Example 3.
  • Questions 3-12 are a review and use the definitions and theorems explained in this section.
  • Questions 13-17 are similar to Example 1 and 2.
  • Question 18 the definitions and theorems explained in this section.
  • Questions 19-22 are similar to Examples 4 and 5.
  • Question 23 is a proof of the Third Angle Theorem.
  • Questions 24-28 are similar to Example 6.
  • Questions 29 and 30 are investigations using congruent triangles, a ruler and a protractor.
  1. If \begin{align*}\triangle RAT \cong \triangle UGH\end{align*}, what is also congruent?
  2. If \begin{align*}\triangle BIG \cong \triangle TOP\end{align*}, what is also congruent?

For questions 3-7, use the picture to the right.

  1. What theorem tells us that \begin{align*}\angle FGH \cong \angle FGI\end{align*}?
  2. What is \begin{align*}m \angle FGI\end{align*} and \begin{align*}m \angle FGH\end{align*}? How do you know?
  3. What property tells us that the third side of each triangle is congruent?
  4. How does \begin{align*}\overline{FG}\end{align*} relate to \begin{align*}\angle IFH\end{align*}?
  5. Write the congruence statement for these two triangles.

For questions 8-12, use the picture to the right.

  1. \begin{align*}\overline{AB}||\overline{DE}\end{align*}, what angles are congruent? How do you know?
  2. Why is \begin{align*}\angle{ACB} \cong \angle{ECD}\end{align*}? It is not the same reason as #8.
  3. Are the two triangles congruent with the information you currently have? Why or why not?
  4. If you are told that \begin{align*}C\end{align*} is the midpoint of \begin{align*}\overline{AE}\end{align*} and \begin{align*}\overline{BD}\end{align*}, what segments are congruent?
  5. Write a congruence statement.

For questions 13-16, determine if the triangles are congruent. If they are, write the congruence statement.

  1. Suppose the two triangles to the right are congruent. Write a congruence statement for these triangles.
  2. Explain how we know that if the two triangles are congruent, then \begin{align*}\angle{B} \cong \angle{Z}\end{align*}.

For questions 19-22, determine the measure of all the angles in the each triangle.

  1. Fill in the blanks in the Third Angle Theorem proof below. Given: \begin{align*}\angle{A} \cong \angle{D}, \ \angle{B} \cong \angle{E}\end{align*} Prove: \begin{align*}\angle{C} \cong \angle{F}\end{align*}
Statement Reason
1. \begin{align*}\angle{A} \cong \angle{D}, \ \angle{B} \cong \angle{E}\end{align*}
2. \begin{align*}\cong\end{align*} angles have = measures
3. \begin{align*}m\angle{A}+m\angle{B}+m\angle{C}=180^\circ\!\\ m\angle{D}+m\angle{E}+m\angle{F}=180^\circ\end{align*}
4. Substitution PoE
5. Substitution PoE
6. \begin{align*}m\angle{C} = m\angle{F}\end{align*}
7. \begin{align*}\angle{C} \cong \angle{F}\end{align*}

For each of the following questions, determine if the Reflexive, Symmetric or Transitive Properties of Congruence is used.

  1. \begin{align*}\angle{A} \cong \angle{B}\end{align*} and \begin{align*}\angle{B} \cong \angle{C}\end{align*}, then \begin{align*}\angle{A} \cong \angle{C}\end{align*}
  2. \begin{align*}\overline{AB} \cong \overline{AB}\end{align*}
  3. \begin{align*}\triangle XYZ \cong \triangle LMN\end{align*} and \begin{align*}\triangle LMN \cong \triangle XYZ\end{align*}
  4. \begin{align*}\triangle ABC \cong \triangle BAC\end{align*}
  5. What type of triangle is \begin{align*}\triangle ABC\end{align*} in #27? How do you know?

Review Queue Answers

  1. \begin{align*}\angle B \cong \angle{H}, \overline{AB} \cong \overline{GH}, \overline{BC} \cong \overline{HI}\end{align*}
  2. \begin{align*}\angle C \cong \angle{M}, \overline{BC} \cong \overline{LM}\end{align*}
  3. The angles add up to \begin{align*}180^\circ\end{align*}
    1. \begin{align*}(5x + 2)^\circ + (4x + 3)^\circ + (3x - 5)^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \qquad \qquad \ 12x = 180^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \qquad \qquad \ \quad x = 15^\circ\end{align*}
    2. \begin{align*}77^\circ, 63^\circ, 40^\circ\end{align*}
    3. acute scalene

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