4.2: Congruent Figures
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.
 Determine the measure of
x . What is the measure of each angle?
 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.
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:
Next match up the angles with the same markings:
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,
Creating Congruence Statements
In Example 1, we determined that
Notice that the congruent sides also line up within the congruence statement.
We can also write this congruence statement five other ways, as long as the congruent angles match up. For example, we can also write
Example 2: Write a congruence statement for the two triangles below.
Solution: Line up the corresponding angles in the triangles:
Example 3: If
Solution: From this congruence statement, we know three pairs of angles and three pairs of sides are congruent.
Third Angle Theorem
Example 4: Find
Solution: The sum of the angles in a triangle is
Notice we were given
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
Example 5: Determine the measure of the missing angles.
Solution: From the Third Angle Theorem, we know
Congruence Properties Recall the Properties of Congruence from Chapter 2. They will be very useful in the upcoming sections.
These three properties will be very important when you begin to prove that two triangles are congruent.
Example 6: In order to say that
Solution: The side
Know What? Revisited The 16 “
Review Questions
 Questions 1 and 2 are similar to Example 3.
 Questions 312 are a review and use the definitions and theorems explained in this section.
 Questions 1317 are similar to Example 1 and 2.
 Question 18 the definitions and theorems explained in this section.
 Questions 1922 are similar to Examples 4 and 5.
 Question 23 is a proof of the Third Angle Theorem.
 Questions 2428 are similar to Example 6.
 Questions 29 and 30 are investigations using congruent triangles, a ruler and a protractor.
 If
△RAT≅△UGH , what is also congruent?  If
△BIG≅△TOP , what is also congruent?
For questions 37, use the picture to the right.
 What theorem tells us that
∠FGH≅∠FGI ?  What is
m∠FGI andm∠FGH ? How do you know?  What property tells us that the third side of each triangle is congruent?
 How does
FG¯¯¯¯¯¯¯¯ relate to∠IFH ?  Write the congruence statement for these two triangles.
For questions 812, use the picture to the right.

AB¯¯¯¯¯¯¯¯DE¯¯¯¯¯¯¯¯ , what angles are congruent? How do you know?  Why is \begin{align*}\angle{ACB} \cong \angle{ECD}\end{align*}? It is not the same reason as #8.
 Are the two triangles congruent with the information you currently have? Why or why not?
 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?
 Write a congruence statement.
For questions 1316, determine if the triangles are congruent. If they are, write the congruence statement.
 Suppose the two triangles to the right are congruent. Write a congruence statement for these triangles.
 Explain how we know that if the two triangles are congruent, then \begin{align*}\angle{B} \cong \angle{Z}\end{align*}.
For questions 1922, determine the measure of all the angles in the each triangle.
 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.
 \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*}
 \begin{align*}\overline{AB} \cong \overline{AB}\end{align*}
 \begin{align*}\triangle XYZ \cong \triangle LMN\end{align*} and \begin{align*}\triangle LMN \cong \triangle XYZ\end{align*}
 \begin{align*}\triangle ABC \cong \triangle BAC\end{align*}
 What type of triangle is \begin{align*}\triangle ABC\end{align*} in #27? How do you know?
Review Queue Answers
 \begin{align*}\angle B \cong \angle{H}, \overline{AB} \cong \overline{GH}, \overline{BC} \cong \overline{HI}\end{align*}
 \begin{align*}\angle C \cong \angle{M}, \overline{BC} \cong \overline{LM}\end{align*}
 The angles add up to \begin{align*}180^\circ\end{align*}
 \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*}
 \begin{align*}77^\circ, 63^\circ, 40^\circ\end{align*}
 acute scalene
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