The Transitive Property of Congruence Explained with Examples

📚 Quick Study Guide

• 📐Definition: The Transitive Property of Congruence states that if one geometric figure is congruent to a second geometric figure, and the second figure is congruent to a third figure, then the first figure is also congruent to the third figure.
• ✨Symbolic Representation (Angles): If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.
• 📏Symbolic Representation (Line Segments): If $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$, then $\overline{AB} \cong \overline{EF}$.
• 💡Key Idea: It's about establishing a chain of congruence. The 'middleman' figure (like $\angle B$ or $\overline{CD}$ above) links the other two.
• ✍️Application: Useful in geometric proofs to establish relationships between different parts of a figure.

Practice Quiz

1. If $\angle P \cong \angle Q$ and $\angle Q \cong \angle R$, then which of the following is true?

   1. $\angle P \cong \angle R$
   2. $\angle P \not\cong \angle R$
   3. $\angle P > \angle R$
   4. $\angle P < \angle R$

2. Given that $\overline{XY} \cong \overline{WZ}$ and $\overline{WZ} \cong \overline{UV}$, what can be concluded?

   1. $\overline{XY} \cong \overline{UV}$
   2. $\overline{XY} \not\cong \overline{UV}$
   3. $\overline{XY} > \overline{UV}$
   4. $\overline{XY} < \overline{UV}$

3. If triangle ABC is congruent to triangle DEF, and triangle DEF is congruent to triangle GHI, what can be said about triangle ABC and triangle GHI?

   1. $\triangle ABC \cong \triangle GHI$
   2. $\triangle ABC \sim \triangle GHI$
   3. $\triangle ABC \not\cong \triangle GHI$
   4. They are unrelated.

4. Suppose angle 1 is congruent to angle 2, and angle 2 is congruent to angle 3. Which property justifies the statement that angle 1 is congruent to angle 3?

   1. Reflexive Property of Congruence
   2. Symmetric Property of Congruence
   3. Transitive Property of Congruence
   4. Addition Property of Congruence

5. If segment AB is congruent to segment CD, and segment CD has a length of 5 cm, and segment EF is congruent to segment CD, what is the length of segment EF?

   1. 5 cm
   2. 10 cm
   3. 2.5 cm
   4. It cannot be determined.

6. Given: $\angle L \cong \angle M$, $\angle M \cong \angle N$. What is the next logical statement based on the Transitive Property of Congruence?

   1. $\angle L \cong \angle N$
   2. $\angle M \cong \angle L$
   3. $\angle N \cong \angle M$
   4. $\angle L + \angle M = \angle N$

7. If polygon QWERTY is congruent to polygon ASDFGH, and polygon ASDFGH is congruent to polygon ZXCVBN, which property allows you to conclude polygon QWERTY is congruent to polygon ZXCVBN?

   1. Transitive Property of Congruence
   2. Reflexive Property of Congruence
   3. Symmetric Property of Congruence
   4. Associative Property of Congruence