Writing a Two-Column Proof Given ?ABC ? ?DEF and ?GHI ? ?DEF Prove m?ABC = m?GHI Angles Reasons given symmetric property transitive property definition of ? angles Statements Reasons 1. ?ABC ? ?DEF 1 given ? 2. ?GHI ? ?DEF 2 given 3. ?DEF ? ?ABC 3 symmetric property
Writing a Two-Column Proof Given ?ABC ? ?DEF and ?GHI ? ?DEF Prove m?ABC = m?GHI Angles Reasons given symmetric property transitive property definition of ? angles Statements Reasons 1. ?ABC ? ?DEF 1 given ? 2. ?GHI ? ?DEF 2 given 3. ?DEF ? ?ABC 3 symmetric property
The Correct Answer and Explanation is:
Two-Column Proof
| Statements | Reasons |
|---|---|
| 1. ∠ABC ≅ ∠DEF | 1. Given |
| 2. ∠GHI ≅ ∠DEF | 2. Given |
| 3. ∠DEF ≅ ∠ABC | 3. Symmetric Property of Congruence |
| 4. ∠GHI ≅ ∠ABC | 4. Transitive Property of Congruence |
| 5. m∠ABC = m∠GHI | 5. Definition of Congruent Angles |
Explanation
In this proof, we are asked to show that the measures of angles ∠ABC and ∠GHI are equal, given that both are congruent to a third angle, ∠DEF. We approach this using a standard two-column proof, a format used in geometry to logically present statements and their justifications side by side.
We begin with the given information: ∠ABC ≅ ∠DEF and ∠GHI ≅ ∠DEF. These are stated explicitly, so they are used as our first two statements with the reason “Given.”
Next, to connect ∠ABC and ∠GHI directly, we use the Symmetric Property of Congruence. This property states that if ∠ABC ≅ ∠DEF, then we can also write ∠DEF ≅ ∠ABC. This switch in order is critical to aligning our angles for the Transitive Property.
Then, we use the Transitive Property of Congruence, which tells us that if ∠GHI ≅ ∠DEF and ∠DEF ≅ ∠ABC, then ∠GHI ≅ ∠ABC. In other words, congruence can be passed through a common third angle to link two other angles.
Finally, we invoke the Definition of Congruent Angles, which states that if two angles are congruent, then their measures are equal. So, if ∠ABC ≅ ∠GHI, then m∠ABC = m∠GHI.
This logical sequence completes our proof. We demonstrated that since both angles are congruent to the same angle, they are congruent to each other, and therefore, their measures are equal. This method of proof showcases the power of properties like symmetry and transitivity in geometry.
