Given: ∠1 and ∠2 form a linear pair; m∠2 + m∠3 = 180 prove: ∠1 is congruent to ∠3
The Correct Answer and Explanation is:
Proof: ∠1 ≅ ∠3
Given:
- ∠1 and ∠2 form a linear pair.
- ( m∠2 + m∠3 = 180^\circ ).
To Prove: ( ∠1 ≅ ∠3 ).
Proof:
- ( ∠1 ) and ( ∠2 ) form a linear pair.
- By definition of a linear pair, ( m∠1 + m∠2 = 180^\circ ).
- ( m∠2 + m∠3 = 180^\circ ) (Given).
- From Step 1 and Step 2, both equations equal 180°.
- ( m∠1 + m∠2 = m∠2 + m∠3 ).
- Subtract ( m∠2 ) from both sides:
- ( m∠1 = m∠3 ).
- By the definition of congruent angles, if their measures are equal, then the angles are congruent:
- ( ∠1 ≅ ∠3 ).
Explanation:
This proof demonstrates the relationship between angles in a linear pair and supplementary angles. A linear pair of angles occurs when two angles are adjacent and their non-common sides form a straight line. In this case, the sum of ( m∠1 ) and ( m∠2 ) is ( 180^\circ ), as a straight line measures ( 180^\circ ).
We also know from the given information that ( m∠2 + m∠3 = 180^\circ ), indicating that ( ∠2 ) and ( ∠3 ) are supplementary. Supplementary angles are two angles whose measures sum to ( 180^\circ ).
The key to the proof is realizing that both equations (Step 1 and Step 2) equal ( 180^\circ ). This allows us to set ( m∠1 + m∠2 ) equal to ( m∠2 + m∠3 ). By subtracting ( m∠2 ) (which is common to both equations), we isolate ( m∠1 = m∠3 ). Since equal angle measures indicate congruence, ( ∠1 ) is congruent to ( ∠3 ).
Thus, the proof relies on the properties of linear pairs and supplementary angles, as well as the transitive property of equality, to establish the congruence of ( ∠1 ) and ( ∠3 ).