Question

statement
1
2
3
4
∠3≅∠4
∠1 and ∠3 form a linear pair
∠2 and ∠4 form a linear pair
∠1 and ∠3 are supplementary
∠2 and ∠4 are supplementary
∠1≅∠2
given
reflexive property
transitive property
substitution property
definition of congruent angles
angle addition property
definition of supplementary angles
definition of complementary angles
definition of linear pair
select
transitive property

Explanation:

Step1: Analyze the given information

We know that \(\angle3\cong\angle4\) (Given).

Step2: Recall the definition of a linear pair

If two angles form a linear pair, then they are supplementary. Since \(\angle1\) and \(\angle3\) form a linear pair, \(\angle1 + \angle3=180^{\circ}\). Similarly, since \(\angle2\) and \(\angle4\) form a linear pair, \(\angle2+\angle4 = 180^{\circ}\) (Definition of Linear Pair).

Step3: Use the substitution property

Because \(\angle3\cong\angle4\), we can substitute \(\angle3\) with \(\angle4\) in the equation \(\angle1+\angle3 = 180^{\circ}\). So \(\angle1+\angle4=180^{\circ}\).

Step4: Apply the transitive property

We have \(\angle2+\angle4=180^{\circ}\) and \(\angle1+\angle4 = 180^{\circ}\). By the transitive property of equality (if \(a + c=b + c\), then \(a = b\)), we can conclude that \(\angle1\cong\angle2\).

Answer:

1. Given
2. Definition of Linear Pair
3. Definition of Supplementary Angles
4. Transitive Property