Question

complete the two - columns proof.
given: x||y
prove: ∠3≅∠5

1. m∠3 + m∠5 = m∠3 + m∠7
2. m∠3 + m∠5 = m∠7
3. m∠3 + m∠7 = m∠5 + m∠7

given
4)
same - side interior angle postulate
linear pair theorem
transitive property of equality
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Explanation:

Step1: Recall given information

We are given some angle - related statements and need to complete the proof.

Step2: Analyze the Transitive Property of Equality

The Transitive Property of Equality states that if \(a = b\) and \(b = c\), then \(a = c\). In the context of angle - measures, if \(m\angle3 + m\angle5=m\angle3 + m\angle7\) and we know some other equalities related to these angles, we can use the property. Since we want to prove \(\angle2\cong\angle5\), and we have angle - sum relationships, we note that if \(m\angle3 + m\angle5=m\angle3 + m\angle7\), we can subtract \(m\angle3\) from both sides of the equation.

Step3: Simplify the equation

By the Subtraction Property of Equality, if \(m\angle3 + m\angle5=m\angle3 + m\angle7\), then \(m\angle5=m\angle7\). Also, if we consider the angle - relationships in the parallel lines (assuming the lines are parallel based on the postulates and theorems mentioned), and we know that \(\angle2\) and \(\angle7\) are corresponding angles (if the lines are parallel), and since \(m\angle5=m\angle7\), we can conclude that \(m\angle2 = m\angle5\) (using corresponding angles and the equality of \(\angle5\) and \(\angle7\)).

Answer:

Since \(m\angle3 + m\angle5=m\angle3 + m\angle7\), by the Subtraction Property of Equality, \(m\angle5=m\angle7\). If the lines are parallel, \(\angle2\) and \(\angle7\) are corresponding angles, so \(m\angle2 = m\angle7\). Then, by the Transitive Property of Equality, \(m\angle2 = m\angle5\), so \(\angle2\cong\angle5\).