Question

complete the two - column proof of the linear pair perpendicular theorem.
given \\( \angle 1 \cong \angle 2 \\)
prove \\( g \perp h \\)

statements	reasons
2. \\( m\angle 1=m\angle 2 \\)	2. definition of congruence
3. \\( m\angle 1 + m\angle 2=180^{\circ} \\)	3. linear pair postulate
4. \\( m\angle 2 + m\angle 2=180^{\circ} \\)	4. substitution property of equality
5. \\( 2(m\angle 2)=180^{\circ} \\)	5. distributive property
6. \\( m\angle 2 = 90^{\circ} \\)	6. division property of equality
7. \\( m\angle 1=90^{\circ} \\)	7.
8. \\( g \perp h \\)	8.

Explanation:

Step1: Find reason for \(m\angle1 = 90^{\circ}\)

Since \(m\angle1=m\angle2\) (from statement 2) and \(m\angle2 = 90^{\circ}\) (from statement 6), by the Transitive Property of Equality (if \(a = b\) and \(b=c\), then \(a = c\)), we can substitute \(b\) with \(m\angle2\), \(a\) with \(m\angle1\) and \(c\) with \(90^{\circ}\).

Step2: Find reason for \(g\perp h\)

By the Definition of Perpendicular Lines (two lines are perpendicular if they intersect to form a right - angle, i.e., an angle of measure \(90^{\circ}\)). Since \(m\angle1=90^{\circ}\) (or \(m\angle2 = 90^{\circ}\)) and \(\angle1\) (or \(\angle2\)) is the angle formed by the intersection of lines \(g\) and \(h\).

Answer:

1. Transitive Property of Equality
2. Definition of Perpendicular Lines