Question

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

1. \\( \angle 1 \cong \angle 2 \\)
2. \\( m \angle 1 = m \angle 2 \\)
3. \\( m \angle 1 + m \angle 2 = 90 ^ { \circ } \\)
4. \\( m \angle 2 + m \angle 2 = 180 ^ { \circ } \\)
5. \\( 2 ( m \angle 2 ) = 180 ^ { \circ } \\)

Explanation:

Step1: Recall Linear Pair Postulate

By the Linear Pair Postulate, if two angles form a linear pair, then their measures add up to \(180^{\circ}\). So, \(m\angle1 + m\angle2=180^{\circ}\) (not \(90^{\circ}\) as wrongly written in the original).

Step2: Substitute using \(m\angle1 = m\angle2\)

Since \(m\angle1 = m\angle2\) (from the definition of congruent angles, given \(\angle1\cong\angle2\)), substitute \(m\angle1\) with \(m\angle2\) in the equation \(m\angle1 + m\angle2 = 180^{\circ}\). We get \(m\angle2+m\angle2=180^{\circ}\).

Step3: Simplify the equation

Using the distributive property \(a + a=2a\) (where \(a = m\angle2\)), the equation \(m\angle2+m\angle2=180^{\circ}\) becomes \(2(m\angle2)=180^{\circ}\).

Step4: Solve for \(m\angle2\)

Divide both sides of the equation \(2(m\angle2)=180^{\circ}\) by \(2\). So, \(m\angle2=\frac{180^{\circ}}{2}=90^{\circ}\).

Step5: Use the definition of perpendicular lines

If the measure of an angle formed by two lines is \(90^{\circ}\), then the two lines are perpendicular. Since \(m\angle2 = 90^{\circ}\) and \(\angle2\) is formed by lines \(g\) and \(h\), by the definition of perpendicular lines (\(g\perp h\) if the angle between them is \(90^{\circ}\)), we have \(g\perp h\).

Answer:

The corrected step - by - step proof is as follows:

STATEMENTS	REASONS
2. \(m\angle1 = m\angle2\)	2. Definition of congruence
3. \(m\angle1 + m\angle2=180^{\circ}\)	3. Linear Pair Postulate
4. \(m\angle2+m\angle2 = 180^{\circ}\)	4. Substitution Property of Equality
5. \(2(m\angle2)=180^{\circ}\)	5. Distributive Property
6. \(m\angle2 = 90^{\circ}\)	6. Division Property of Equality (\(m\angle2=\frac{180^{\circ}}{2}\))
7. \(g\perp h\)	7. Definition of perpendicular lines (if the measure of the angle between two lines is \(90^{\circ}\), the lines are perpendicular)