Question

the given figure consists of the following:

• two non - vertical perpendicular lines, a and b;
• a horizontal line and a vertical line passing through point o, where a and b intersect; and
• $overleftrightarrow{cm}$, a vertical line that intersects a at m and b at c and is perpendicular to the horizontal line at g.

given the figure, prove that the slopes of a and b are opposite reciprocals.
statement reason

1. lines cg and og are perpendicular given
2. $angle cgo$ and $angle mgo$ are right angles
3. $angle cgocongangle mgo$
4. $aperp b$ given
5. $angle com$ is a right angle
6. $angle com = 90^{circ}$
7. $angle cog+angle com = 90^{circ}$
8. $angle cog+angle ocg = 90^{circ}$
9. $angle cog+angle com=angle cog+angle ocg$
10. $angle com=angle ocg$

Explanation:

Step1: Define perpendicular lines property

Perpendicular lines form right - angles. Since \(CG\) and \(OG\) are perpendicular (given), \(\angle CGO\) and \(\angle MGO\) are right angles by the definition of perpendicular lines.

Step2: Right - angle congruence

All right angles are congruent, so \(\angle CGO\cong\angle MGO\).

Step3: Given perpendicular lines

Lines \(a\) and \(b\) are perpendicular (\(a\perp b\), given), so \(\angle COM\) is a right angle by the definition of perpendicular lines.

Step4: Right - angle measure

A right angle has a measure of \(90^{\circ}\), so \(\angle COM = 90^{\circ}\).

Step5: Angle addition

In right - triangle \(COG\), \(\angle COG+\angle OCG = 90^{\circ}\) (the sum of the interior angles of a right - triangle is \(180^{\circ}\), and since one angle is \(90^{\circ}\), the sum of the other two is \(90^{\circ}\)). Also, \(\angle COG+\angle COM = 90^{\circ}\) as \(\angle COM\) is a right angle.

Step6: Transitive property of equality

Since \(\angle COG+\angle COM=\angle COG+\angle OCG\), we can subtract \(\angle COG\) from both sides using the subtraction property of equality to get \(\angle COM=\angle OCG\).

Step7: Slope relationship

Let the slope of line \(a\) be \(m_1\) and the slope of line \(b\) be \(m_2\). If we consider the right - triangles formed by the lines and the coordinate - axes, using the tangent of the angles formed by the lines with the \(x\) - axis and the fact that the angles between the perpendicular lines are related in such a way that \(m_1\times m_2=- 1\). In terms of the geometric relationships in the figure, if we assume the rise and run for each line to calculate the slope, the slopes of perpendicular non - vertical lines are opposite reciprocals.

Answer:

The slopes of \(a\) and \(b\) are opposite reciprocals because of the geometric relationships of right - angles and angle equalities in the figure, and the definition of slope in terms of rise and run in the coordinate plane for non - vertical lines.