Question

for each graph, decide if the two lines are parallel, perpendicular, or neither.graph 1graph 2graph 3graph 4parallelparallelparallelparallelperpendicularperpendicularperpendicularperpendicularneitherneitherneitherneither

Explanation:

Graph 1

• Step1: Check parallel condition

Parallel lines have the same slope. Visually, these lines do not have the same slope (they are not “running” in the same “steepness” direction).

• Step2: Check perpendicular condition

Perpendicular lines have slopes \(m_1\) and \(m_2\) such that \(m_1\times m_2=- 1\). Visually, the angle between them is not \(90^{\circ}\).

Graph 2

• Step1: Check angle

One line is horizontal (slope \(m_1 = 0\)) and the other is vertical (slope \(m_2\) is undefined in the traditional \(y = mx + b\) sense, but geometrically, the angle between a horizontal and a vertical line is \(90^{\circ}\).

Graph 3

• Step1: Check parallel condition

Parallel lines are lines in a plane that do not intersect. Visually, these two lines are “running” in the same direction and will never meet.

Graph 4

• Step1: Check angle

One line is vertical. Let's assume we can calculate slopes (for non - vertical line, pick two points \((x_1,y_1)\) and \((x_2,y_2)\) on the non - vertical line. If the non - vertical line has a slope \(m\) and the vertical line has an “infinite” slope in the geometric sense, the angle between them is not \(90^{\circ}\). But wait, no:

• Correct Step1: Re - evaluate

Wait, no. One line is vertical. Let's use the geometric definition. If we consider the direction of the non - vertical line. The vertical line and the non - vertical line: pick a point of intersection (by extending if necessary). Using the grid, we can see that they form a \(90^{\circ}\) angle.

Answer:

Graph 1: Neither
Graph 2: Perpendicular
Graph 3: Parallel
Graph 4: Perpendicular