Answer:
See below where I explain line by line.
Step-by-step explanation:
If two lines are parallel, their slopes are equal.
Look at line 1: Both lines have slope 1/2. Since both slopes are 1/2, the slopes are equal, and the lines are parallel.
Look at line 7: slope 4/8, slope 1/2. Reduce 4/8. It reduces to 1/2. Both lines have slope 1/2. The lines have equal slopes, so they are parallel.
If two lines are perpendicular, then their slopes are opposite reciprocals. Reciprocals are two numbers that when when written as a fraction, when you flip one, you get the other. For example, 4/5 and 5/4. They are reciprocals because is you flip 4/5, you get 5/4. For two lines to be perpendicular, their slopes need to be reciprocals and opposites. That means their signs must be different (one positive and one negative) in addition to being reciprocals. For example, two lines with slope 5/8 and -8/5 are perpendicular since the slopes are both reciprocals and opposites.
Look at line 2: -2/3 and 3/2. If you flip -2/3, you get -3/2. The opposite of -3/2 is 3/2. That means each slope is the opposite reciprocal of the other, so they are perpendicular.
Line 5: slope 4, slope -1/4. Write 4 as a fraction. Now you have 4/1. Flip 4/1 to get 1/4. Then make it the opposite, -1/4. The second line does have slope -1/4, so they are perpendicular.
Line 6: slope -1/6, slope 6. Flip -1/6. You get -6/1. Find its opposite: 6/1. 6/1 is the same as 6. That means that -1/6 and 6 are opposite reciprocals and the lines are perpendicular.
Lines that are neither parallel nor perpendicular.
Look at line 3: 5/2 and 2/5. If you flip 5/2, you get 2/5. The slopes are reciprocals, but they are not opposites since they are both positive. The lines are not perpendicular. Also 5/2 is not equal to 2/5, so the lines are not parallel either.
Look at line 4: slope -3; slope -1/3. Start with -3. Write it as a fraction: -3/1. Now flip it: -1/3. Now find its opposite: 1/3. If the slopes were -3 and 1/3, the lines would be perpendicular because they'd be reciprocal and opposite. The second slope is -1/3, not 1/3, so they are not opposite, and the lines are not perpendicular. Also, -3 and -1/3 are not equal, so the lines are also not parallel.
