Polygons Examples

All regular polygons are made of pairs of parallel line segments. True or false?

False. This applies only to regular polygons with an even number of sides. Polygons with an odd number of sides can't be made of pairs of parallel sides because, quite simply, odd numbers aren't divisible by 2.

A rectangle is an example of a regular quadrilateral. True or false?

This one's not so clear-cut. In general, we'll say false because while a rectangle has four congruent angles, it doesn't have to have four congruent sides. Only special rectangles called squares (have you heard of them?) are regular quadrilaterals because they have four congruent angles and four congruent sides. So while rectangles might be regular quadrilaterals (if they're squares), they aren't always.

What is the maximum number of perpendicular intersections possible in a pentagon?

The easiest way to answer this question is probably to draw out a bunch of pentagons trying to stick in as many right angles as possible. The more right angles we have, the more perpendicular line segments there are.

As you can see we can only include a maximum of 3 right angles, which means 3 perpendicular intersections. We can't add any more, or else we'll have a quadrilateral.

We can look at this a little more algebraically too, for you variable-minded folks. If the total interior angles in a polygon with n sides is 180(n – 2), a pentagon would need to have an interior angle sum of 540°. Three right angles already add up to 270°, so let's focus on the other two angles.

If we take away 270° from 540°, we'll have 270° left over. The last two angles would have to split 270° among themselves. So if one angle is 135°, the other will have to be 135°. If one is 269°, the other will be 1°. If one is 90°, the other will be 180°.

Hold on. But 180° is just a straight line. And that's exactly why we can't have four 90° angles: a pentagon with four right angles means the fifth angle will be a straight line. So yeah, a pentagon with four 90° angles is a rectangle. Pretty crazy, right?