Basic Math and Pre-Algebra
PART 3. The Shape of the World
CHAPTER 14. Quadrilaterals and Other Polygons
As you work your way deeper into geometry, you’ll begin to encounter more complex figures. We started with just lines, then angles, and in the last chapter, we worked with triangles. In this chapter we’ll begin looking at polygons-shapes with more sides and angles.
The primary focus in this chapter is on four-sided figures, and we’ll look at several different subgroups of the family of four-sided polygons. For each of the families, we’ll consider the special properties of sides, angles, and diagonals that belong to that family. After you’ve gotten to know all the members of the family, we’ll look at finding their perimeters and areas. Before moving on, we’ll explore polygons with even more sides and learn some interesting facts about them.
The term quadrilateral is used for any four-sided polygon, but most of the attention falls on the members of the family called parallelograms. The name parallelogram comes from the fact that these quadrilaterals are formed by parallel line segments. A parallelogram is a quadrilateral with two pairs of opposite sides parallel. Whenever you look at one pair of parallel sides, the other sides can be thought of as transversals. You know a bit about the angles that are formed when parallel lines are cut by a transversal, and you can see that you have consecutive interior angles that are supplementary.
A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel and congruent.
In parallelogram ABCD, and in any parallelogram, consecutive angles are supplementary. In ABCD, that means m∠A + m∠B = 180°, m∠B + m∠C = 180°, m∠C + m∠D = 180°, and m∠D + m∠A = 180°. If you do a little algebra, saying that m∠A + m∠B = m∠B + m∠C, and subtracting m∠B from both sides, you can show that LA and LC are the same size. In any parallelogram, opposite angles are congruent.
Draw a diagonal in any parallelogram and you form two triangles. Let’s draw diagonal in parallelogram ABCD. That will form ∆ACD and ∆CAB. Because alternate interior angles ∠DCA and ∠BAC are congruent. Because ∠DAC and ∠BCA are congruent. That gives you two pairs of congruent angles, and AC is between those angles in both triangles and equal to itself. If you rotate ∆CAB so that ∠BAC sits on top of ∠DCA and ∠BCA sits on top of ∠DAC, not only will the shared side match itself, but you’ll see matching and matching . That tells you that the opposite sides are not only parallel, but also congruent.
Drawing one diagonal in a parallelogram divides it into two matching triangles. When both diagonals are drawn in the parallelogram, it makes four triangles. If we call the point where the diagonals intersect point E, we can show that ∆ADE matches ∆CBE. AD = BC, because the opposite sides of the parallelogram are congruent. m∠ADE = m∠CBE and m∠DAE = m∠BCE because the opposite sides are parallel, so alternate interior angles are congruent. That tells you how to match up the parts of the triangles, and you’ll see the other parts match up as well. If the triangles are the same size and shape, DE = EB and AE = EC, so the diagonals of the parallelogram bisect each other.
The family of parallelograms is made up of many different types of parallelograms. Some have only the properties of parallelograms we’ve covered so far, but others are special in one or more ways.
For each quadrilateral described, decide if there is enough information to conclude that the quadrilateral is a parallelogram.
1. In quadrilateral ABCD, and
2. In quadrilateral PQRS, with diagonal ∠QRP ≅ ∠SPR and ∠QPR ≅ ∠SRP.
3. In quadrilateral FORK, ∠F ≅ ∠K and FO = RK.
4. In quadrilateral LAMP, with diagonals and intersecting at S, ∆ALS ≅ ∆PMS and ∆AMS ≅ ∆PLS.
5. In quadrilateral ETRA, with diagonals and intersecting at X, TX = RX and EX = AX.