In This Chapter
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The properties of parallelograms•
Identifying special types of parallelograms•
The properties of trapezoids•
How to find the area and perimeter of quadrilaterals•
Polygons with more than four sidesAs 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.
Parallelograms
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.
DEFINITION
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 mA + mB =180r, mB + mC =180r, mC + mD =180r, and mD + mA =180r. If you do a little algebra, saying that mA + mB = mB + mC, and subtracting mB from both sides, you can show that A and C 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 AC in parallelogram ABCD. That will form 'ACD and 'CAB. Because DC AB, alternate interior angles DCA and BAC are congruent. Because AD BC, 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 AB matching CD and BC matching AD. That tells you that the opposite sides are not only parallel, but also congruent.
B
A
C
D
D C
B A
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. mADE = mCBE and mDAE = mBCE 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.
D C
B A
E
CHECK POINT
For each quadrilateral described, decide if there is enough information to conclude that the quadrilateral is a parallelogram.
1. In quadrilateral ABCD, AB CD and BC AD.
2. In quadrilateral PQRS, with diagonal PR, QRP # SPR and QPR # SRP.
3. In quadrilateral FORK, F # K and FO = RK.
4. In quadrilateral LAMP, with diagonals LM and AP intersecting at S, 'ALS # 'PMS and 'AMS # 'PLS.
5. In quadrilateral ETRA, with diagonals ER and TA intersecting at X, TX = RX and EX = AX.
Rectangles
A rectangle is a parallelogram in which adjacent sides are perpendicular, and so it has four right angles. Because the rectangle is a parallelogram, it has all the properties of a parallelogram, but also the special property that all four of the angles are the same size. It’s an equiangular parallelogram.
A rectangle is a parallelogram with four right angles.
You’ve probably worked with rectangles before, because there are so many rectangles all around us. You’re reading this on a rectangular page, and you could be sitting in a room built from rectangles.
Every rectangle is a parallelogram, so its opposite sides are parallel and congruent, its consecu-tive angles are supplementary (90r + 90r = 180r), and its opposite angles are congruent. Those are properties of every parallelogram. The rectangle also has consecutive congruent angles, because all angles are congruent.
In any parallelogram, a diagonal makes two congruent triangles. That’s still true in a rectangle, but those congruent triangles will both be right triangles. In any parallelogram, the diagonals bisect each other, and because the rectangle is a parallelogram, that’s still true. But the rectangle has another special property. The diagonals of a rectangle are congruent. In most parallelograms, which sort of slant to one side, there’s a long diagonal and a short one, but in a rectangle, both diagonals are the same length.
Suppose that in rectangle ABCD, the diagonals intersect at E. If BE = 8, how long is AE? Because the diagonals are congruent and bisect each other, AE = EC = BE = ED. So if BE = 8, AE = 8 as well.