Parallel lines with a transversal

Check out Figure 6-1, which shows three lines that kind of resemble a giant not-equal sign. The two horizontal lines are parallel, and the third line that crosses them is called a transversal. As you can see, the three lines form eight angles.

The eight angles formed by parallel lines and a transversal are either congruent or supplementary. The following theorems tell you how various pairs of angles relate to each other.

Proving that angles are congruent: If a transversal intersects two parallel lines, then the following angles are congruent:

• Alternate interior angles: The pair of angles 3 and 6 (as well as 4 and 5) are alternate interior angles. These angle pairs are on opposite (alternate) sides of the transversal and are in between (in the interior of) the parallel lines.
• Alternate exterior angles: Angles 1 and 8 (and angles 2 and 7) are called alternate exterior angles.
• Corresponding angles: The pair of angles 1 and 5 (also 2 and 6, 3 and 7, and 4 and 8) are corresponding angles. Angles 1 and 5 are corresponding because each is in the same position (the upper left-hand corner) in its group of four angles.

Proving that angles are supplementary: If a transversal intersects two parallel lines, then the following angles are supplementary (see Figure 6-1 again):

• Same-side interior angles: Angles 3 and 5 (and 4 and 6) are on the same side of the transversal and are in the interior of the parallel lines, so they’re called (ready for a shock?) same-side interior angles.
• Same-side exterior angles: Angles 1 and 7 (and 2 and 8) are called same-side exterior angles.

You can sum up the definitions and theorems above with the following simple idea. When you have two parallel lines cut by a transversal, you get four acute angles and four obtuse angles (except when you get eight right angles). All the acute angles are congruent, all the obtuse angles are congruent, and each acute angle is supplementary to each obtuse angle.

Proving that lines are parallel: All the theorems in this section work in reverse. You can use the following theorems to prove that lines are parallel. That is, two lines are parallel if they’re cut by a transversal such that

• Two corresponding angles are congruent.
• Two alternate interior angles are congruent.
• Two alternate exterior angles are congruent.
• Two same-side interior angles are supplementary.
• Two same-side exterior angles are supplementary.

The transversal theorems

Let’s take a look at some of the theorems in action: Given that lines m and n are parallel, find the measure of .

The angle and the angle are alternate exterior angles and are therefore congruent. Set them equal to each other and solve for x:

This equation has two solutions, so take them one at a time and plug them into the x’s in the alternate exterior angles. Plugging into gives you for that angle. And because is its supplement, must be , or . The solution gives you for the angle and for . So and are your answers for .

When you get two solutions (such as and ) in a problem like this, you do not plug one of them into one of the x’s (like ) and the other solution into the other x (like ). You have to plug one of the solutions into all x’s, giving you one result for both angles ( and ); then you have to separately plug the other solution into all x’s, giving you a second result for both angles ( and ).

Angles and segments can’t have measures or lengths that are zero or negative. Make sure that each solution for x produces positive answers for all the angles or segments in a problem. (In the preceding problem, you should check both the angle and the angle with each solution for x.) If a solution makes any angle or segment in the diagram zero or negative, it must be rejected even if the angles or segments you care about end up being positive. However, do not reject a solution just because x is zero or negative: x can be zero or negative as long as the angles and segments are positive (, for example, works just fine in the problem above).

Now here’s a proof that uses some of the transversal theorems:

You may want to extend the lines in transversal problems. Doing so can help you see how the angles are related.

For instance, if you have a hard time seeing that and are indeed alternate interior angles (for step 3 of the proof), rotate the figure (or tilt your head) until the parallel segments and are horizontal; then extend ,, and in both directions, turning them into lines (you know, with the little arrows). After doing that, you’re looking at the familiar parallel-line scheme shown in Figure 6-1.

Properties of the parallelogram

The parallelogram has the following properties:

• Opposite sides are parallel by definition.
• Opposite sides are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.

If you just look at a parallelogram, the things that look true (namely, the things on this list) are true and are thus properties, and the things that don’t look like they’re true aren’t properties.

If you draw a picture to help you figure out a quadrilateral’s properties, make your sketch as general as possible. For instance, as you sketch your parallelogram, make sure it’s not almost a rhombus (with four sides that are almost congruent) or almost a rectangle (with four angles close to right angles).

Imagine that you can’t remember the properties of a parallelogram. You could just sketch one (as in Figure 6-6) and run through all things that might be properties.

Tables 6-1, 6-2, and 6-3 concern questions about what might or might not be properties of a parallelogram (refer to Figure 6-6).

Look at your sketch carefully. When I show students a parallelogram like the one in Figure 6-6 and ask them whether the diagonals look congruent, they often tell me that they do despite the fact that one is literally twice as long as the other! So when asking yourself whether a potential property looks true, don’t just take a quick glance at the quadrilateral, and don’t let your eyes play tricks on you. Look at the segments or angles in question very carefully.

The sketching-quadrilaterals method and the questions in the preceding three tables bring me to an important tip about mathematics in general.

Whenever possible, bolster your memorization of rules, formulas, concepts, and so on by trying to see why they’re true or why they make sense. Not only does this make the ideas easier to learn, but it also helps you see connections to other ideas, and that fosters a deeper understanding of mathematics.

Properties of the three special parallelograms

Figure 6-7 shows you the three special parallelograms, so-called because they’re, as mathematicians say, special cases of the parallelogram. (In addition, the square is a special case or type of both the rectangle and the rhombus.) The three-level hierarchy you see with parallelogram rectangle square or parallelogram rhombus square in the quadrilateral family tree (Figure 6-2) works just like mammal dog Dalmatian. A dog is a special type of a mammal, and a Dalmatian is a special type of a dog.

Before reading the properties that follow, try figuring them out on your own. Using the shapes in Figure 6-7, run down the list of possible properties from the beginning of “The Properties of Quadrilaterals,” asking yourself whether they look like they’re true for the rhombus, the rectangle, and the square.

Here are the properties of the rhombus, rectangle, and square. Note that because these three quadrilaterals are all parallelograms, their properties include the parallelogram properties.

• The rhombus has the following properties:
  • All the properties of a parallelogram apply.
  • All sides are congruent by definition.
  • The diagonals bisect the angles.
  • The diagonals are perpendicular bisectors of each other.
• The rectangle has the following properties:
  • All the properties of a parallelogram apply.
  • All angles are right angles by definition.
  • The diagonals are congruent.
• The square has the following properties:
  • All the properties of a rhombus apply.
  • All the properties of a rectangle apply.
  • All sides are congruent by definition.
  • All angles are right angles by definition.

Let’s try a problem: Find the perimeter of rhombus RHOM.

Here’s the solution: All the sides of a rhombus are congruent, so HO equals . And because the diagonals of a rhombus are perpendicular, is a right triangle. You finish with the Pythagorean Theorem:

You can reject because that would result in having legs with lengths of and 0. So x equals 3, which gives a length of 5. Because rhombuses have four congruent sides, RHOM has a perimeter of , or 20 units.

Properties of the kite

Check out the kite in Figure 6-8 and try to figure out its properties before reading the list that follows.

The properties of the kite are as follows:

• Two disjoint pairs of consecutive sides are congruent by definition ( and ). Note: Disjoint means that one side can’t be used in both pairs — the two pairs are totally separate.
• The diagonals are perpendicular.
• One diagonal (, the main diagonal) is the perpendicular bisector of the other diagonal (, the cross diagonal). (The terms “main diagonal” and “cross diagonal” are quite useful, but don’t look for them in other geometry books because I made them up.)
• The main diagonal bisects a pair of opposite angles ( and ).
• The opposite angles at the endpoints of the cross diagonal are congruent ( and ).

The last three properties are called the half properties of the kite.

Grab an energy drink and get ready for another proof. Due to space considerations, I’m going to skip the game plan this time. You’re on your own — egad!

Properties of the trapezoid and the isosceles trapezoid

Practice your picking-out-properties proficiency with the trapezoid and isosceles trapezoid in Figure 6-9. Remember: What looks true is likely true, and what doesn’t, isn’t.

• The properties of the trapezoid are as follows:
  • The bases are parallel by definition.
  • Each lower base angle is supplementary to the upper base angle on the same side.
• The properties of the isosceles trapezoids are as follows:
  • The properties of trapezoids apply by definition.
  • The legs are congruent by definition.
  • The lower base angles are congruent.
  • The upper base angles are congruent.
  • Any lower base angle is supplementary to any upper base angle.
  • The diagonals are congruent.

Proving you’ve got a parallelogram

The five methods for proving that a quadrilateral is a parallelogram are among the most important proof methods in this section. One reason they’re important is that you often have to prove that a quadrilateral is a parallelogram before going on to prove that it’s one of the special parallelograms (a rectangle, a rhombus, or a square).

Five ways to prove that a quadrilateral is a parallelogram: There are five ways to prove that a quadrilateral is a parallelogram. The first four are the converses of parallelogram properties. Make sure you remember the oddball fifth one — which isn’t the converse of a property — because it often comes in handy:

• If both pairs of opposite sides of a quadrilateral are parallel, then it’s a parallelogram (reverse of the parallelogram definition).
• If both pairs of opposite sides of a quadrilateral are congruent, then it’s a parallelogram.
• If both pairs of opposite angles of a quadrilateral are congruent, then it’s a parallelogram.
• If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram.
• If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it’s a parallelogram.

Here’s a proof to give you some practice with one of the parallelogram proof methods.

Because all quadrilaterals (except for the kite) contain parallel lines, be on the lookout for opportunities to use the parallel-line theorems from early in this chapter. And always keep your eyes peeled for congruent triangles.

Your game plan might go something like this:

• Look for congruent triangles. This diagram takes the cake for containing congruent triangles — it has six pairs of them! Don’t spend much time thinking about them — except the ones that might help you — but at least make a quick mental note that they’re there.
• Consider the givens. The given congruent angles, which are parts of and , are a huge hint that you should try to show these triangles congruent. You have those congruent angles and the congruent sides and from parallelogram HEJG so you need only one more pair of congruent sides or angles to use SAS or ASA.

• Think about the end of the proof. To prove that DEFG is a parallelogram, it would help to know that so you’d like to be able to prove the triangles congruent and then get by CPCTC. That eliminates the SAS option for proving the triangles congruent because to use SAS, you’d need to know that — the very thing you’re trying to get with CPCTC. (And if you knew there’d be no point to showing that the triangles are congruent, anyway.) So you should try the other option: proving the triangles congruent with ASA.

  The second angle pair you’d need for ASA consists of and . They’re congruent because they’re alternate exterior angles using parallel lines and and transversal . Okay, so the triangles are congruent by ASA, and then you get by CPCTC. You’re on your way.

• Consider parallelogram proof methods. You now have one pair of congruent sides of DEFG. Two of the parallelogram proof methods use a pair of congruent sides. To complete one or the other of these methods, you need to show one of the following:

  • That the other pair of opposite sides are congruent
  • That and are parallel as well as congruent

  Ask yourself which approach looks easier or quicker. Showing would probably require showing a second pair of triangles congruent, and that looks like it’d take a few more steps, so try the other tack.

  Can you show ? Sure, with one of the parallel-line theorems. Because angles GDH and EFJ are congruent (by CPCTC), you can finish by using those angles as congruent alternate interior angles, or Z-angles, to give you . That’s a wrap!

Now take a look at the formal proof:

Proving that you’ve got a rectangle, rhombus, or square

Three ways to prove that a quadrilateral is a rectangle: Note that the second and third methods require that you first show (or be given) that the quadrilateral in question is a parallelogram:

• If all angles in a quadrilateral are right angles, then it’s a rectangle (reverse of the rectangle definition).
• If the diagonals of a parallelogram are congruent, then it’s a rectangle.
• If a parallelogram contains a right angle, then it’s a rectangle.

Six ways to prove that a quadrilateral is a rhombus: You can use the following six methods to prove that a quadrilateral is a rhombus. The last three methods require that you first show (or be given) that you’ve got a parallelogram:

• If all sides of a quadrilateral are congruent, then it’s a rhombus (reverse of the rhombus definition).
• If the diagonals of a quadrilateral bisect all the angles, then it’s a rhombus.
• If the diagonals of a quadrilateral are perpendicular bisectors of each other, then it’s a rhombus.
• If two consecutive sides of a parallelogram are congruent, then it’s a rhombus.
• If either diagonal of a parallelogram bisects two angles, then it’s a rhombus.
• If the diagonals of a parallelogram are perpendicular, then it’s a rhombus.

Four methods to prove that a quadrilateral is a square: In the last three of these methods, you first have to prove (or be given) that the quadrilateral is a rectangle, rhombus, or both:

• If a quadrilateral has four congruent sides and four right angles, then it’s a square (reverse of the square definition).
• If two consecutive sides of a rectangle are congruent, then it’s a square.
• If a rhombus contains a right angle, then it’s a square.
• If a quadrilateral is both a rectangle and a rhombus, then it’s a square.

Proving that you’ve got a kite

Two ways to prove that a quadrilateral is a kite: Proving that a quadrilateral is a kite is pretty easy. Usually, all you have to do is use congruent triangles or isosceles triangles:

• If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it’s a kite (reverse of the kite definition).
• If one of the diagonals of a quadrilateral is the perpendicular bisector of the other, then it’s a kite.

When you’re trying to prove that a quadrilateral is a kite, the following tips may come in handy:

• Check the diagram for congruent triangles. Don’t fail to spot triangles that look congruent and to consider how using CPCTC might help you.
• Keep the first equidistance theorem in mind. When you have to prove that a quadrilateral is a kite, you might have to use the equidistance theorem in which two points determine a perpendicular bisector.
• Draw in diagonals. One of the methods for proving that a quadrilateral is a kite involves diagonals, so if the diagram lacks either of the two diagonals, try drawing in one or both of them.

Now get ready for a proof:

Here’s how your game plan might work for this proof.

• Note that one of the kite’s diagonals is missing. Draw in the missing diagonal, .
• Check the diagram for congruent triangles. After drawing in , there are six pairs of congruent triangles. The two triangles most likely to help you are and .
• Prove the triangles congruent. You can use ASA.
• Use the equidistance theorem. Use CPCTC with and to get and . Then, using the equidistance theorem, those two pairs of congruent sides (with points R and H) determine the perpendicular bisector of the diagonal you drew in. Over and out.

I’m skipping the formal proof, but give it a try. It should take you about 12 steps.