Lesson 5-5: Properties of Trapezoids (Day 55)

Today I subbed in an eighth grade special ed English class. This is my third visit to this class -- I described my first visit in my September 12th post. My second visit was on October 14th, but that was the day that my old district was closed, hence it was a non-posting day.

As I wrote in my September 12th post, there are three periods with an aide (one Study Skills and the two eighth grade English classes) and two with a co-teacher (both seventh grade English). Thus, as usual, there is no "A Day in the Life" today.

Of course, today is Halloween. The eighth graders had a special Halloween assignment where they read an article about spiders, answer questions, and then color a spider where the correct colors are determined by their answers.

In my September 12th post, I write that the seventh graders were reading some short story I'd never heard of, titled "Lambert." (But it had nothing to do with quadrilaterals with three right angles.) In today's seventh grade classes, they take a quiz on another short story. I don't remember anything about it.

The eighth graders also watch the following video about a ghost ship:

I continue to hand out candy and pencils for the holidays. I don't always sing music for classes when there is an aide or co-teacher, but I can't resist singing "Ghost of a Chance" today. I start singing in the Study Skills class, then continue again in the English classes when students from the Study Skills class requests the song again.

Some students are dressed in pajamas. This is part of Red-Ribbon "Week." Actually, Red-Ribbon Week is supposed to be the last full week in October -- but some interpret this to mean that the entire week from Monday-Friday must be in October, while others interpret this to mean that Halloween must be included in the week. Still others take advantage and consider the last eleven days of October this year, from the 21st to today, to be Red-Ribbon "Week." (A week "with" eleven days -- hmm, they must be using the Eleven Calendar!)

Chapter 12 of Ian Stewart's The Story of Mathematics is called "Impossible Triangles: Is Euclid's Geometry the Only One?" Here's how it begins:

"Calculus was based on geometric principles, but the geometry was reduced to symbolic calculations, which were then formalized as analysis."

This chapter is all about one of my favorite topics -- different geometries, including both spherical and projective geometry. Stewart writes:

"The discovery that Euclid was not alone, that there can exist logically consistent types of geometry in which many of Euclid's theorems fail to hold, emerged from a renewed interest in the logical foundations of geometry, debated and developed from the middle of the 18th century to the middle of the 19th."

And so we begin with a little history:

"As far as Europe was concerned, geometry was becalmed in the doldrums between the years 300 and 1600."

But one application of different geometries was in Renaissance art. Stewart continues:

"Ordinary Euclidean geometry is about features that remain unchanged by rigid motions -- lengths, angles."

And of course, by "rigid motions" the author means "isometries" -- Common Core transformations such as reflections, rotations, and translations. (He reminds us that Euclid himself used congruent triangles -- and it's still a debate today whether to use isometries or congruent triangles.) On the other hand, projective geometry is different. Parallel lines appear to intersect at the horizon:

"They behave like this on an ideal infinite plane, not just on a slightly rounded Earth. In fact, they only behave exactly like this on a plane."

Stewart mentions a projective theorem, proved by Desargues in 1648. (I blogged about this theorem more than two years ago.) It can be proved by considering projections of 3D figures onto the plane:

"So we can use Euclidean methods to prove projective theorems. Projective geometry differs from Euclidean geometry as far as its viewpoint goes (pun intended)."

On the other hand, non-Euclidean geometry is different. One mathematician who worked with Euclid's axioms should be familiar to us -- Adrien Legendre. I spent two summers on the blog focusing on Legendre's work. He proved that Euclid's Fifth Postulate was equivalent to the existence of similar triangles:

"But he, and most other mathematicians, wanted something even more intuitive. In fact, there was a feeling that the Fifth Postulate was simply superfluous -- a consequence of the other axioms."

By the way, it's only fitting that I'm writing about Euclidean and non-Euclidean geometry on the day that I'm posting Lesson 5-5. During the first two years of this blog, I considered Lesson 5-5 to be the first lesson that required the Fifth Postulate, since it's all about trapezoids, which, of course, have a pair of parallel sides. (I had intentionally delayed Lesson 3-5, on parallel lines, until covering Lesson 5-5 back then.) Two lessons later is a brief mention of non-Euclidean geometry in our text.

(And so I'm also adding the "Spherical Geometry" label to this post.)

Indeed, Lesson 5-7 is on Triangle-Sum, and Legendre was the one who proved that the defect of a triangle (180 minus the angle sum) is proportional to its area:

"This seemed promising: if he could construct a triangle whose sides were twice those of a given triangle, but with the same angles, then he would obtain a contradiction, because the larger triangle would not have the same area as the smaller one."

But of course, he failed to prove the Fifth Postulate. We move on to another familiar name, Gerolamo Saccheri, who tried to use the quadrilaterals named after him to prove the Fifth Postulate. (Recall that a Saccheri quadrilateral is defined as one with two base angles that are right, and two congruent sides that are opposite each other and each adjacent to one base angle.) Indeed, he considered Saccheri quads whose summit angles are acute:

"Eventually he proved a rather complicated theorem about a family of lines all passing through one point, which implied that two of these lines would have a common perpendicular at infinity."

As it turned out, this wasn't really a contradiction. Well, since we already mentioned Saccheri and his quadrilaterals, of course we're going to mention Lambert and his quadrilaterals. (We know Lambert quadrilaterals to be those with at least three right angles.) He also proved that if the fourth angle of a Lambert quad is acute, something interesting happens with the angles of any polygon:

"Add all the angles, and subtract this from 2n - 4 right angles: the result is proportional to the polygon's area."

This is, of course, the defect of the polygon. The author also mentions a result of spherical geometry that was known to Lambert -- the excess, rather than the defect, of a spherical triangle or other polygon is proportional to the area. (On the sphere, the summit angles of a Saccheri quad and the fourth angle of a Lambert quad are obtuse, but we consider the acute case as well.)

By now, mathematicians were starting to suspect that Euclid's Fifth Postulate is independent of the other four postulates. We consider Nikolai Lobachevsky:

"By 1840 Lobachevsky was publishing a book on the topic, in which he complained about the lack of interest."

Even though Stewart doesn't give all the details, we already know them. Eventually, two new types of non-Euclidean geometry were discovered -- spherical (elliptic) and hyperbolic geometry. Indeed, hyperbolic geometry is the one where the summit angles of a Saccheri quad, and the fourth angle of a Lambert quad, are acute:

"It has numerous intriguing features, which distinguish it from Euclidean geometry. If the surface has zero curvature, like a Euclidean plane, then it is the Euclidean plane, and we get Euclidean geometry."

Non-Euclidean geometry led scientists to wonder, what is the shape of the universe? Before other geometries were discovered, it was naturally assumed that space was Euclidean:

"What else could it be? This question ceased to be rhetorical when logically consistent alternatives to Euclid's geometry began to appear."

And of course, the answer to that question is still unknown:

"Thanks to some imaginative and unorthodox thinking, often viciously contested by a less imaginative majority, it is now understood -- by mathematicians and physicists, at least -- that there are many alternatives to Euclid's geometry, and that the nature of physical space is a question for observation, not thought alone."

On that note, Stewart concludes the chapter as follows:

"For that matter, much of mathematics bears no obvious relation to reality at all -- but is useful, all the same."

Unlike previous chapters, this one has no biographical sidebars -- only what non-Euclidean geometry did for them and what it does for us.

I found a link to a puzzle about Halloween and spherical geometry (on a pumpkin, of course):

https://fivethirtyeight.com/features/can-you-carve-the-perfect-pumpkin/

I want to carve the perfect eye for my pumpkin this Halloween, but I can’t seem to make it the right size. Since symmetry is the key to beauty, the triangular eye should be equilateral and equiangular, which is easier said than done on the surface of a spherical pumpkin!

Anyway, the triangular eye that I made is way too big. Its sides all meet at right angles, and the resulting eye takes up a whole eighth of the pumpkin’s surface.

Instead, I want an eye that’s precisely half that size, or one-sixteenth of the pumpkin’s surface. For such an ideal pumpkin eye, at what angle should each of the sides meet?

Lambert's theorem about the spherical excess is the key to solving this puzzle. Try it!

By the way, there's been another recent Numberphile video on non-Euclidean geometry. But first, let's look at the definition of rectangle:

• A rectangle is a four-sided polygon, all of whose angles are right.

We know that rectangles, as defined above, exist only in Euclidean geometry. But there are several generalizations that do exist in non-Euclidean (either spherical or hyperbolic) geometry:

1. Eliminate right: A rectangle is a four-sided polygon, all of whose angles are congruent.
2. Eliminate all: A rectangle is a four-sided polygon, most of whose angles are right.

Choice 1) refers to equiangular quads, while choice 2) leads to Lambert quads. It can be proved that a Saccheri quad is a compromise between 1) and 2) -- it is a four-sided polygon with two right base angles and two congruent (not necessarily right) summit angles.

But there is a fourth possible definition as well:

• Eliminate four: A rectangle is a polygon, all of whose angles are right.

This is the definition implied by the following Numberphile video. (Actually it discusses squares, but we can define these as equilateral rectangles, whatever those are.)

Notice that by this fourth definition, a rectangular n-gon has n right angles. We already stated above that the angles of a Euclidean n-gon has 2n - 4 right angles, and the difference between 2n - 4 and the actual angle sum of a non-Euclidean n-gon is the defect (or excess) of the n-gon.

It's easy to solve 2n - 4 > n (defect), 2n - 4 = n (Euclidean), and 2n - 4 < n (excess) to show that it's a defect (hyperbolic) when n > 4, Euclidean when n = 4, and an excess (spherical) when n < 2. This shows us how many sides a "rectangle" can have in each of the three geometries.

We already know that a Euclidean rectangle can only have four sides. In the spherical case, at n = 3, there is a three-sided rectangle -- this is our well-known triangle with three right angles (the much too big pumpkin eye in the puzzle above). Each side of this triangle is a quadrant of the sphere. Notice that n = 2 case also makes sense on the sphere -- it is a 90-degree lune. Two of the n = 3 triangles can be glued together to create this lune.

The video shows us n = 5 for hyperbolic geometry. Not all five-sided hyperbolic rectangles are squares, though. We can glue two of these square pentagons to form n = 6, a six-sided rectangle. Of course, the resulting hexagon won't be square, since two of the sides would be twice as long as the other four. But I believe we can adjust the sides to form a square hexagon. All rectangular hexagons (square or not) have the same area, since they all have the same defect (two right angles).

I believe it's possible to glue together two square pentagons to form a hexagon, two hexagons to form an octagon, and two octagons to form a dodecagon (12-gon), in such a way that the resulting dodecagon is square (with all sides twice as long as those of the original pentagon).

The Numberphile video is the first time I've seen the square/rectangle definition loosened so that they no longer have to be quadrilaterals. The "equiangular" definition 1) is the only other loosening I've ever seen in a text. We don't need to loosen "rectangle" to mean a Lambert or Saccheri quad because they already have the established names, Lambert or Saccheri quad.

Lesson 5-5 of the U of Chicago text is called "Properties of Trapezoids." In the modern Third Edition of the text, trapezoids appear in Lesson 6-6.

This is what I wrote last year in describing this worksheet:

The first theorem that we have is the Isosceles Trapezoid Symmetry Theorem:
The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and is a symmetry line for the trapezoid.

Notice that this theorem is similar to the Kite Symmetry Theorem. In many ways, there is a sort of dualism between the kite and the isosceles trapezoid. A kite is defined by having consecutive equal sides, while an isosceles trapezoid has consecutive equal angles. The symmetry line for the kite bisects two of the angles, and the symmetry line for the isosceles trapezoid bisects two of the sides.

Here is the proof of the Isosceles Trapezoid Symmetry Theorem as given by the U of Chicago. This time, since Section 5-5 gives a paragraph proof, let me post a two-column proof here on the blog.

Given: ZOID is an isosceles trapezoid with angles I and D equal in measure.
           m is the perpendicular bisector of ID
Prove: m is the perpendicular bisector of ZO
           m is a symmetry line for ZOID

Proof:
Statements                           Reasons
1. m is the perp. bis. of ID  1. Given
2. D' = I, I' = D                    2. Definition of reflection
3. angle I = angle D               3. Given
4. ray DZ reflected is IO      4. Reflections preserve angle measure.
5. Z' lies on ray IO               5. Figure Reflection Theorem
6. ZOID is a trapezoid         6. Given
7. ZO | | DI                           7. Definition of trapezoid
8. ZO perpendicular to m     8. Fifth Postulate
9. Z' lies on line ZO              9. Definition of reflection
10. Z' = O                             10. Line Intersection Theorem
11. O' = Z                             11. Flip-Flop Theorem
12. ZOID reflected is OZDI 12. Figure Reflection Theorem
13. m symm. line of ZOID   13. Definition of symmetry line
14. m is the perp. bis. of OZ 14. Definition of reflection

This proof is quite long and can be intimidating for students. A teacher can break it down by actually folding the isosceles trapezoid along line m. The teacher can ask, "Where does D fold to?" The students will probably answer I, only to have the teacher ask "Why?"

Then the tougher part is to show that Z folds to O. We do it one step at a time -- first we show that since reflections preserve angle measure, Z' must be somewhere on ray OI -- although not necessarily O (but of course students will want to jump to that conclusion).

Now we have another theorem, the Isosceles Trapezoid Theorem:
In an isosceles trapezoid, the non-base sides are equal in measure.

In other words, our definition of isosceles trapezoid implies what we usually think of when we hear the word isosceles. The text states that this is merely a corollary of the previous theorem -- since the reflection of one of the non-base sides is the other.

Returning as a text, another corollary given is the Rectangle Symmetry Theorem:
Every rectangle has two symmetry lines, the perpendicular bisectors of its sides.

This corollary follows from the Isosceles Trapezoid Symmetry Theorem in the same way that the Rhombus Symmetry Theorem (which shows that a rhombus also has two symmetry lines) follows from the Kite Symmetry Theorem. It shows again the beauty of using inclusive definitions.

So far in this section, we've mentioned the symmetries of the kite, rhombus, isosceles trapezoid, as well as the rectangle. A general trapezoid, as we've mentioned before, has no symmetry. But there's one special quadrilateral that's missing -- the parallelogram. So does a parallelogram have symmetry?

Also, so far in this section we've mentioned reflections and reflection symmetry, but there's another transformation that we've learned about -- rotations and rotational symmetry. So do any of our special quadrilaterals have rotational symmetry.

We can answer both of these questions at once: A parallelogram has rotational symmetry!

Notice that the U of Chicago text doesn't discuss rotational symmetry of any figure, much less that of the parallelogram. Yet we see in the Common Core Standards:

CCSS.MATH.CONTENT.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe therotations and reflections that carry it onto itself.
(emphasis mine)

The rotational symmetry of the parallelogram appears in the Third Edition. Therefore, the many of the changes I made three years ago can also be viewed as "fixing" my Second Edition to include material from the Third Edition. The tricky part is that in adhering to the Second Edition order, the students don't actually see rotations until Chapter 6. So my old worksheet mentions rotations, but that means there's Chapter 6 material on a Chapter 5 worksheet. I'll keep this discussion of rotations here on the blog, but teachers may want to save rotations for Chapter 6.

And so we should discuss rotational symmetry in general and that of the parallelogram in particular.

But let's begin with the rhombus and rectangle before going on to the general parallelogram. We recall that our definition of rotation is the composite of reflections in intersecting lines -- and if those lines are symmetry lines of a figure, each one maps the figure to itself. Therefore, the entire rotation must map the figure to itself. It follows that any figure that contains two lines of symmetry must automatically have rotational symmetry as well! And the Two Reflection Theorem for Rotations tells us exactly what the center and magnitude of the rotation are. The center is where the two symmetry lines intersect, and the magnitude is double the angle between the intersecting lines.

Corollary to Two Reflection Theorem for Rotations:
If l and m are intersecting symmetry lines of a figure, then it also has rotational symmetry, where the magnitude is twice the non-obtuse angle between l and m and the center is the point of intersection of l and m.

Applying this to the rhombus, we see that both its diagonals are symmetry lines. Because a rhombus is a kite, its diagonals are perpendicular. So the magnitude of the rotation is twice 90, or 180 degrees, about the point where the diagonals intersect.

And now let's look at the rectangle. Its symmetry lines are the perpendicular bisectors of its sides, and we can use the Two Perpendiculars Theorem and the Fifth Postulate to show that these symmetry lines are also perpendicular. So the magnitude of the rotation is twice 90, or 180 degrees, about the point where the diagonals intersect.

But as it turns out, all parallelograms -- not just the rhombuses and rectangles -- can be shown to have 180-degree rotational symmetry. The U of Chicago doesn't prove this, but Hung-Hsi Wu does. In his Theorem 4, he proves that the opposite sides of a parallelogram are equal, by showing that the parallelogram has rotational symmetry.

Here is Wu's proof of his Theorem 4, in paragraph form (as Wu himself gives it):

Given: ABCD is a parallelogram
Prove: AD = BC

Proof:
Let M be the midpoint of the diagonal AC and we will use Theorem 1 -- Wu's First Theorem (that a 180-degree rotation maps a line to a parallel line, already proved on this blog) to explore the implications of the 180-degree rotation R around M.

Because MA = MC and rotations preserve distance, we have C" = A, so that R maps line BC to a line passing through A and parallel to line BC. Since the line AD has exactly the same two properties by assumption, Playfair implies that R maps line BC to line AD. Similarly, R maps line AB to CD. Thus since B lies on both AD and CD, its image B" is a point that lies on both AD and CD. By the Line Intersection Theorem, this point is exactly D.

Recall we also have C" = A. Therefore R maps the segment BC to the segment joining D (which is B") to A (which is C") by the property that a rotation (like a reflection) maps segments to segments. The latter segment has to be the segment DA, by the Unique Line Assumption. Thus by the Figure Reflection Theorem, R maps ABCD to CDAB, so ABCD has rotational symmetry. Since rotations preserve distance, BC = AD, as desired. QED

This proof is very similar to the Isosceles Trapezoid Symmetry Theorem. Both theorems depend on what Wu calls "Lemma 6" -- identifying the reflection or rotation image of a point by finding two lines that intersect at the preimage point and noting that the images of these two lines must intersect at the image of the point.

Like the Isosceles Trapezoid Symmetry Theorem and its corollary the Isosceles Trapezoid Theorem, we should call part of the proof the Parallelogram (Rotational) Symmetry Theorem and then derive that opposite sites are equal as a corollary. Wu also derives as a corollary that opposite angles are equal -- but we can also derive this by applying the Trapezoid Angle Theorem twice -- since a parallelogram is a trapezoid. Even the third major property of parallelograms -- that their diagonals bisect each other -- can be derived as a corollary (since rotations preserve distance, we must also have MB = MD in the above proof).

So we can prove all the major properties of parallelograms using only rotations and not using triangle congruence at all. The U of Chicago derives the properties of other figures using reflection symmetry but reverts to triangle congruence for the parallelogram properties (in Chapter 7).

Finally, notice that since rhombuses and rectangles are parallelograms, we can show that they have 180-degree rotational symmetry without looking at their lines of symmetry. But I still like using the Corollary to the Two Reflection Theorem for Rotations because it can be used to determine the rotational symmetry for figures other than quadrilaterals. We earlier proved that an equilateral triangle has three lines of symmetry -- so it also has rotational symmetry.

Lesson 5-4: Properties of Kites (Day 54)

Chapter 11 of Ian Stewart's The Story of Mathematics is called "Firm Foundations: Making Calculus Make Sense." Here's how it begins:

"By 1800 mathematicians and physicists had developed calculus into an indispensable tool for the study of the natural world."

This chapter is all about proving the theorems of Calculus -- something that is often taught nowadays in upper division college-level classes called "Real Analysis." The plan is to satisfy Bishop Berkeley and his objection that Calculus is all about "the ghosts of departed quantities."

We began with the late 18th century mathematicians, who began to grow concerned:

"In particular, they were uncertain how to handle discontinuous functions, such as f (x) = 0 if x < 0, f (x) = 1 if x > 0. This function suddenly jumps from 0 to 1 as x passes through 0."

And the problems were exposed by Joseph Fourier:

"And it was Fourier who really got up their noses, with his amazing ideas about writing any function as an infinite series of sines and cosines, developed in his study of heat flow."

The functions expressible as a Fourier series include, believe it or not, discontinuous functions:

"Slowly the mathematicians of the 19th century started separating the difficult conceptual issues in this difficult area."

And the first important concept that needed to be defined was "continuous function":

"The person who made the first serious start on sorting out this mess was a Bohemian priest, philosopher, and mathematician."

That mathematician was Bernhard Bolzano. He came up with a rigorous definition of continuity and proved, for example, that polynomial functions are continuous. Stewart also writes about Niels Abel (another mathematician I discussed in side-along reading a few years ago), who complained about the lack of rigor up until that point:

"His criticisms struck home, and gradually order emerged from the chaos. Bolzano's ideas set these improvements in motion. He made it possible to define the limit of an infinite sequence of numbers, and from that the series, which is the sum of an infinite sequence."

(I'm sorry, but considering that today is Game 7 of the World Series, I can't avoid the joke -- Bolzano proved that the World Series converges. OK, replace "World Series" with 1 + 1/2 + 1/4 + 1/8 + ....)

So Bolzano rigorously defined the limits that are still taught today in Calculus AB. But his definition was more formal that anything that appears on the AP: the sequence a_n tends to limit a as n tends to infinity if for every epsilon > 0, there exists N > 0 such that for any n > N, |a - a_n| < epsilon. (A few years ago I wrote about Erdos, who called little children "epsilons." This is where that name comes from -- "epsilon" is supposed to be a small number.)

Stewart tells us that this idea extends to Fourier series:

"There are several variations on the basic theme, and for Fourier series you have to pick the right ones. Weierstrauss realized that the same ideas work for complex numbers as well as real numbers."

It was this mathematician, Weierstrauss, who first came up with the idea of power series (as in Taylor or Maclaurin series), as well as some other interesting functions:

"One of his most surprising theorems proves that there exists a function f (x) of a real variable x, which is continuous at every point, but differentiable at no point."

Recall the math teacher blogger Sam Shah, who actually named his blog after this function -- "Continuous Everywhere but Differentiable Nowhere." Here's a link to his latest post:

https://samjshah.com/2019/10/08/a-short-whiteboard-activity-to-check-understanding/

(In this October 8th post, Shah has his Algebra II class draw some functions on whiteboards. Notice that these functions are definitely not continuous everywhere.)

But writing down power series without worrying about convergence leads to problems:

"Even Euler stated some pretty stupid things. For instance, he started from the power series 1 + x + x^2 + x^3 + x^4 + ... which sums to 1/(1 - x), set x = -1 and deduced that 1 - 1 + 1 - 1 + 1 - 1 + ... = 1/2 which is nonsense."

There was a recent debate involving series that don't converge -- recall that Numberphile claimed in a video that 1 + 2 + 3 + 4 + ... = -1/12, and Mathologer corrected him.

On this note, Stewart concludes the chapter by discussing infinitesimals -- a nonstandard system in which Berkeley's "ghost" numbers actually exist:

"In spirit, it is close to the way Cauchy thought. It remains a minority speciality -- but watch this space."

The sidebars in this chapter are all about the famous Riemann hypothesis (named after the Riemann who also formalized spherical geometry), what analysis did for them, and what analysis does for us.

Lesson 5-4 of the U of Chicago text is called "Properties of Kites." In the modern Third Edition of the text, kites appear in Lesson 6-5.

This is what I wrote last year about today's topic:

Section 5-4 of the U of Chicago text covers kites. The kite is a relatively new quadrilateral classification. Not only did Euclid never define kite, but many texts made no mention of kites -- including my class geometry textbook from 20 years ago. Nowadays most texts define kite, but some include kites only in bonus questions, not in the main text.

Here's what John Conway wrote about the kite -- over 20 years ago, right around the time that I was taking my geometry class:

In fact it's not quite true, either, because "kite" is not
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.

Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles
trapezoid.

David Joyce, meanwhile, doesn't find kites to be necessary at all:

Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.

Notice that both Conway and Joyce want to exclude trapezoids -- but Joyce is the only writer I know who wants to omit rhombuses. Of course, for Joyce, the emphasis should be on triangles and parallel lines, not quadrilaterals like rhombuses. Also, notice that kites are not specifically mentioned in the Common Core Geometry Standards. But I find that students can identify a kite more readily than a rhombus or trapezoid.

Now every kite contains two special vertices, known as its "ends." As defined by the U of Chicago:

"The common vertices of the equal sides of a kite are the ends of the kite."

Notice that Michael Serra doesn't define ends, but does give a name to the two angles whose vertices are the ends of the kite -- the vertex angles, in analogy with the vertex angle of an isosceles triangle. I see that from a proof standpoint, this makes sense, since the first thing that we do in the proof of our main theorem is divide the kite into two isosceles triangles.

And now here is our main theorem: the Kite Symmetry Theorem. As I mentioned back in the lesson on isosceles triangles, we use symmetry to determine the properties of kites. (A pre-Common Core proof might divide the kite into two triangles to be proved congruent by SSS.) I will post a proof here of the Kite Symmetry Theorem. It is taken directly from the U of Chicago, except that I, as always, add a Given step to the beginning of the proof. Since the U of Chicago's proof has eight steps, mine has nine:

Kite Symmetry Theorem:

The line containing the ends of a kite is a symmetry line for the kite.

Given: ABCD is a kite with ends B and D.

Prove: Line BD is a symmetry line for ABCD.

Proof:

Statements                                           Reasons

1. ABCD is a kite with ends B and D  1. Given

2. AB = BC, AD = DC                         2. Definition of ends of kite (meaning)

3. Tri. ABC and ADC are isosceles      3. Definition of isosceles triangle (sufficient)

4. Let m be the perp. bis. of AC          4. A segment has exactly one perp. bisector

5. A' = C, C' = A                                  5. Definition of reflection (sufficient)

6. m contains B and D                         6. The perp. bis. of the base = angle bis. of the vertex angle

                                                                 (so it contains the vertex)

7. B' = B, D' = D                                 7. Definition of reflection (sufficient)

8. ABCD reflected over m is CBAD     8. Figure Reflection Theorem

9. m (Line BD) is a symmetry line      9. Definition of symmetry line (sufficient)

Notice that more than half of the reasons in this proof are definitions. This underlines how important definitions are to the study of quadrilaterals.

As for the other theorems in this lesson, the Kite Diagonal Theorem follows directly from lines 4 and 6 of the above proof. It makes the symmetry diagonal the perpendicular bisector of the other diagonal and so the diagonals of a kite are perpendicular, and the symmetry diagonal bisects the other one.

Finally, we have the Rhombus Symmetry Theorem. It states that a rhombus has two symmetry lines, as both of the lines containing its diagonals are symmetry lines. This follows directly from the classification of a rhombus as a kite. In texts that define kite exclusively, the theorems "The diagonals of a kite are perpendicular" and "the diagonals of a rhombus are perpendicular" are two separate theorems, often in two separate sections. But here we can easily see why the diagonals of both the kite and the rhombus are perpendicular -- because the rhombus is a kite! Also, since we are defining kite inclusively, we don't need any extra steps in the proof to ensure that our kite isn't a rhombus.