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CHAPTER 4
Deductive Geometry
Deductive geometry is the art of deriving new geometric facts from previously-known facts by using logical reasoning. In elementary school, many geometric facts are introduced by folding, cutting, or measuring exercises, not by logical deduction. But as we have seen, fifth and sixth grade students are already practicing — and enjoying — deductive reasoning as they solve unknown angle problems. In geometry, a written logical argument is called a proof. Section 4.1 introduces one type of proof: “unknown angle proofs”. Unknown angle proofs are natural continuations of stu- dents’ experience in solving unknown angle problems; the transition is a small step that re- quires no new concepts. Indeed, as you will see, unknown angle proofs are almost identical to the “Teacher’s Solutions” that you wrote in the previous chapter! Section 4.2 describes how congruent triangles are introduced in middle school. Congruence is a powerful geometric tool that opens a door to new aspects of geometry; some of this is covered in Sections 4.3 and 4.4. These sections also describe how the facts about triangles and quadrilaterals that students learned in grades 5 and 6 are revisited at a higher level in middle school. In this chapter we reach the last stage in the preparation of students for high school geom- etry. As you read and do problems, think about how these problems are part of a story line that goes back to learning to measure angles in grade 4 and learning to measure lengths in grade 1. You will be teaching part of this story, and it is important to know how it unfolds. Background Knowledge Here is a list of the geometric facts at our disposal at this point. These facts will be used in the examples and homework problems in this chapter. Several additional facts will be added to this list in Section 4.2.
- The measures of adjacent angles add. ( c = a + b .) Abbreviation: ∠ s add. b a c 73
74 • CHAPTER 4. DEDUCTIVE GEOMETRY
- The sum of adjacent angles on a straight line is 180◦. (If L is a line then a + b = 180 ◦.) Abbreviation: ∠ s on a line. b a L
- The sum of adjacent angles around a point is 360◦. ( a + b + c + d = 360 ◦.) Abbreviation: ∠ s at a pt. b c d a
- Vertically opposite angles are equal. (At the intersection of two straight lines, a = c and b = d ). Abbreviation: vert. ∠ s. b c d a
- When a transversal intersects parallel lines, corresponding angles are equal. (If AB ∥ CD then a = b .) Abbreviation: corr. ∠ s, AB ∥ CD.
- Conversely, if a = b then AB ∥ CD. Abbreviation: corr. ∠ s converse. b a A B C D
- When a transversal intersects parallel lines, alternate interior angles are equal. (If AB ∥ CD then a = c .) Abbreviation: alt. ∠ s, AB ∥ CD.
- Conversely, if a = c then AB ∥ CD. Abbreviation: alt. ∠ s converse. a c A B C D
- When a transversal intersects parallel lines, interior angles on the same side of the transversal are supplementary. (If AB ∥ CD then a + d = 180 ◦.) Abbreviation: int. ∠ s, AB ∥ CD.
- Conversely, if a + d = 180 then AB ∥ CD. Abbreviation: int. ∠ s converse. a d A B C D
- The angle sum of any triangle is 180◦. (*) ( a + b + c = 180 ◦.) Abbreviation: ∠ sum of ∆. b a c
- Each exterior angle of a triangle is the sum of the opposite interior angles. (*) ( e = a + b ). Abbreviation: ext. ∠ of ∆. (^) b e a
76 • CHAPTER 4. DEDUCTIVE GEOMETRY EXAMPLE 1.1. In the figure, angles A and C are right angles and angle B is 78 ◦. Find d. A d ° 78° B C D Teacher’s Solution: 90 + 78 + 90 + d = 360 ∠ sum in 4-gon 180 + 78 + d = 360 78 + d = 180 ∴ d = 102. Example 1.1 is a fact about one particular shape. But if we replace the specific measurement 78 ◦^ by an unspecified angle measure b ◦, then the identical reasoning yields a general fact about quadrilaterals with two 90◦^ interior angles. EXAMPLE 1.2. In the figure, angles A and C are right angles. Prove that d = 180 − b. A d ° b ° B C D Proof. 90 + b + 90 + d = 360 ∠ sum in 4-gon 180 + b + d = 360 b + d = 180 ∴ d = 180 − b. Notice the distinction between the above examples. Example 1.1 is an unknown angle problem because its answer is a number: d = 102 is the number of degrees for the unknown angle. We call Example 1.2 an unknown angle proof because the conclusion d = 180 − b is a relationship between angles whose size is not specified. EXAMPLE 1.3. In the figure, AB ∥ EC and BD ∥ EF. Find b. b° 43° A B E D C^ F 43 ° c° b° A B E F D C (^) Teacher’s Solution: Extend the lines as shown. Mark angle c as shown. c = 43 alt. ∠s, BC ∥ EF b = c alt. ∠s, BA ∥ ED ∴ b = 43. There is nothing special about the number 43. The same reasoning shows that b = a in the picture on the right. The proof below is a Teacher’s Solution with two embellishments. First, it is “launched” by a preamble that states, in very few words, what we are assuming as known and what we wish to show. Second, the solution involves two auxiliary lines, as we explain to the reader on a line labeled “construction”. b° a°
SECTION 4.1 UNKNOWN ANGLE PROOFS • 77 Given: AB ∥ ED and BC ∥ EF. To prove: b = a. a° c° b° A B E F D C Construction: Extend sides BC and ED. Mark angle c as shown. Proof. c = a alt. ∠s, BC ∥ EF b = c alt. ∠s, BA ∥ ED ∴ b = a. We have just turned Example 1.3 into a proof. The proof requires no additional effort! EXAMPLE 1.4. In the figure, AB ∥ CD. Prove that z = x + y. C (^) y ° D x ° z ° A B You have seen this problem before: it is almost identical to Example 2.3 on page 65. The previous version was an unknown angle problem: two of the three angles x , y , and z were given (one was 37◦^ and another was 75◦) and the problem was to find the third. The version above leaves x , y , and z unspecified and the problem is to prove that they are related. Just as before, the proof requires a construction, and several different constructions will work. Here is one proof. C (^) y° D x ° A (^) B z° a° (^) b° Given: AB ∥ CD. To prove: z = x + y. Construction: Extend line as shown. Mark angles a and b as shown. Proof. a = x corr. ∠s, AB ∥ CD b = y vert. ∠s z = a + b ext. ∠s of a ∆ ∴ z = x + y. A polygon whose vertices lie on a circle is said to be inscribed in the circle. The next example proves a famous fact about inscribed triangles. EXAMPLE 1.5. Any inscribed triangle with a side passing through the center of the circle is a right triangle. A B C O
SECTION 4.1 UNKNOWN ANGLE PROOFS • 79 teachers, asking students to use a simple, clear format is a matter of self-interest: it makes student work easier to read. In fact, this applies to everything written by mathematics students. In this book, we will adopt the format used in the examples in this section. We will call a proof written in this format an Elementary Proof. You will be expected to use this in all home- work problems that ask for an Elementary Proof. The following template shows the features of an Elementary Proof. Given : AD ⁄⁄ CE and AB BC. To Prove : x+ y = 90 °. Construction: Extend AB to meet E at angle z.
. .. x = z corr. s , AD ⁄⁄ CE z+y = 90 ° ext. of ∆ x+y = 90 °. Proof: A B C x z^ y D E Preamble states the given information and the goal.
Elementary Proof
Last line identical to the“To Prove’’ line. Facts used are recorded using our abbreviations. Diagram shows all points, lines and angles used. } Only 1 fact per line. Any needed construction is explained before the proof starts. Hints. Clear proofs are short and simple. To that end,
- Do not label the two columns “statement” and “reason” (everyone already knows this!).
- Do not include reasons for simple arithmetic and algebra steps.
- To avoid cluttering the picture, label only those points, lines and angles used in the proof.
Establishing Facts Using Proofs
The next two examples show how two familiar facts about triangles follow from the proper- ties of parallel lines. In fifth grade, students justified the “angle sum of triangle” fact by cutting and re-arranging paper triangles. Middle school students can give a purely geometric argument for this fact. These are very important proofs: learn them. THEOREM 1.7. In any triangle, the sum of the interior angles is 180 ◦. a b c A B C x y L Given: ∆ ABC. To prove: a + b + c = 180 ◦. Construction: Draw line L through C parallel to AB. Mark angles x and y as shown. Proof. a = x corr. ∠s, L ∥ AB b = y alt. int. ∠s, L ∥ AB c + x + y = 180 ◦^ ∠s on a line ∴ a + b + c = 180 ◦.
80 • CHAPTER 4. DEDUCTIVE GEOMETRY THEOREM 1.8. Each exterior angle of a triangle is the sum of the opposite interior angles. a b c e Given: ∆ ABC. To prove: e = a + b. Proof. e = 180 ◦^ − c ∠s on line a + b = 180 ◦^ − c ∠ sum of ∆ ∴ e = a + b. EXERCISE 1.9. Compare both theorems with the “picture proof” of the same fact described in Section 2.2 of Chapter 2. Are they compatible? Do these proofs elaborate on the picture proofs? The next proof is different from the fifth-grade paper-folding explanation, but is still easy to understand. THEOREM 1.10. Opposite angles in a parallelogram are equal. x a c A B (^) C D Given: ABCD is a parallelogram. To prove: a = c. Construction: Extend sides AB , DC , and BC. Mark angle x as shown. Proof. a = x corr. ∠s, BC ∥ AD x = c alt. int. ∠s, CD ∥ AB ∴ a = c.
Theorems and Proofs in the Classroom
Students often acquire misconceptions about the meaning of the words “theorem” and “proof”. Many believe that a theorem is “a mathematical fact” and a proof is “an explana- tion of why a fact is true.” This viewpoint embodies a subtle misunderstanding that teachers should try to prevent from taking root. Theorems are not statements of universal truths. Rather, they are “if-then” statements: If certain assumptions are true then a stated conclusion is true. The assumptions are of two types: those explicitly stated as hypotheses (after the word Given in our format), and a collection of “background knowledge” facts (often not explicitly mentioned) that have already been accepted or proven true. A proof is a sequence of deductive steps that explain how the conclusion follows from the hypothesis. This can be said more compactly as follows: DEFINITION 1.11. A proof of a mathematical statement is a detailed explanation of how that statement follows logically from other statements already accepted as true. A theorem is a mathematical statement with a proof. Theorems are the building blocks of geometry. Once a theorem has been proved, it can be added to the list of background facts and used in subsequent proofs. For example, after proving
82 • CHAPTER 4. DEDUCTIVE GEOMETRY
- In the figure, AB ∥ XY and BC ∥ YZ. Prove that ∠ ABC = ∠ XYZ. A B C X Y Z
- In the figure, PQ ∥ T S. Prove that ∠ QRS = q + s. S R q° s° Q T P
4.2 Congruent Triangles
Two line segments are called congruent if they have equal lengths. Two angles are congruent if they have equal measures. Similarly, two triangles are called congruent if their sides have the same lengths and their angles have the same measures. The first part of this section explores how curricula build up to this notion of congruent triangles. The second part focuses on the criteria that ensure that two triangles are congruent.
A Curriculum Sequence
Early Grades. In geometry, figures that are duplicates – exact copies of one another – are called congruent. Duplicate figures can be drawn in different positions and orientations. Thus, to check whether two figures are congruent one must realign them to see if they match. Young children are taught to match shapes, first with figures made of cardboard or thin plastic, then visually with pictures. The phrase “same size and same shape” is often used to mean “congruent”. The grade 1 Primary Mathematics books contain exercises like this: a) b) (^) c) d) Do the figures have the same size and same shape? These are exercises in visualization only; students are not expected to make measurements. The goal is to convey the idea that matching requires first sliding and rotating, and then com- paring lengths and angles. The term congruence is not used, but the seed of the idea is planted. Teaching comment: Children may misinterpret the term “same shape” to mean “shapes with the same name”. But the word “shape” refers to the angles and proportions in a figure, not just its name. Thus, the figures in b) above are both triangles, but do not have the same shape. Matching exercises also introduce ideas that are used later for defining area and for modeling fractions. For example, a regular hexagon, decomposed into 6 congru- ent triangles, can be used to illustrate fractions whose denominator is 6. fractional unit whole unit 65^ units
SECTION 4.2 CONGRUENT TRIANGLES • 83 Middle School Introduction. The Primary Mathematics curriculum introduces congruence in grade 7 (some curricula start as early as grade 5). One approach starts with see-through tracing paper or overhead transparency sheets as in the following exercise. EXERCISE 2.1. Trace triangle A on a transparent sheet and lay it over figures B and C. Can you make them match? Flip the sheet over and try again. A B C This exercise makes the idea of “same size, same shape” more precise, yet it still relies on visual matching. But notice that “matching” can be described purely in terms of the 3 side lengths and 3 angle measures: two triangles match if they can be aligned so that all six of these measurements exactly agree. In diagrams, such matching can be shown by marking the triangles as in the figure below. A B C P Q R But it is more efficient to use symbols. This requires (i) a symbolic way of describing how to align two triangles, and (ii) a definition for what is meant by “exactly match”. To align triangles, one pairs up their vertices. Such a pairing is called a correspondence. More precisely, a correspondence between two triangles pairs each vertex of one triangle with one and only one vertex of the other. The figure shows triangles ∆ PQR and ∆ XYZ aligned in a way that suggests the correspondence P ↔ X , Q ↔ Y , and R ↔ Z. Z Y X R Q P Once a correspondence is chosen, we can talk about corresponding sides and corresponding angles. For the correspondence P ↔ X , Q ↔ Y , and R ↔ Z , ∠ P corresponds to ∠ X , side QR corresponds to side YZ , etc. In this way we can compare the six measurements of two triangles, even if they don’t have the same shape or size. When the corresponding measurements are equal, the correspondence gives us a precise way of stating that the triangles are exact copies of each other. DEFINITION 2.2. Two triangles are congruent if, under some correspondence,
- all pairs of corresponding sides are equal, and
- all pairs of corresponding angles are equal.
SECTION 4.2 CONGRUENT TRIANGLES • 85
Congruence Tests for Triangles: a Teaching Sequence
Congruent triangles have six pairs of equal measurements (3 pairs of angles and 3 pairs of side lengths). However, to show that triangles are congruent, it isn’t necessary to check all six pairs. Four congruence criteria make the task easier. The criteria are often called “tests” and are named by triples: the Side-Side-Side Test, Angle-Side-Angle Test, Side-Angle-Side Test and the Right-Hypotenuse-Side Test. Each test states that two triangles are congruent whenever the named measurements match under some correspondence. Congruence tests are often introduced in middle school by activities that ask students to construct triangles with specified measurements. The aim is to make the meaning of the tests clear and convince students of their validity. Side-Side-Side Test. Construction 6 on page 53 showed how to use the 3 side-lengths of a triangle to create a duplicate triangle with the same side-lengths. To verify that the two triangles are congruent, one must also know that corresponding angles have equal measure. EXERCISE 2.5. Using a compass and a straightedge, copy ∆ ABC onto a blank paper as in Construction 6 on page 53. Then make the following checks: A C B First Check: Measure the angles in the copied triangle with a protractor. Are they equal to the corresponding angles in ∆ ABC? Are the two triangles congruent? Second Check: Cut out your triangle and place it on top of the original triangle in this book. Align the sides. Are all angle measures equal? Are the triangles congruent? In Exercise 2.5, both checking methods refer directly to the definition of congruence — both require verifying that all corresponding angles and sides are equal. The teaching goal at this stage is to make students explicitly aware of the definition: congruence means 6 matching measurements. Side-Side-Side Test Two triangles are congruent if corresponding sides are equal under some correspondence, i.e., If AB = PQ , BC = QR , CA = RP , then ∆ ABC " ∆ PQR. (Abbreviation: SSS .) A B C P Q R
86 • CHAPTER 4. DEDUCTIVE GEOMETRY Angle-Side-Angle Test. In the next example, students duplicate a triangle by copying two angles and the side between them. EXERCISE 2.6. Using a straightedge and protractor, copy ∆ ABC onto a blank paper as described below. Then make the two checks stated beneath the picture. A 4 cm 64° (^) 35° C B P 4 cm R T^ S ? ? Q ? 64° 35° Ruler and Protractor Construction:
- Draw PQ of length AB.
- Draw ray
PS so ∠ S PQ = 64 ◦.
QT so that ∠ T QP = 35 ◦, as shown.
- Mark R where the two rays intersect and draw PR and QR. First Check: Use a protractor to measure ∠ R; is it equal to ∠ C? Use your compass to compare lengths QR and PR; are they equal to BC and AC? Are the two triangles congruent? Second Check: Cut out your triangle and place it on top of ∆ ABC. Align the sides. Are all angles and side lengths equal? Are the two triangles congruent? Angle-Side-Angle Test Two triangles are congruent if two pairs of corresponding angles and their included sides are equal, i.e., If ∠ A = ∠ P , ∠ B = ∠ Q , AB = PQ , then ∆ ABC " ∆ PQR. (Abbreviation: ASA .) A B C P Q R The ASA Test should really be called the “two angles and a side” test. After all, if two pairs of angles are equal, then the third pair is also equal (because the angles of a triangle total 180◦). Consequently, each side lies between two pairs of corresponding angles, and we can apply the ASA Test. Some books distinguish between the ASA and AAS conditions, but we will the term “ASA Test” for both.
88 • CHAPTER 4. DEDUCTIVE GEOMETRY Likewise, triangles with one pair of equal corresponding angles and two pairs of equal corresponding sides needn’t be congruent, as the following “swinging girl” picture shows. x º 20 cm 13 cm 13 cm The shaded triangle (base 11 cm) and the large triangle (base 21 cm) have the same Angle-Side-Side data but are not congruent. 11 cm (^) 10 cm The fact that Angle-Side-Side measurements do not determine a unique triangle is easily said: there is no A.S.S. test. This phrasing is not recommended for classroom use. Typically, textbooks tell students that “A.A.A. and S.S.A. do not confirm congruence of triangles.” Right-Hypotenuse-Leg Test. For pairs of right triangles, it is enough to compare two pairs of sides. In a right triangle, the side opposite the 90◦^ angle is called the hypotenuse and the other sides are called legs. The following exercise shows that two right triangles are congruent if they have equal hypotenuses are equal and one leg of equal length. EXERCISE 2.8. Copy the right triangle ∆ ABC onto some tracing paper using the construction below. B C 2 cm 3 cm A R Q L 2 cm 3 cm P S Construction:
- Draw a segment PQ of length 2 cm.
- Draw line L perpendicular to PQ passing through Q.
- Draw circle, center P , radius 3 cm.
- Mark as R one of the two points where the circle intersects L. Measure angles ∠ P and ∠ R; are they equal to ∠ A and ∠ C? Use your compass to compare the lengths QR and BC; are they equal? What do you conclude about the two triangles? Right-Hypotenuse-Leg Test If two right triangles have hypotenuses of the equal length and a pair of legs with equal length, then the triangles are congruent, i.e., If ∠ B = ∠ Q = 90 ◦, AC = PR , AB = PQ , then ∆ ABC " ∆ PQR. (Abbreviation: RHL .) B C A Q R P
SECTION 4.2 CONGRUENT TRIANGLES • 89 The RHL Test uses two sides and a non-included angle — Angle-Side-Side measurements. In general, for such measurements, the swinging girl picture produces two non-congruent trian- gles with the same Angle-Side-Side measurements. But the RHL Test is true because, when the angle x is a right angle, the swinging girl picture becomes the diagram in Exercise 2.8, and the triangles are actually congruent. Homework Set 14
- Do Exercise 2.3 in this section (just fill in the blanks in your textbook). Then fill in all of the blanks in the ex- ercises below. (On your homework sheet, just state the correspondence for each part.) a) B A C Y Z X ∠ A = ∠ B = ∠ C = AB = BC = CA = Correspondence: ∆ ABC " ∆ b) L E^ D F (^) M N ∠ D = ∠ E = ∠ F = DE = EF = FD = Correspondence: ∆ DEF " ∆ c) S (^) R T W U V ∠ R = ∠ S = ∠ T = RS = S T = T R = Correspondence: ∆ RS T " ∆ 2. Do Exercise 2.5 on page 85 of this section. Answer the “First Check” and “Second Check” questions beneath the picture. 3. Is there an “SSSS Test” for quadrilaterals? That is, is it true that two quadrilateral with four pairs of correspond- ing sides of equal length are necessarily congruent? Ex- plain and illustrate. 4. Do Exercise 2.6 in this section. First do the construction (in the first step, use your compass to carry the “4 cm” length from your book to your HW paper). Then answer the “First Check” and “Second Check” questions. 5. Do Exercise 2.7 on page 87, this time using ruler and pro- tractor. Then answer the “Check” questions. 6. For each figure, name the congruent triangles and state the reason they are congruent (i.e., SSS, ASA, SAS, RHL). a) A B (^) D C b) E F H G c) I L J K d) N M O^ P Q
SECTION 4.3 APPLYING CONGRUENCES • 91 X Y A B C Given: AB = BC and AX = YC. To Prove: BX = BY. Proof. AB = BC given ∠ A = ∠ C base ∠s of isos. ∆ AX = YC given ∴ ∆ AXB " ∆ CY B SAS. ∴ BX = BY corr. sides of " ∆s. The triangle tests can also be used in unknown angle proofs. EXAMPLE 3.2. In the figure, ∠ A = ∠ D and AE = DE. Prove ∠ ECB = ∠ EBC. B C E A D To prove this, mark angles as shown and concentrate your attention on the two shaded triangles. B C E D A (^) a (^) x b d y c Given: a = d and AE = DE. To Prove: b = c. Proof. a = d given AE = DE given x = y vert. ∠s ∴ ∆ AEB " ∆ DEC ASA. ∴ EB = EC corr. sides of " ∆s. ∴ b = c base ∠s of isos. ∆. Study Examples 3.1 and 3.2 for a moment. In each proof, Line 4 states that two triangles are congruent by a congruence test, Lines 1-3 are the facts needed to apply the test, and Line 5 is a conclusion based on this congruence. In this chapter, almost all of the proofs will have this format. It is simple, but it has many applications! As students learn to construct proofs, it is easy for them to make a “false start” — their first approach doesn’t work. This is completely normal! Proofs are like puzzles: the fun lies false starts in proofs in trying different strategies to find a solution. The reward, like the reward in solving a tricky puzzle, is a feeling of accomplishment. In fact, geometric proofs were a common amusement of educated people in the 19th century, just as crossword and sudoku puzzles are today. The next example is a problem in which it is easy to make a false start. The figure contains several pairs of congruent triangles; which pair should be used in the proof? Try to find a strategy before you look at the proof written below. Here is a strategy that helps: color, highlight or shade the segments that appear in the “Given” and the “To Prove” statements. Look for a pair of congruent triangles that contain these sides.
92 • CHAPTER 4. DEDUCTIVE GEOMETRY EXAMPLE 3.3. In the figure, CA = CB. Prove that AS = BR. B (^) S C R A Given: CA = CB. To Prove: AS = BR. Proof. In ∆ ACS and ∆ BCR , CA = CB given ∠ C = ∠ C common angle ∠ R = ∠ S = 90 ◦^ given ∴ ∆ ACS " ∆ BCR AAS. ∴ AS = BR corr. sides of " ∆s. EXERCISE 3.4. This proof used the congruence ∆ ACS " ∆ BCR. Name (without proof) two other pairs of congruent triangles in this figure (use the letter T to label the intersection point in the middle of the figure). These examples indicate that there are two levels of congruent triangle proofs: ones that are especially simple because the figure contains only one pair of congruent triangles, and ones in which students must find the appropriate congruent triangles from among several pairs. Careful textbooks (and careful teachers!) provide plenty of practice at the first level before challenging students with problems at the second level. The problems in the homework for this section (HW Set 15) are similar to a well-written eighth grade textbook. As you do these proofs, notice how they are arranged so that they slowly increase in difficulty.
Proofs for Symmetry Explanations
Congruence tests can be used to prove many of the facts that were introduced in fifth and sixth grade using symmetry arguments. We will give three examples. As you will see, the elementary school “folding proofs” contain the key ideas of a complete mathematical proof. For example, the following proof is just the detailed explanation of the fifth grade picture proof shown on page 62 of Primary Math 5B and on page 44 of this book. THEOREM 3.5. In an isosceles triangle, base angles are congruent. (Abbreviation: base ∠ s of isos. ∆ .)
