Geometry, You Can Do It!

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Congruence Theorems - ! ’s

In math, the word congruent is used to describe objects that have the same size and shape. When you traced things when you were a little kid, you were using congruence. Neat, don’t you think? Stop signs would be examples of congruent shapes. Since a stop sign has 8 sides, they would be congruent octagons. We are going to look specifically at triangles. To determine if two triangles are congruent, they must have the same size and shape. They must fit on top of each other, they must coincide. Mathematically, we say all the sides and angles of one triangle must be congruent to the corresponding sides and angles of another triangle. A D B C E F In other words, we would have to show angles A, B, and C were congruent (! ) to angles D, E and F, then show!" , BC , and AC were ! to DE ,EF , and DF respectively. We would have to show those six relationships. Ah, but there is good news. If I gave everyone reading this three sticks of length 10”, 8”, and 7”, then asked them to glue the ends together to make triangles, something interesting happens. When I collect the triangles, they all fit very nicely on top of each other, they coincide. They are congruent! Why’s that good news? Because rather than showing all the angles and all the sides of one triangle are congruent to all the sides and all the angles of another triangle (6 relationships), I was able to determine congruence just using the 3 sides. A shortcut. That leads us to the side, side, side congruence postulate. SSS Postulate If three sides of one triangle are congruent, respectively, to three sides of another triangle, then the triangles are congruent.

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A D B C E F Using the same type of demonstration as before, we can come up with two more congruence postulates. The side, angle, side postulate is abbreviated SAS Postulate. SAS Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, respectively, then the two triangles are congruent. A D B C E F A third postulate is the angle, side, angle postulate. ASA Postulate If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. A D B C E F If you are going to be successful, you need to memorize those three postulates and be able to visualize that information. Combining this information with previous information, we will be able to determine if triangles are congruent. So study and review!

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Proofs: Congruent ! ’s

To prove other triangles are congruent, we’ll use the SSS, SAS and ASA congruence postulates. We also need to remember other theorems that will lead us to more information. For instance, you should already know by theorem the sum of the measures of the interior angles of a triangle is 180º. A corollary to that theorem is if two angles of one triangle are congruent to two angles of another triangle; the third angles must be congruent. OK, that’s stuff that makes sense Let’s write it formally as a corollary. Corollary If 2 angles of one triangle are congruent to two angles of another triangle, the third angles are congruent. Using that information, let’s try to prove this congruence theorem. AAS Theorem If two angles and the non included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. First let’s draw and label the two triangles. C F A B D E Given : !A ! !D, !C ! ! F, !" ! DE

Prove: ! ABC ! ! DEF

1. !A ! !D Given !C ! ! F

!" ! DE 2. !B ! ! E 2 ! ’s of a ! congruent 2 ! ’s of another ! , 3rd ! ’s congruent 3. ! ABC ! ! DEF ASA

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That was too easy. Now we have 4 ways of proving triangles congruent: SSS, SAS, ASA, and AAS. You need to know those. I know what you are thinking; you want to try another one. OK, we’ll do it! Here’s what you need to be able to do. First, label congruences in your picture using previous knowledge. After that, look to see if you can use one of the four methods (SSS, SAS, ASA, AAS) of proving triangles congruent. Finally, write those relationships in the body of proof. You are done. Piece of cake! A

X C D B Given: AC || BD AB bisects CD

Prove: ! ACX ! ! BDX Remember to mark up your picture with that information as I did.

A

X C D B

1. AC || BD Given AB bis. CD

2. CX ! DX Def. of bisector 3. !C ! !D || lines cut by tran, alt int ! ’s are ! 4. !ACX ! !BDX Vert. ! ’s are ! 5. ! ACX ! ! BDX ASA

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I can’t tell you how important it is to fill in the picture by labeling the information given to you and writing other relationships you know from previous theorems, postulates, and definitions. In order to prove that, we had to know the definition of a bisector and the subsequent mathematical relationship. And, even though vertical angles were not part of the information given, we could see from the diagram vertical angles were formed and we could then use the theorem that all vertical angles are congruent.

Right Triangles While the congruence postulates and theorems apply for all triangles, we have postulates and theorems that apply specifically for right triangles. HL Congruence Postulate If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. Since the HL is a postulate, we accept it as true without proof. The other congruence theorems for right triangles might be seen as special cases of the other triangle congruence postulates and theorems. LL Congruence Theorem If two legs of one right triangle are congruent to two legs of another right triangle, the triangles are congruent. That’s a special case of the SAS Congruence Theorem. HA Congruence Theorem If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of another right triangle, the triangles are congruent. This congruence theorem is a special case of the AAS Congruence Theorem. And finally, we have the Leg Angle Congruence Theorem. LA Congruence Theorem If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, the triangles are congruent. If you drew and labeled the picture of the LA Congruence Theorem, you would see that could be derived from the ASA or AAS congruence theorems.

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Proofs: cpctc

When two triangles are congruent, each part of one triangle is congruent to the corresponding part of the other triangle. That’s referred to as corresponding parts of congruent triangles are congruent, thus cpctc. You can determine if two line segments or two angles are congruent by showing they are the corresponding parts of two congruent triangles. One way to prove two segments or two angles are congruent is by: 1. Identify two triangles in which segments or angles are the corresponding parts. 2. Prove the triangles are congruent. 3. State the two parts are congruent, supporting the statement with the reason; “corresponding parts of congruent triangles are congruent” That reason is normally abbreviated “cpctc” Let’s see if we can prove two line segments are congruent.

A D P C B Given: AB and CD bisect each other Prove: AD ! BC Now the strategy to prove the segments are congruent is to first show the triangles are congruent. So, let’s fill in the picture showing the relationships based upon the information given and other relationships that exist using our previously learned definitions, theorems, and postulates. A D P C B

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Using the definition of bisector, I can determine that AP ! PB and CP ! PD . I also notice I have a pair of vertical angles. Notice how I marked the drawing to show these relationships. Let’s go ahead and fill in the body of the proof Statements Reasons 1. AB and CD bisect Given each other 2. AP ! PB Def. of bisector

CP ! PD

3. !APD ! !PBC Vert ! ’s !

4. ! APD ! ! PBC SAS

5. AD ! BC cpctc Filling in the body of the proof is easy after you mark the congruences, in your picture. The strategy to show angles or segments are congruent is to first show the triangles are congruent, then use cpctc. Try this one on your own. C A X B C A X B

1 2 Give Given: AC ! BC , AX ! BX

Prove: !1 ! !2

1 2

Mark the picture with the parts that are congruent based on what’s given to you. Then mark the relationships based upon your knowledge of geometry.

In this case CX ! CX

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From the picture we can see three sides of one triangle are congruent to three sides of another triangle, therefore the triangles are congruent by SSS Congruence Postulate.

! AXC ! ! BXC If the triangles are congruent, then the corresponding parts of the triangles are congruent by cpctc. That means !1 ! !2, just what we wanted to prove. We can develop quite a few relationships based upon knowing triangles are congruent. Let’s look at a few. Theorem. If 2 ! ’s of ∆ are ! , the sides opposite those ! ’s are !

C

G: ∆ ABC !A ! ! B P: AC ! BC A X B B X Statements Reasons 1. Draw ! bisector CX Construction

2. ! 1 ! ! 2 Def ! bisector

3. !A ! ! B Given

4. CX ! CX Reflexive Prop

5. ∆CAX ! ∆CBX AAS

6. AC ! BC cpctc

The converse of that theorem is also true.

Theorem. If 2 sides of a triangle are congruent, the angles opposite those sides are congruent.

1 2

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Theorem. A diagonal of a ||ogram separates the ||ogram into 2 ! ∆’s

W T

G: RSTW

P: ∆RST ! ∆TWR

R S

Statements Reasons

1. RSTW is a ||ogram Given

2. RS || WT Def - ||ogram

3. ! 1 ! ! 2 2 || lines cut by t, alt int ! ’s ! 4. RT ! RT Reflexive

5. RW || ST Def - ||ogram

6. ! 3 ! ! 4 2 || lines cut by t, alt int ! ’s ! 7. ∆RST ! ∆TWR ASA

Knowing the diagonal separates a parallelogram into 2 congruent triangles suggests some more relationships. Theorem. The opposite sides of a parallelogram are congruent. Theorem. The opposite angles of a parallelogram are congruent. Both of those can be proven by adding CPCTC to the last proof. Many students mistakenly think the opposite sides of a parallelogram are equal by definition. That’s not true, the definition states the opposites sides are parallel. This theorem allows us to show they are also equal or congruent. The idea of using cpctc after proving triangles congruent by SSS, SAS, ASA, and ASA will allow us to find many more relationships in geometry. Let’s look at some more proofs.

4 2

3 1

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Theorem: The diagonals of a ||ogram bisect each other.

M L

G: JKLM

P: JE ! LE; KE! ME

J K

Statements Reasons 1. JKLM is ||ogram Given

2. JK || ML Def - ||ogram

3. ! 1 ! ! 2 2 || lines cut by t, alt int ! ’s

4. JK ! ML opposite sides ||ogram !

5. ! 3 ! ! 4 2 || lines cut by t, alt int ! ’s

6. ∆JEK ! ∆LEM ASA

7. JE ! LE cpctc KE ! ME

Notice to prove this theorem, I first drew the parallelogram, then I drew in the diagonals. In order to prove triangles congruent, I had to add angles to the picture, so I labeled angles 1, 2, 3, and 4 and developed the relationships based upon my previous knowledge of geometry. Using my triangle congruence knowledge, I can further develop my knowledge of parallelograms, rectangles and rhombi. Theorem The diagonals of a rectangle are congruent.

E

1 3

4 2

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Theorem: The diagonals of a rhombus are ! G: Rhombus RSTW R P: RT ! SW 1

S W

T

Statements Reasons

1. RSTW – Rhombus Given

2. RS ! RW Def – Rhombus

3. SA ! WA Diagonals ||ogram bisect each other 4. RA = RA Reflexive

5. ∆RSA ! ∆RWA SSS

6. ! 1 = ! 2 cpctc

7. RT ! SW 2 intersecting lines form ! adj

! ’s

The following theorem follows from this proof and the theorem that states if two sides of a triangle are congruent, the angles opposite those sides are congruent. Theorem Each diagonal of a rhombus bisects a pair of opposite angles. As you can see, the congruence theorems allow us to determine other mathematical relationships by going one step further and using cpctc. Using this strategy, we have now shown a number of relationships.

1 2 A

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Summary Congruence Theorems SSS SAS ASA AAS HL LL HA LA If two sides of a triangle are congruent, the angles opposite those sides are congruent. If two angles of a triangle are congruent, the sides opposite those angles are congruent. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. The diagonals of a rectangle are congruent. The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects a pair of opposite angles.

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Coordinate Geometry We’ve been using deductive reasoning in a t-proof to prove theorems thus far. Sometimes, theorems might be better proven using coordinate geometry. In a nutshell, coordinate geometry allows us to express our knowledge of geometry using algebra. We are going to look at a couple of proofs using coordinate geometry. Theorem: The median of a trapezoid is || to the bases and is equal to half the sum of the bases. R W G: RSTW is a trap P: MN || ST M N MN || RW

MN = 12

(ST + RW)

In this theorem, we are required to prove two things; first lines are parallel and second a mathematical relationship. From the geometry we have already learned, we know how to show lines are parallel by looking at angles. In your algebra class, you learned that parallel lines have the same slope Place trapezoid on coordinate axes, label points, carefully keeping relationships. Find slopes, || lines have = slopes. Find distances. That sounds easy enough, but it takes a little extra thinking to label things so the arithmetic does not get in the way of what we are trying to prove. (2a, 2b) (2c, 2b) R W (a, b) (c +d, b) M N S T (0, 0) (2d, 0) I have very conveniently placed the trapezoid on the coordinate axis having coordinates (0, 0). Now, I could have labeled the coordinates of R as (a, b). The reason I did not do

º º

T S

º º

� �

� �

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that was because I know I have to find the midpoint of RS and that would have led to a fraction. So I got a little tricky, I labeled R as (2a, 2b). That way the midpoint M is (a, b). Using the same type of logic, I will label W as (2c, 2b) and T as (2d, 0). The slope is the change in y over the change is x, so the slope of MN = (b–b)/(c+d–a) or 0. The slope of RW is (2b–2b)/(2c–2a) or 0. The slope of ST is (0–0)/(2d–0) or 0. Since MN, RW, and ST have the same slope, those lines are all ||. Or more precisely

MN || ST and MN || RW

We have shown the lines are parallel. Now we have to show the median is half the sum of the bases. Since these are all horizontal lines, all I have to do is subtract to find their distances. MN = c + d – a RW = 2c – 2a ST = 2d RW + ST = (2c – 2a) + 2d = 2(c –a + d) From the algebra we can see that RW + ST is twice MN. Another way to say that is MN is half of RW + ST

MN = 12

(STW + RW)

Now we have shown both parts, the lines being parallel and the median being half te sum of the bases. I can not stress enough the importance of setting this up by labeling points conveniently. We have proven another theorem, but this time we used coordinate geometry. As you become more comfortable with t-proofs and coordinate geometry, you will have to decide which method to use. When you are not able to prove a theorem using one method, you now have another way at getting at the proof.

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Let’s look at another theorem and prove it using coordinate geometry. Theorem The segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of the third side. The last proof might suggest to show lines are parallel in coordinate geometry, we need to show they have the same slope. We also found distances in the last proof, knowing how to find that will help us find the distances in this problem. Let’s look how I positioned the triangles on the coordinate axes. Which positioning do you think might help us with our proof.

Since we are going to be looking for distances and midpoints again, using the origin comes in handy because the coordinates are (0, 0). Let’s label the diagram using the positioning on the left. C (2m, 2n) R(m, n) S (m+p, n) A (0, 0) B (2p, 0) By labeling C (2m, 2n), the midpoint R is easy to find for AC. The same is true finding the midpoint S for BC.

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The slope of RS and AB are zero. Since they have the same slope, the lines are parallel. Let’s find the distances - RS = (m + p) – m = p AB = 2p – 0 = 2p AB is twice RS, that means that RS is half AB. Mathematically, we have RS = ½ AB