Geometry Notes - Unit 4 Congruence

Transcription

1 Geometry Notes - Unit 4 ongruence Triangle is a figure formed by three noncollinear points. lassification of Triangles by Sides Equilateral triangle is a triangle with three congruent sides. Isosceles triangle is a triangle with at least two congruent sides Scalene triangle is a triangle with no congruent sides. lassification of Triangles by ngles cute triangle is a triangle with three acute angles. Right triangle is a triangle with one right angle. Obtuse triangle is a triangle with one obtuse angle. djacent sides of a triangle are two sides sharing a common vertex. djacent sides ommon Vertex Hypotenuse of a right triangle is the side opposite the right angle. Hypotenuse Right ngle pg. 1

2 If I asked an entire class to draw a triangle on a piece of paper, then had each person cut out their triangle, we might see something interesting happen. Let s label the angles 1, 2, and 3 as shown y tearing each angle from the triangle, then placing them side by side, the three angles always seem to form a straight line That might lead me to believe the sum of the interior angles of a triangle is 180º. While that s not a proof, it does provide me with some valuable insights. The fact is, it turns out to be true, so we write it as a theorem. Triangle Sum Theorem: The sum of the interior angles of a triangle is 180º. Example Find the measure of Since the sum of the two angles given is 110, 3 must be pg. 2

3 rawing a triangle and cutting out the angles suggests the sum of the interior angles is 180 is not a proof. Let s see what that proof might look like. Theorem Proof - The sum of the measures of the angles of a triangle is 180º. R 4 F 2 5 S Given: Δ EF Prove: m 1+ m 2+ m 3= E The most important part of this proof will be our ability to use the geometry we have already learned. If we just looked at the three angles of the triangle, we d be looking for an awfully long time without much to show for it. What we will do is use what we just learned we were just working with parallel lines, so what we will do is put parallel lines into our picture by constructing RS parallel to E and labeling the angles formed. Now we have parallel lines being cut by transversals, we can use our knowledge of angle pairs being formed by parallel lines. Statements Reasons 1. Δ EF Given 2. raw RS E onstruction 3. 4& FS are supp. Ext. sides of 2 adj. 's form a line 4. m 4+ m FS = 180 ef Supp 's 5. m FS = m 2+ m 5 dd Postulate 6. m 4+ m 2+ m 5= 180 Substitution lines cut by trans., alt. int. 's 8. m 1+ m 2+ m 3= 180 Substitution pg. 3

4 Notice how important it is to integrate our knowledge of geometry into these problems. Step 3 would not have jumped out at you. It looks like 4, 2 & 5 form a straight angle, but we don t have a theorem to support that so we have to look at 4& FS first. theorem that seems to follow directly from that theorem is one about the relationship between the exterior angle of a triangle and angles inside the triangle. If we drew three or four triangles and labeled in their interior angles, we would see a relationship between the two remote interior angles and the exterior angle. Theorem: The exterior of a is equal to the sum of the 2 remote interior 's Given: Δ Prove: m 1 = m + m 2 1 Statements Reasons 1. m + m + m 2= 180 Int s of = & 2 are supp. 's Ext. sides 2 adj. 's 3. m 1+ m 2= 180 ef. supp. 's 4. m + m + m 2= m 1+ m 2 Substitution 5. m + m = m 1 Subtr. Prop. of Equality orollary Triangle Sum Theorem: The acute angles of a right triangle are complementary. pg. 4

5 Example) Find the value of x. The sum of the interior angles is (2x + 10) + (3x) = 180 (combine like terms) 5x + 40 = 180 (subtract 40) 5x = (2x + 10) 3x (divide by 5) x = 28 Example) Find the value of x. 60 x (2x + 10) Exterior angle is equal to the sum of the two remote interior angles (2x+10) = x + 60 (subtract x) x + 10 = 60 (subtract 10) x = 50 ongruence 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. pg. 5

6 E F In other words, we would have to show angles,, and were congruent ( ) to angles, E and F, then show,, and were to E, EF, and F respectively. We would have to show those six relationships. h, 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 is that good news? ecause 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. 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. E F pg. 6

7 Using the same type of demonstration as before, we can come up with two more congruence postulates. The Side, ngle, Side postulate is abbreviated SS Postulate. SS 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. E F third postulate is the ngle, Side, ngle postulate. S Postulate: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. E F If you are going to be successful, you need to memorize those three postulates and be able to visualize that information. ombining this information with previous information, we will be able to determine if triangles are congruent. So study and review! pg. 7

8 Proofs: ongruent Δ 's To prove other triangles are congruent, we ll use the SSS, SS and S 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. 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. orollary: 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. S 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. F E Given:, F, E Prove: Δ Δ EF pg. 8

9 Statements Reasons 1., F, E Given 2. E 2 's of a Δ congruent 2 3. Δ Δ EF S 's of another Δ, 3 rd 's congruent Now we have 4 ways of proving triangles congruent: SSS, SS, S, and S. You need to know these. Here s what you need to be able to do. First, label congruences in your picture using previous knowledge. fter that, look to see if you can use one of the four methods (SSS, SS, S, S) of proving triangles congruent. Finally, write those relationships in the body of proof. You are done. Piece of cake! Given: bisects Prove: ΔX Δ X X Remember to mark up your picture with that given information as I did. X pg. 9

10 Statements Reasons 1. bisects Given 2. X X ef. of bisector 3. lines cut by trans., alt. int. 's are 4. X X Vert. 's are 5. ΔX Δ X S 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 those triangles congruent, we had to know the definition of a bisector and the subsequent mathematical relationship. nd, 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. Proofs: PT 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 orresponding Parts of ongruent Triangles are ongruent, thus PT. One way you can determine if two line segments or two angles are congruent is by showing they are the corresponding parts of two congruent triangles. 1. Identify two triangles in which the 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. pg. 10

11 **That reason is usually abbreviated PT. Let s see if we can prove two line segments are congruent. P Given: and bisect each other Prove: 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. P Using the definition of bisector, I can determine that P P and P P. I also notice I have a pair of vertical angles. Notice how I marked the drawing to show these relationships. Even though the vertical angles are marked in my diagram, I must write the statement in my proof. Let s go ahead and fill in the body of the proof. pg. 11

12 Statements Reasons 1. and bisect each other Given 2. P P, P P ef. of bisector 3. P P Vert. 's 4. ΔP Δ P SS 5. PT 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 PT. Try this one on your own. Given:, X X 1 2 Prove: 1 2 X 1 2 Mark the picture with the parts that are congruent based on what s given to you. X Then mark the relationships based upon your knowledge of geometry. In this case X X pg. 12

13 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 ongruence Postulate. ΔX Δ X If the triangles are congruent, then the remaining corresponding parts of the triangles are congruent by PT. 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. Given: Δ, Prove: 1 2 Statements Reasons X 1. raw bisector X onstruction ef. bisector 3. Given 4. X X Reflexive Prop. 5. ΔX Δ X S 6. PT pg. 13

14 The converse of that theorem is also true. Theorem: If 2 sides of a triangle are congruent, the angles opposite those sides are congruent. Theorem: diagonal of a ogram separates the ogram into 2 Δ 's. Given: RSTW W 1 3 T Prove: ΔRST Δ TWR R 4 2 S Statements Reasons 1. RSTW is a ogram Given 2. RS WT ef. of ogram lines cut by trans., alt. int. 's 4. RT RT Reflexive 5. RW ST ef. of ogram lines cut by trans., alt. int. 's 7. ΔRST Δ TWR S Knowing the diagonal separates a parallelogram into 2 congruent triangles suggests some more relationships. pg. 14

15 Theorem: The opposite sides of a parallelogram are congruent. Theorem: The opposite angles of a parallelogram are congruent. oth of those can be proven by adding PT 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 PT after proving triangles congruent by SSS, SS, S, and S will allow us to find many more relationships in geometry. Right Triangles While the congruence postulates and theorems apply for all triangles, we have postulates and theorems that apply specifically for right triangles. HL 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 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 SS ongruence Theorem and could be considered redundant. H 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. pg. 15

16 This congruence theorem is a special case of the S ongruence Theorem and can also be considered redundant. nd finally, we have the Leg ngle ongruence Theorem. L 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 L ongruence Theorem, you would see that could be derived from either the S or S congruence theorems depending on which set of legs or angles are congruent. Once again these theorems could be considered redundant. s marked in this first set, S. s marked in this second set, S. pg. 16