Lesson 15 Proofs involving congruence

Transcription

1 1 Lesson 15 Proofs involving congruence Congruent figures are objects that have exactly the same size and shape One figure would lie exactly on top of the other figure (Don t confuse congruency with similarity Similar figures have equal angles while the sides are not equal but are in proportion) To show two triangles are congruent, we would need to show that all the angles and all the sides of one triangle are congruent (equal) to the corresponding angles and sides of a second triangle However, we have congruence postulates that give us shortcuts to showing triangle congruence These postulates were discussed in lesson 9: SSS, SAS, AAAS, and HL Example 1: Given: AD BD, ED CD ; Prove: ADE BDC Step 1: Our first step is to outline the proof We do this by first marking what we were given Since we are given AD BD, we will mark these sides with one tick showing that they are congruent This means we now have one side, S Since we are given ED CD, we will mark these sides with two ticks showing that they are congruent This gives us a second side so we now have SS Step 2: Look at the drawing and see what else we can defer from the information given If you are stuck, look again at the four congruence postulates: SSS, SAS, AAAS, and HL Usually you can rule a couple of them out We see we know two sides already so probably we will use either SSS or SAS We know we can rule out HL postulate because this postulate requires that we have a right triangle We were not told that we had a right angle, nothing is said about having perpendicular lines nor about having an altitude Therefore, we can not have a right triangle so we know we can not use the HL postulate Also the chances of using AAAS when given two sides would be very slim although possible Therefore we need to know either a side or an angle Many students incorrectly believe that two congruent sides implies that the third

2 2 sides must also be congruent This is VERY, VERY WRONG!!!! Two congruent angles does imply that the third angle is congruent because the triangles must each measure 180 o However, this can NOT be carried over to sides!!!! What do we notice about the angles? We see that ADE is directly across from BDC as the angles share the same lines These angles are called vertical angles and all vertical angles are congruent Therefore, we will mark this angle Step 3: Notice that the angle we just marked was between the two sides marked as being congruent Therefore we have the SAS postulate We are now ready to write our formal proof Step 4: We use a two column proof where statements are on the left and the reasons why we can make those statements are on the right What we want to prove should always be our very last statement on our proof or we did something wrong Be sure to number each statement AND each reason In this proof, we will begin with one of the given statements S 1 AD BD (Write S beside the number one because we are showing a side of the triangle is congruent to another side This will help us remember to include all the information from our outline we need in our formal proof) 1 Given Now we will follow with another statement Typically it will be the second thing you marked in your outline

3 3 S 1 AD BD (Write S beside the number one because we are showing a side of the triangle is congruent to another side This will help us remember to include all the information from our outline we need in our formal proof) 1 Given S 2 ED CD (Again we place the letter S beside this statement to show we have a side congruent to another side for our congruency postulate) A 3 ADE BDC (This time we will write the letter A beside the number 3 to show we have just showed an angle is congruent to another angle) 2 Given 3 Vertical angles are congruent (The reason listed must be why we were able to say ADE was congruent to BDC In our outline, we said they were vertical angles and all vertical angles are equal Notice we now have a side, another side, and the angle included between the two sides listed so this is the SAS congruency postulate The SAS postulate shows that triangles are congruent so we must be able to write which triangle is congruent to which one Remember order is important as the first letter of one triangle must be congruent to the first letter of the second triangle and so forth We see that we wanted to show ADE is congruent to BDC so we will write only ADE down in that order Then we will look back at our sketch to see that the corresponding parts of the second triangle match the same order as what we want to show Notice that angle E corresponds to angle C and angle A corresponds to angle B so we are correct We will now add our last step to our proof 4 ADE BDC 4 SAS Postulate This is what we wanted to prove so we are now finished

4 4 Answer: SAS congruency postulate S 1 AD BD S 2 ED CD A 3 ADE BDC 4 ADE BDC 1 Given 2 Given 3 Vertical angles are congruent 4 SAS Postulate Example 2: Given Q S, PQ SR Write a two-column proof to prove: QR SP Step 1: Our first step is to outline the proof We do this by first marking what we were given, Q S Since PQ SR, we know that QPR and SRP are alternate-interior angles because they lie inside the parallel lines and are on alternating sides of the transversal Please note that three letters must be used to name the angle and we can not use just letter P or just letter R because there is more than one angle formed from the vertex at P and vertex at R unlike angles Q and S above

5 5 Also note that in this case QRP and SPR are NOT also alternate-interior angles because the angle must connect the parallel line to the transversal and not just another side of the quadrilateral See the diagram below We now have two pairs of congruent angles in our outline Since two angles of one triangle are congruent to two angles of another triangle, then the third angles must be congruent We can also see that the triangles share the side PR, thus it is congruent to itself We now have enough to show the AAAS congruency postulate so we will begin to write our formal proof

6 6 1 PQ SR A 2 QPR SRP A 3 Q S A 4 QRP SPR S 5 PR PR 6 QPR SRP 7 QR SP (To say statement 7 you must have just previously stated that you had congruent triangles) 1 Given 2 If two parallel lines are cut by a transversal, then the alternate-interior angles are congruent (Line 2 must follow line 1 because our reason for line 2 refers to parallel lines which was shown in the line above) 3 Given 4 AA AAA (This notation means if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent) 5 Reflexive Axiom 6 AAAS congruency postulate 7 CPCTC (This means corresponding parts of congruent triangles are congruent) Answer: Outline: AAAS congruency postulate 1 PQ SR A 2 QPR SRP A 3 Q S A 4 QRP SPR S 5 PR PR 6 QPR SRP 7 QR SP 1 Given 2 If two parallel lines are cut by a transversal, then the alternate-interior angles are congruent 3 Given 4 AA AAA 5 Reflexive Axiom 6 AAAS congruency postulate 7 CPCTC