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ProvingTrianglesCongruentTopicPages in PacketAngles in Triangles/Definition ofCongruent TrianglesPages 2-6Identifying Congruent TrianglesCongruent Triangles ProofsC.P.C.T.C.Pages 7- 13Pages 16-21Pages 25-29C.P.C.T.C. and BEYONDIsosceles TrianglePages 30 - 33Pages 34 - 37Proving Triangles Congruentwith hy.legRight Angle Theorem &Equidistance; TheoremsPages 38-43Detour ProofsPage 51- 57Missing Diagram ProofsPages 58- 62Pages 44-50Assignment:(Honors TXTBK)HOLT TXTBK:Page 227#9-14,19-22,4142,45,49This Packet pages 14- 15This Packet pages 22-24Pages 127-129 #’s6,12,13,18,21Pages 135 #’s #2, 5, 7-11, 15Page 155 #’s 20,21, 23, 24,25Page 160 # 16Page 158 #’s 5, 12, 17Pgs 182-183 #’s 4, 9, 14Pg 189-190 #’s 14,15,16,17, 20Pages 174 – 175 #’s11,13,14,17Page 141 #4Page 179 #’s 8, 11, 12, 14Answer Keys Start on page 631

Day 1SWBAT: Use properties of congruent triangles. Provetriangles congruent by using the definition of congruence.2

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5.6.The angle measures of a triangle are in the ratio of 5:6:7. Find the angle measures of thetriangle.7. Solve for m4

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Day 2 - Identifying Congruent Triangles7

Geometric figures are congruent if they are the same size and shape. Corresponding angles andcorresponding sides are in the same in polygons with an equal number of .Two polygons are polygons if and only if their sides are .Thus triangles that are the same size and shape are congruent.Ex 1: Name all the corresponding sides and angles below if the polygons are congruent.Corresponding SidesCorresponding AnglesEx 2:8

Identifying Congruent Triangles9

An included side is the common side of twoconsecutive angles in a polygon. The followingpostulate uses the idea of an included side.10

The pair of triangles below has two corresponding parts marked ascongruent.1.4.Answer: 2.Answer: 5.Answer: 3.Answer: Answer: 6.Answer: 11

Using the tick marks for each pair of triangles, name the method {SSS, SAS, ASA, AAS}that can be used to prove the triangles congruent. If not, write not possible.(Hint: Remember to look for the reflexive side and vertical angles!!!!)12

ChallengeSolve for x.SUMMARYExit Ticket13

Homework14

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Day 3 – Proving Congruent TrianglesWarm - Up1.2.Congruent Triangle Proofs16

2)GivenSeg bisector 17

3)18

LEVEL B4)19

5. Given: ̅̅̅̅̅̅̅̅̅̅̅̅, ̅̅̅̅̅̅̅̅̅̅̅̅̅,20

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Practice with Congruent TrianglesAB1243EDCCADB3.22

BEDCASXRYT23

DCABDC12AB24

Day 4 - CPCTCSWBAT: To use triangle congruence and CPCTC to prove that parts of two triangles arecongruent.25

You Try It!26

Example 1:Z27

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SUMMARYWarm - Up29

C.P.C.T.C. and BEYONDAuxiliary LinesA diagram in a proof sometimes requires lines, rays, or segments that do notappear in the original figure. These additions to diagrams are auxiliary lines.Ex 1: Consider the following problem.This proof would be easy ifTheorem:Ex 2:30

Ex 3: CPCTC and BeyondMany proofs involve steps beyond CPCTC. By using CPCTC first, we canprove altitudes, bisectors, midpoints and so forth. NOTE: CPCTC is notalways the last step of a proof!Theorem: All radii of a circle are congruent!31

Example 4:Given: Q, ̅̅̅̅Prove: ⃗⃗⃗⃗⃗̅̅̅̅SExample 5:Given:,Prove: C is the midpoint of ̅̅̅̅32

SUMMARYExit Ticket33

Day 6 - Isosceles Triangle Proofs34

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Summary of Isosceles TrianglesExit Ticket37

Day 7 - Hy-LegWarm – Up38

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Given: ̅̅̅̅ is an altitude in Circle O.Prove: ̅̅̅̅ ̅̅̅̅OEFG40

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4.42

SUMMARYExit Ticket43

Day 8 –Right Angle Theorems & Equidistance TheoremTheorem: If two angles are both supplementary andcongruent, then they are right angles.(ACB*** Proving that lines are perpendicular dependson you proving that they form .1. Given:Prove:̅̅̅̅̅̅̅̅̅̅̅̅.44

EQUIDISTANCE THEOREMDefinition: The distance between two objects is the length of the shortestpath joining them.Postulate: A line segment is the shortest path between two points.If two points P and Q are the same distance from a third point, X, they aresaid to be equidistant from X.Picture:Statement1. ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅Means .̅̅̅̅̅̅̅̅ , and̅̅̅̅45

Definition: The perpendicular bisector of a segment is the linethat bisects and is perpendicular to the segment.Equidistance Theorem –If two points are each equidistant from the endpoints of a segment, thenthe two points determine the perpendicular bisector of that segment.46

2.3.Given:Prove:̅̅̅̅̅̅̅̅47

WHY the Equidistance Theorem?48

Converse of the Equidistance Theorem –If a point is on the perpendicular bisector a segment, then it is equidistantfrom the endpoints of that segment.4.Given: ̅̅̅̅̅̅̅̅Prove:49

SUMMARYExit Ticket50

Day 9 - Detour ProofsWarm - UpGiven:Prove:̅̅̅̅̅̅̅̅̅51

Example 1:Prove:Whenever you are asked to prove that triangles or parts oftriangles are congruent and you suspect a detour may beneeded, use the following procedures.52

Procedure for Detour Proofs1.2.3.4.5.Determine which triangles you must provecongruent to reach the desired conclusionAttempt to prove those triangles congruent – ifyou cannot due to a lack of information – it’stime to take a detour Find a different pair of triangles congruent basedon the given informationGet something congruent by CPCTCUse the CPCTC step to now prove the trianglesyou wanted congruent.Example 2:Given: 1 2 , 3 4Prove:53

Example 3:54

Example 4:55

SUMMARY(3,4,5)(7,9,10)56

Exit Ticket57

Day 10 - Missing Diagram ProofsWarm - Up58

Many proofs we encounter will not always be accompanied by adiagram or any given information. It is up to us to find the importantinformation, set up the problem, and draw the diagram all byourselves!!!Procedure for Missing Diagram Proofs1. Draw the shape, label everything.2. The “if” part of the statement is the “given.”3. The “then” part of the statement is the “prove.”4. Write the givens and what you want to prove.Example 1: If two altitudes of a triangle are congruent, then thetriangle is isosceles.Given:Prove:59

Example 2: The medians of a triangle are congruent if the triangle is equilateral.Given:Prove:60

Example 3: the altitude to the base of an isosceles trianglebisects the vertex angle.Given:Prove:61

SUMMARYExit Ticket62

ANSWER KEYS63

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Day 2 Answers65

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Day 3 – Answers67

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Answers to Day 470

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Answers to Day 573

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Answers to Isosceles HW Day 620.21.1.1. Given2.2.3.Prove:3. Transitive Prop. (1, 2)4.4. Linear Pair Thm5.5. Congruent Suppl. Thm6.6. Transitive Prop. (3, 5)7.7.8. CDG is Isosceles8. Definition of

23. Given:Prove: Figure AOBP is equilateral.1. Given1.2. Definition of angle bisector(A) 2.(A)(S)3. Reflexive Property3.4.APB4. ASA (2, 3, 2)5.5. CPCTC6.6. All radii of a7.7. Transitive Prop. (5, 6)8. Figure AOBP is equilateral8. If a figure has all sidesare

24.(S)(S)(A)(2, 4, 3)

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Page 160 #161.1. Given2.2. Def of(A)3.3. all right(A)4.4. Reflexive Property(S)5.DEB5. AAS (3, 4, 1)6.6. CPCTC7.7. Transitive Prop. (1, 6)8.is equilateral8. If a figure has all sides

Geometry HonorsAnswer KeyProving Triangles Congruent with Hypotenuse LegPage 158 #’s 5 , 12 and 1712)

Right Angle Theorem and Equidistance TheoremsPages 182 – 183 #’s 4, 9, 14

Page 189 – 190 #’s 14, 15, 16, 17, and 20

Answers to Detour ProofsDetour Proofs pages 174- 175 #’s 11, 13, 14, 17

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Answers to Missing Diagram ProofsPage 179 #8, 11, 12, 14All Right Angles are Congruent

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