1. Objectives: To recognize congruent figures and their corresponding parts

2. Congruent figures -> have the same size and shape. When two figures are congruent, you can move one so that it fits exactly on the other one. Congruent polygons -> have congruent corresponding parts (matching sides and angles). Matching vertices are corresponding vertices. *When you name congruent polygons, always list corresponding vertices in the same order.*

3. Ex: List the congruent corresponding parts D T J F R C Sides: TJ congruent to RC JD congruent to CFDT congruent to FR Angles: T congruent to R J congruent to CD congruent to F

4. The fins of the Space Shuttle suggest congruent pentagons. Find m<B. A C E SP RW B DT 132 88

5. Theorem 4-1 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. A BC D E F <C congruent to <F

6. Example of proving triangles congruent: Given: PQ congruent PSQR congruent SR <Q congruent <S<QPR congruent <SPR Prove: Triangle PQR congruent Triangle PSR P QS R

7. Homework #19 Due Thursday (Oct 17) Page 200 – 201 #1 – 27 odd

8. Section 4.2 – Triangle Congruence by SSS and SAS Objectives: To prove two triangles congruent using the SSS and SAS Postulates

9. In the last section, we learned that if two triangles have three pairs of congruent corresponding angles and three pairs of congruent corresponding sides, then the triangles are congruent. However, you do not need to know that all six corresponding parts are congruent in order to conclude that two triangles are congruent.

10. Postulate 4.1 -> Side-Side-Side (SSS) Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. A B C D EF Triangle ABC is congruent to triangle DEF

11. Example using SSS Given: HF congruent HJFG congruent JK H is the midpoint of GK Prove: Triangle FGH congruent triangle JKH H F G J K

12. Postulate 4.2 -> Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. A B C D E F Triangle ABC congruent to triangle DEF

13. Example using SAS Given: AB congruent BEBC congruent BD Prove: Triangle ABC congruent triangle DBE A B E C D

14. Homework #20 Due Monday (Oct 22) Page 208 – 209 # 2 – 24 Even

15. Section 4.3 – Triangle Congruence by ASA and AAS Objectives: To prove two triangles congruent using the ASA postulate and the AAS theorem

16. In the last section, we learned that two triangles are congruent if two pairs of sides are congruent and the included angles are congruent (SAS). Today, we will find out that two triangles are also congruent if two pairs of angles are congruent and the included sides are congruent (ASA).

17. Postulate 4.3 – Angle-Side-Angle (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. A BC D EF Triangle ABC congruent to triangle DEF

18. Example using ASA Given: NM congruent NP<M congruent <P Prove: Triangle NML congruent triangle NPO L M N P O

19. Theorem 4.2 – Angle-Angle-Side (AAS) Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non- included side of another triangle, then the triangles are congruent. A B C D E F Triangle ABC congruent to triangle DEF

20. Examples using AAS Given: <S congruent <QRP bisects <SRQ Prove: Triangle SRP congruent to triangle QRP P Q S R

21. Given: XQ parallel to TRXR bisects QT Prove: triangle XMQ congruent triangle RMT X Q M T R

22. Homework #21 Due Tuesday (Oct 23) Page 215 – 217 # 1 – 23 odd

23. Section 4.4 – Using Congruent Triangles: CPCTC Objective: To use triangle congruence and CPCTC to prove that parts of two triangles are congruent.

24. With SSS, SAS, ASA, and AAS, we learned how to use three parts of triangles to show that the triangles are congruent. Once you have triangles congruent, you can make conclusions about their other parts because, by definition, Corresponding Parts of Congruent Triangles are Congruent. This is abbreviated as CPCTC.

25. Ex: In an umbrella frame, the stretchers are congruent and they open to angles of equal measure. Given: SL congruent to SR<1 congruent to <2 Prove: <3 congruent to <4 C RL S 1 2 3 4 5 6

26. Ex: Given: <Q congruent <R<QPS congruent <RSP Prove: SQ congruent to PR P Q SR

27. According to legend, one of Napoleon’s officers used congruent triangles to estimate the width of a river. On the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted the spot on his side of the river that was in line with his eye and the tip of his visor. Given: <DEG and <DEF are right angles<EDG congruent <EDF Prove: EF congruent EG F D E G

28. Homework # 22 Due Wed (Oct 24) Page 222 – 224 # 1 – 19 odd Quiz Thursday/Friday Section 4.1 – 4.4

29. Section 4.5 – Isosceles and Equilateral Triangles Objectives: To use and apply properties of isosceles triangles

30. Isosceles triangles are common in the real world. They can be found in everything from bridges to buildings. Legs -> the congruent sides of an isosceles triangle Base -> third side (not one of the legs) Vertex angle -> formed by the two congruent sides (legs) Base angles -> other two angles in an isosceles triangle

31. Legs Vertex angle Base Base angles

32. Theorem 4.3 -> Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. C A B <A congruent to <B

33. Theorem 4.4 -> Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite the angles are congruent. C A B AC congruent to BC

34. Theorem 4.5 -> Isosceles Bisector Theorem The bisector of the vertex angles of an isosceles triangle is the perpendicular bisector of the base. C AB D CD perpendicular to AB CD bisects AB

35. Ex: Explain why each statement is true. <WVS congruent to <S TR congruent to TS T W S V R U

36. Ex: Using Algebra Find the value of y M N L O y 63

37. Corollary -> a statement that follows immediately from a theorem. Corollary to Theorem 4.3 If a triangle is equilateral, then the triangle is equiangular Corollary to Theorem 4.4 If a triangle is equiangular, then the triangle is equilateral

38. Homework #23 Due Tuesday (Oct 30) Page 230 – 232 #1 – 7 all #10 – 13 all #19 – 22 all

39. Section 4.6 – Congruence in Right Triangles Objectives: To prove triangles congruent using the HL Theorem

40. In a right triangle, the side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called legs.

41. Theorem 4.6 -> Hypotenuse-Leg (HL) Theorem If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Three conditions must be met in order to use the HL Theorem: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

42. Ex: Given: AB congruent to AC Prove: triangle ADB congruent to triangle ADC A BC D

43. Ex: Given: CD congruent to EAAD is perpendicular bisector of CE Prove: triangle CBD congruent to triangle EBA C D A E B

44. Ex: Given: WZ congruent to YX<W and <Y are right angles Prove: triangle XWZ congruent to triangle ZYX W X YZ

45. Homework # 24 Due Wednesday/Thursday Page 237 – 239 # 1 – 4 all #6 – 14 even #20 – 21 all

46. Section 4.7 – Using Corresponding Parts of Congruent Triangles Objectives: To identify congruent overlapping triangles To prove two triangles congruent by first proving two other triangles congruent

47. Some triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a c0mmon side or angle. You can simplify your work with overlapping triangles by separating and redrawing triangles.

48. Example: Identifying Common Parts Separate and redraw triangle DFG and EHG. Identify the common angle. G HF DE G D F G H E <G is the common angle

49. Ex: Using Common Parts Given: <ZXW congruent <YWX <ZWX congruent <YXW Prove: ZW congruent to YX Z Y W X

50. Given: triangle ACD congruent to triangle BDC Prove: CE congruent to DE AB CD E

51. Sometimes you can prove one pair of triangles congruent and then use their congruent corresponding parts to prove another pair congruent. Ex: Given: PS congruent to RS<PSQ congruent to <RSQ Prove: Triangle QPT congruent to triangle QRT Q R S P T

52. Ex: Separating Overlapping Triangles Given: <CAD congruent <EAD<C congruent <E Prove: BD congruent to FD A B C D E F

53. Homework #25 Due Thurs/Friday (Nov 1/Nov 2) Page 243 – 245 #1 – 6 all #8 – 11 all #17-18 and #20-21 all Chapter 4 Test Tuesday