Congruent Polygons and Congruent Parts

Size: px

Start display at page:

Download "Congruent Polygons and Congruent Parts"

Derrick Ray
6 years ago
Views:

1 Congruent Polygons and Congruent Parts Two polygons are congruent if and only if there is a one-to-one correspondence between their vertices such that corresponding angles are congruent and corresponding sides are congruent Corresponding parts of congruent polygons are congruent. Congruent Triangles The following also holds true for Triangles: Reflexive Property ΔABC ΔABC Symmetric Property If ΔABC ΔDEF then ΔABC ΔDEF Transitive Property If ΔABC ΔDEF and ΔDEF ΔRST then ΔABC ΔRST Proving Triangles Congruent Side Angle Side Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other. SAS SAS Angle Side Angle Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of the other. ASA ASA Side Side Side Two triangles are congruent if three sides of one triangle are congruent respectively, to three sides of the other. SSS SSS Congruence Based on Triangles An Altitude of a triangle is a line segment drawn from any vertex of the triangle perpendicular to and ending in the line contains the opposite side. A Median of a triangle is a line segment that joins any vertex of the triangle to the midpoint of the opposite side. An Angle Bisector of a triangle is a line segment that bisects any angle of the triangle and terminates in the side opposite that angles. Using congruent triangles to prove line segments congruent and angles congruent. (Corresponding parts of congruent triangles are congruent)(cpctc) 1

2 Isosceles Triangles If two sides of a triangle are congruent, the angles opposite these sides are congruent. The median from the vertex angle of an isosceles triangle bisects the vertex angle. The median from the vertex angle of an isosceles triangle is perpendicular to the base. Equilateral Triangle Every equilateral triangle is equiangular. Perpendicular Bisector The Perpendicular Bisector of a line segment is any line or subset of a line that is perpendicular to the line at its midpoint. If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment. If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of a line segment. The perpendicular bisectors of the sides of a triangle are congruent. Methods of Proving Lines or Line Segments Perpendicular 1. The two lines form right angles at their point of intersection. 2. The two lines form congruent adjacent angles at their point of intersection. 3. Each of two points on one line is equidistant from the endpoints of a segment of the other. 2

3 page

4 4

5 5

6 6

7 #1 Given: ΔABC CD bisects AB CD AB Prove: ΔACD ΔBCD 1. ΔABC CD bisects AB CD AB 1. Given 7

8 #2 Given: ABC and DBE bisect each other. Prove: ΔABD ΔCBD 8

9 #3 Given: AB = CD and BC = AD DAB, ABC, BCD and CDA are rt Prove: ΔABC ΔADC #4 Given: PQR RQS PQ QS Prove: ΔPQR ΔRQS 9

10 #5 Given: AEB & CED intersect at E E is the midpoint AEB AC AE & BD BE Prove: ΔAEC ΔBED 10

11 #6 Given: AEB bisects CED AC CED & BD CED Prove: ΔEAC ΔEBD 11

12 #7 Given: ΔABC is equilateral D midpoint of AB Prove: ΔACD ΔBCD 12

13 #8 Given: m A = 50, m B = 45, AB = 10cm, m D = 50 m E = 45 and DE = 10cm Prove: ΔABC ΔDEF #9 Given: GEH bisects DEF m D = m F Prove: ΔGFE ΔDEH 13

14 #10 Given: PQ bisects RS at M R S Prove: ΔRMQ ΔSMP #11 Given: DE DG EF GF Prove: ΔDEF ΔDFG 14

15 #12 Given: KM bisects LKJ LK JK Prove: ΔJKM ΔLKM #13 Given:. PR QR P Q RS is a median Prove: ΔPSR ΔQSR 15

16 #14 Given: EG is bisector EG is an altitude Prove: ΔDEG ΔGEF 16

17 #15 Given: A and D are a rt AE DF AB CD Prove: EC FB 17

18 #16 Given: AC BC D midpoint of AB Prove: A B #17 Given:. AB CD CAB ACD Prove: AD BC 18

19 #18 Given: AEB & CED bisect each Other Prove: C D 19

20 #19 Given: KLM & NML are rt KL NM Prove: K N 20

21 #20 Given: AB BC CD PA PD & PB PC Prove: a) APB DPC b) APC DPB 21

22 #21 Given: PM is Altitude PM is median Prove: a) ΔLNP is isosceles b) PM is bisector 22

23 #22 Given: AC BC Prove: CAD CBE #23 Given: AB BC & AD CD Prove: BAD BCD 23

24 #24 Given: ΔABC ΔDEF M is midpoint of AB N is midpoint DE Prove: ΔAMC ΔDNF 24

25 #25 Given: ΔABC ΔDEF CG bisects ACB FH bisects DFE Prove: CG FH 25

26 #26 Given: ΔAME ΔBMF DE CF Prove: AD BC 26

27 #27 Given: AEC & DEB bisect each other Prove: E is midpoint of FEG 27

28 #28 Given: BC BA BD bisects CBA Prove: BD bisects CDA 28

29 #29 Given: AE FB AD BC A and B are Rt. Prove: ΔADF ΔCBE DF CE 29

30 #30 Given: SPR SQT PR QT Prove: ΔSRQ ΔSTP R T 30

31 #31 Given: AD BC AD AB & BC AB Prove: ΔDAB ΔCBA AC BD 31

32 #32 Given: BAE CBF BCE CDF AB CD Prove: AE BF E F 32

33 #33 Given: TM TN M is midpoint TR N is midpoint TS Prove: RN SM 33

34 #34 Given: AD CE & BD EB Prove: ADC CEA 34

35 #35 Given: AE BF & AB CD ABF is the suppl. of A Prove: ΔAEC ΔBFD 35

36 #36 Given: AB BC BD bisects ABC Prove: AE CE #37 Given: PB PC Prove: ABP DCP 36

37 #38 Given: AC and BD are bisectors of each other. Prove: AB BC CD AD 37

38 #39 Given: AEFB, 1 2 CE DF, AE BF Prove: ΔAFD ΔBEC #40 Given: SX SY, XR YT Prove: ΔRSY ΔTSX 38

39 #41 Given: AD BC AD AB, BC AB Prove: ΔDAB ΔCBA 39

40 #42 Given: AF CE 1 2, 3 4 Prove: ΔABE ΔCDF 40

41 #43 Given: AB BF, CD BF 1 2, BD FE Prove: ΔABE ΔCDF 41

42 #44 Given: BAC BCA CD bisects BCA AE bisects BAC Prove: ΔADC ΔCEA 42

43 #45 Given: TR TS, MR NS Prove: ΔRTN ΔSTM #46 Given: CEA CDB, ΔABC AD and BE intersect at P PAB PBA Prove: PE PD 1. 43

44 #47 Given: AB AD and BC CD Prove:

45 #48 Given: BD is both median and altitude to AC Prove: BA BC 45

46 #49 Given: CDE CED and AD EB Prove: ACD BCE 46

47 #50 Given: Isosceles triangle CAT CT AT and ST bisects CTA Prove: SCA SAC 47

48 #51 Given: 1 2 BD AC Prove: ΔABD ΔCBD 48

49 #52 Given: P S R is midpoint of PS Given: ΔPQR ΔSTR 49

50 #53 Given: FG DE G is midpoint of DE Given: ΔDFG ΔEFG 50

51 #54 Given: AC BC D is midpoint of AB Prove: ΔACD ΔBCD 51

52 #55 Given: PT bisects QS PQ QS and TS QS Prove: ΔPQR ΔRST 52

53 #56 Given: AB DE and FE BC FE AD and BC AD Prove: ΔAEF ΔCBD 53

54 #57 Given: SM is bisector of LP RM MQ a b Prove: ΔRLM ΔQPM 54

55 #59 Given: AC BC CD AB Prove: ΔACD ΔBCD 55

56 #60 Given: FQ bisects AS A S Prove: ΔFAT ΔQST 56

57 #61 Given: A D and BCA FED AE CD AEF BCD Prove: ΔABC ΔDFE 57

58 #62 Given: SU QR, PS RT TSU QRP Prove: ΔPQR ΔSTU Q U 58

59 #63 Given: M D ME HD THE SEM Prove: ΔMTH ΔDSE 59

60 #64 Given; SQ bisects PSR P R Prove: ΔPQS ΔQSR 60

61 #65 Given: PQ SQ and TS QS R midpoint of QS Prove: P T 61

62 #66 Given: BC FB, BT BV DV TS, CD FS Prove: D S 62

63 #67 Given: PQ DE and PB AE QA PE and BD PE Prove: D Q 63

64 #68 Given: TS TR P Q Prove: PS QR 64

65 #69 Given: HY and EV bisect each other Prove: HE VY 65

66 #70 Given: E D and A C B is the midpoint of AC Prove: AE DC #71 Given: E is midpoint of AB AD AB and BC AB 1 2 Prove: AD CB 66

67 67

Picture: Picture: Picture:

Picture: Picture: Picture: Postulate - Side-Side-Side (SSS) Congruence: If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. Picture: Postulate - Side-Angle-Side (SAS)