Postulates and Theorems in Proofs

Size: px

Start display at page:

Download "Postulates and Theorems in Proofs"

Hugo O’Brien’
6 years ago
Views:

1 Postulates and Theorems in Proofs A Postulate is a statement whose truth is accepted without proof A Theorem is a statement that is proved by deductive reasoning. The Reflexive Property of Equality: a = a A quantity is equal to itself. The Symmetric Property of Equality: If a = b then b = a An Equality may be expressed in either order. The Transitive Property of Equality: If a = b and b = c then a = a Quantities equal to the same quantity are equal to each other. The Substitution Postulate: A quantity may be substituted for its equal in any statement of equality. The Partition Postulate: A whole is equal to the sum of all its parts A segment or angle Is congruent to the sum of its parts The Addition Postulate: If a = b and c = d then a + c = b + d If equal quantities are added to equal quantities, the sums are equal. The Subtraction Postulate: If a = b and c = d then a - c = b - d If equal quantities are subtraction to equal quantities, the difference are equal. The Multiplication Postulate: If a = b and c = d then ac = bd If equal quantities are multiplied to equal quantities, the products are equal The Division Postulate: If a = b and c = d then a/c = b/d If equal quantities are divided by nonzero equals quantities, the quotients are equal Halves of equal quantities are equal. The Power Postulate: If a = b then a 2 = b 2 The Roots Postulate: If a = b and a > 0 then a = b. 1

2 #1 Given: AM CN MB ND Prove: AB CD #2 Given: m 1 m 2 m 3 m 4 Prove: m QPS m QRS 2

3 #3 Given: AD BE DC EC Prove: AC BC #4 Given: m XWZ m WXZ m ZWY m ZXY Prove: m XWY m WXY 3

4 #5 Given: DC DE CB EA Prove: DB DA #6 Given: m QST m PTS m QSP m PTQ Prove: m PST m QTS 4

5 #7 Given: m ABC m DCB m a m b Prove: m x m y #8 Given: AD CB AE CF Prove DE BF 5

6 #9 Given: AD BC AE BF Prove: ED FC #10 Given: m WZY m WXY m RZY m RXS Prove: m WZR m YXS 6

7 #11 Given: m FCD m GDC m GCD m FDC Prove: m FCG m GDF #12 Given: m ADC m ABC m ADE m CBD Prove: m EDC m DBA 7

8 #13 Given: AB CD AF ½ AB EC ½ CD Prove: AF EC #14 Given: m DAB m DCB m 2 ½ m DAB m 1 ½ m DCB Prove: m 1 m 2 8

9 #15 Given: m SRX m RSY m SRY ½ m SRX m RSX ½ m RSY Prove: SRY RSX #16 Given: AC DB AE ½ AC DE ½ DB Prove: AE DE 9

10 #17 Given: AB CD Prove: AC BD #18 Given: AC BD Prove: AB CD 10

11 #19 Given: AD DC DC DB Prove: AD DB #20 Given: AB CD CD DE Prove: AB DE #21 Given: AB DE Prove: AD BE 11

12 #22 Given: AEB CED Prove: AEC BED #23 Given: AEB BEC DEC BEC Prove: AEB DEC 12

13 #24 Given: AEC BED Prove: AEB DEC #25 Given: Prove: ABC ACB 13

14 #26 Given: AD BE Prove: AC CE #27 Given: AB CD CD DE Prove: AB DE #28 Given: AB CD Prove: AC BD 14

15 #29 Given: BD bisects ABC Prove: 1 2 #30 Given: AB BC DF bisects AB EF bisects BC Prove: AD EC 15

16 #31 Given: DCB DAB AC bisects DCB AC bisects DAB Prove: r s #32 Given: AE DB Prove: AD EB 16

17 #33 Given: DA bisects CAB EB bisects CBA CAB CBA Prove: 1 2 #34 Given: DE FB Prove: DF EB 17

18 #35 Given: m LQM m NQP Prove: m LQN m MQP #36 Given: AD BC E is the midpoint of AD F is the midpoint of BC Prove: AE CF 18

19 #37 Given: AC BD AC and BD bisect each other Prove: AE EB #38 Given: RD bisects CDA Prove:

20 Vertical Angles in Proofs - 2 steps 1. State Intersecting lines form vertical angles. 2. Vertical angles are congruent. Perpendicular lines and Right angles in Proofs 2 steps 1. State Perpendicular lines form right angles. 2. All right angles are congruent. #39 Given: AB and CD intersect at E Prove: AEC BED #40 Given: AB and CD intersect at E Prove: AED BEC 20

21 #41 Given: AB and CD intersect at E Prove: AED BEC #42 Given: AC and DE intersect at B Prove: CBE ABD 21

22 #43 Given: DE and AC intersect at B BC bisects EBF Prove:

23 #44 Given: DB AC Prove: 1 2 #45 Given: CA AB, DE AB Prove: CAB DBA 23

24 #46 Given: CE ED, AD ED Prove: CEB ADB #47 Given: AE EC, BE ED Prove:

25 #48 Given: FB AD, GC AD 1 4 Prove:

26 #1 6 In the diagram below, ΔABC ΔDEF 1) Complete the statement A A) F C) D B) C D) E 2) Complete the statement B A) A C) F B) E D) D 3) Complete the statement DFE A) CBA C) CAB B) ABC D) ACB 4) Complete the statement AC A) AB C) BC B) DF D) DE 5) Complete the statement AB A) AB C) BC B) DF D) DE 6) Complete the statement EF A) AB C) BC B) DF D) DE #7 12 In the diagram below ΔPQR ΔSTR 7) Complete the statement P A) Q C) S B) T D) TRS 8) Complete the statement Q A) T C) TRS B) S D) PRQ 9) Complete the statement PRQ A) TRS C) Q B) S D) T 11) Complete the statement TR A) SR C) TR B) QP D) RQ 10) Complete the statement PR A) SR C) TR B) QP D) RQ 12) Complete the statement TS A) SR C) TR B) QP D) RQ 26

27 13) Given: ΔSUV ΔEVU, Name the three pairs of congruent angles and three pairs of Congruent sides. 14) Given: ΔABD ΔBAC, Name the three pairs of congruent angles and three pairs of Congruent sides. 15) Two triangles are congruent if A) corresponding sides and corresponding angles are congruent. B) corresponding sides are in proportion C) The angles in each triangle have sum of 180. D) corresponding angles are congruent. Proving Triangles Congruent SSS - SideSideSide If 3 sides in one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent. SAS - SideAngleSide If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA - AngleSideAngle If two angles and included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS - AngleAngleSide If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL - HL HypLeg If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent (CPCTC) Corresponding parts of congruent triangles are congruent. 27

28 #16 28 Which method, if any, will prove the triangles are congruent? 16) 17) A) SSS C) none B) SAS D) ASA 18) A) SSA C) none B) SAS D) AAS 19) A) ASA C) none B) SAS D) SSS 20) A) SAS C) ASA B) none D) AAS 21) A) SAS C) ASA B) none D) SSS 22) A) none C) SAS B) SSS D) AAS 23) A) AAS C) HL B) SAS D) none 24) A) SSS C) SAS B) none D) ASA 25) A) none C) SAS B) SSA D) AAS A) ASA C) SSS B) SSA D) none 28

29 26) 27) A) SSS C) AAS B) none D) ASA A) SAS C) AAS B) ASA D) none 28) A) AAA C) ASA B) None D) AAS 29) In the accompanying diagram ΔABC and ΔRST are right triangles with right angles at B and S, respectively: A R & BC ST 30) In the accompanying diagram, B is the midpoint of AC, DA AC, EC AC, and DA EC. Which statement can be used to justify that ΔABC ΔRST? A) ASA ASA C) AAS AAS B) HL HL D) SAS SAS Which method of proof may be used to prove ΔDAB ΔECB? A) ASA ASA C) SAS SAS B) HL HL D) AAS AAS 29

30 31) In the accompanying diagram LP LR, TR LR, and M is the midpoint of LR. 32) In the accompanying diagram, PR RQ, SQ RQ, and PQ SR. Which method could be used to prove ΔTMR ΔPML? A) SSS SSS C) SAS SAS B) HL HL D) ASA ASA 33) In the accompanying diagram of quadrilateral QRST, QR RS, QT TS, and RS TS. Which statement can be used to prove that ΔPQR ΔSRQ A) HL HL C) SSS SSS B) AAS AAS D) SAS SAS 34) In the accompanying diagram of quadrilateral ABCD, diagonal AC bisects BAD and BCD. Which method of proof may be used to prove ΔQRS ΔQTS? A) HL C) SAS B) ASA D} AAS 35) In the accompanying diagram, MN NP, QP NP, and O is the midpoint of NP and MQ. Which statement can be used to prove that ΔABC ΔADC? A) ASA ASA C) SAS SAS B) SSS SSS D) HL HL 36) The pair of triangles below have two corresponding parts marked as congruent. What additional information is needed for SAS congruence correspondence? Which reason would be least likely to be used to prove ΔMNO ΔQPO? A) SAS SAS C) ASA ASA B) AAS AAS D) HL HL A) NA MO C) DA TO B) A O D) N M 30

31 37) The pair of triangles below have two corresponding parts marked as congruent. What additional information is needed for ASA congruence correspondence? 38) The pair of triangles below have two corresponding parts marked as congruent. What additional information is needed for SSS congruence correspondence? A) A M C) N M B) A O D) N O 39) The pair of triangles below have two corresponding parts marked as congruent. What additional information is needed for SAS congruence correspondence? A) L A C) CE KA B) E K D) EL KA 40) The pair of triangles below have two corresponding parts marked as congruent. What additional information is needed for ASA congruence correspondence? A) E K C) EL KA B) C J D) L A A) R U C) PR ST B) PQ ST D) P S 41) The pair of triangles below have two corresponding parts marked as congruent. What additional information is needed for AAS congruence correspondence? A) R U B) RQ TU, only C) RP SU, only D) either RQ TU or RP SU 31

32 32

IB MYP Unit 6 Review

IB MYP Unit 6 Review Name: Date: 1. Two triangles are congruent if 1. A. corresponding angles are congruent B. corresponding sides and corresponding angles are congruent C. the angles in each triangle have a sum of 180 D.