Using Definitions and Theorems in Proofs

A midpoint divides a segment into 2 ≅ segments

A midpoint divides a segment in half

A bisector intersects a segments at its midpoint

An angle bisector divides an angle into 2 ≅ angles

An angle bisector divides an angle in half

Perpendicular Lines intersect at right angles

Right angles measure 90

An altitude is a segment from a vertex to the opposite side of a

A median is a segment from a vertex to the opposite midpoint of a

1. Given: E is the midpoint of ̅̅ ̅̅ Prove: ̅̅ ̅̅ ≅ ̅̅ ̅̅

Statement Reasons

2. Given: ⃡ bisects ̅̅ ̅̅ at X Prove: ̅̅ ̅̅ = ̅̅ ̅̅

Statement Reasons

3. Given: bisects ABC Prove: ABD ≅ CBD

Statement Reasons

4. Given: ⃡ ̅̅ ̅̅ ̅̅ Prove: mDEB = 90

Statement Reasons

A BE

T RX

L

M

AB

C D

A B

C

D

E

5. Given: ⃡ is the perpendicular bisector of ̅̅ ̅̅ ̅̅ Prove: mCEA = 90 and ̅̅ ̅̅ ̅̅ ̅̅

Statement Reasons

6. Given: ̅̅̅̅ is a median of RQT Prove: ̅̅̅̅ ≅ ̅̅̅̅

Statement Reasons

7. Given: ̅̅̅̅ is an altitude of RQT Prove: is a measures 90

Statement Reasons

8. Given: ̅̅ ̅̅ ̅̅ ̅̅ ̅ Prove: ̅̅ ̅̅ is an altitude of ∆SRA

Statement Reasons

9. Given: O is the midpoint of ̅̅ ̅̅ ̅̅ Prove: ̅̅ ̅̅ is a median of ∆FLY

Statement Reasons

10. Given: FYO ≅ LYO Prove: ̅̅ ̅̅ is an angle bisector

Statement Reasons

A B

C

D

E

Q

R S T

Q

R S T

S T A

R

Some more Definitions • A ∆ with a right angle is a right ∆ • A ∆ with two ≅ sides is an isosceles triangle 11. Given: ̅̅ ̅̅ ≅ ̅̅ ̅̅ Prove: is isosceles

Statements Reasons 12. Given: ̅̅ ̅̅ ̅ Prove: ∆HJL is a right triangle Statements Reasons

S B

K

Proofs Using the Properties of Equality

Postulate – A statement that is an obvious truth and does not require a proof. Theorem – A true statement that has been proven.

The Addition and Subtraction Postulates of Equality When equal quantities are added (or subtracted) from equal quantities, the sums (or differences) are equal.

Write out the two equalities in two separate statements, then add or subtract them in a third statement. The reason is the Addition Postulate or Subtraction Postulate. We also have multiplication and division postulates which work the same way! 15. Given: mABC = 42 and mDEF = 48 Prove: ABC and DEF are complementary Statements Reasons 16. Given: mDLP = 107 and mMEF = 73 Prove: DLP and MEF are supplementary Statements Reasons

17. Given: 3x + 4 = 19 2x + 10 = 20 Prove: 5x + 14 = 39 Statements Reasons 18. Given: x = 12 3y = 14 Prove: x + 3y = 26 Statements Reasons

The Reflexive Postulate of Equality/Congruence Any quantity is equal to itself Any figure is congruent to itself

Obvious – but we need to use this when adding or subtracting the same quantity from both sides of an equation.

The Substitution Property of Equality A quantity can be substituted for any equal quantity

It’s common sense, but we will use this one a lot. Now things are going to get interesting !!! 19. Given: 2x + 12 = 20 Prove: x = 4 Statements Reasons 20. Given: 6x - 3 = x + 27 Prove: x = 6 Statements Reasons

21. Given: mPDQ = 84

bisects PDQ Prove: mDPS = 42 Statements Reasons 22. Given: AB = CD Prove: AB + BC = BC + CD

Statements Reasons

23. Given: G is the midpoint of ̅̅ ̅̅ H is the midpoint of ̅̅ ̅ Prove: ̅̅ ̅̅ ≅ ̅̅̅̅

Statements Reasons 24. Given: B is the midpoint of ̅̅ ̅̅ E is the midpoint of ̅̅ ̅̅ Prove: ̅̅ ̅̅ ̅̅ ̅̅

25. Given: bisects RST mRSX = 3x – 2 mRST = 9x – 76 Prove: mRST = 140

Statements Reasons

26. Given: ⃡ , bisects CBE

bisects ABE Prove: ABF ≅ EBD (you can use the postulate “all right angles are congruent”)

Statements Reasons

F HG I

A B C

D E F

RS

T

X

A B C

D

E

F

27. Given: ∆ABC, X is the midpoint of ̅̅ ̅̅ , Z is the midpoint of ̅̅ ̅̅ , AB = 10, AC = 12, and ZX = 4. Prove: the perimeter of ∆AXZ = 15

Statements Reasons

28.Given: RD AE , DE RA Prove: perimeter of ∆RDE = perimeter of ∆ERA

Statements Reasons

29. Given: Y is the midpoint of ̅̅ ̅̅ ̅

and X is the midpoint of ̅̅ ̅̅ ̅

Prove: WX = ¼ WZ

(you can use the reason “simplify” when

doing basic algebraic steps like combining

like terms or multiplying two coefficients)

Statements Reasons

AB

C

X

Z

D E

AR

W X Y Z

The Partition Postulate The whole equals the sum of the parts

AB + BC AC ABD + DBC = ABC

AB + BC = AC mABD + mDBC = mABC

Partition is often combined with the addition or subtraction postulates.

30. Given: AB DE

Prove: DE + BC = AC

Statements Reasons

31. Given: QR = ST

Prove: QS = RT

Statements Reasons

32. Given: mGFH = 40 and

mEFH = 50

Prove: EFG is a right angle

Statements Reasons

33. Given: mSOG = 37

mSOB = 98

Prove: mBOG = 135

Statements Reasons

A B C

D

B A

C

A B C

D E

E

F

G

H

Q

R

S

T

34. Given: ̅̅̅̅ ̅̅ ̅̅ , ̅̅̅̅ ̅̅ ̅̅ ̅

Prove: ̅̅ ̅̅ ̅̅ ̅̅ ̅

Statements Reasons

35. Given: ⃡ , ACE ≅ BCE

Prove: DEA ≅ DEB

Statements Reasons

36. Given: mCBD = 60, mABD = 30

Prove: ∆ABC is a right triangle

Statements Reasons

37. Challenge

Given: MO JL , KN JL , KN MO

Prove: The area of ∆JLM equals the area of ∆JLK

Statements Reasons

38. Challenge

Given: DEI VER

Prove: DER VEI

Statements Reasons

J

K

L

M

N O

D R I

E

V

Proofs Using Theorems

Vertical angles are congruent Linear pairs are supplementary Angle sum theorem Exterior angle theorem Isosceles triangle theorem

39.Given: ACE , BAC ECD

Prove: BAC ACB

Statements Reasons

40. Given: AB and CD intersect at point O.

Prove: AOC BOD

(use the theorem “supplements of congruent

angles are congruent”

41. Given: ̅̅ ̅̅ ̅̅ intersects ̅̅ ̅̅ ̅̅ at E

B ≅ D

Prove: A ≅ C

42.Given: BOC and DOF are complementary

Prove: EOC and AOD are complementary

A

BC

D

O

A B

C

D

E

F

O

A

B

C

D

E

43. Given: BEu ruu

bisects FBD, ABF CBD

Prove: ABE is a right angle

44. Given: RT ST

Prove: 1 3

45. Given: DAB ABD, BC AD

Prove: BCD BDC

46. Given: ∆ABC, ACE , and BCD

Prove: mDCE + mBAC + mABC =

180

A B C

D

E

F

RS

T

1

2 3

A B

C

D

D

C

A B

E

47. Given: ∆ABC, ∆BED, ABE , CBD

Prove: m1 + m2 = m4 + m5

48. Given: JM JK , KM bisects JKL and JML Prove:∆KLM is isosceles

49. Given: ∆ABC with AB extended to D,

AC = BC

Prove: m1 = m2 + m4

50. Given DABE , CBF , mACB = 50, and

mFBE = 40

Prove: AC DABE

1

2

3

4

5

6

A

B

C

D

E

J

K

L

M

A B

C

D

1 2 3

4

A B ED

C

F