Given: Prove: x = 104y ;17y3x21

1. __________ 1. ___________

2. __________ 2. ___________

3. __________ 3. ___________

4. __________ 4. ___________

Statements Reasons

4y ;17y3x21

x = 10

1712x21

5x21

Given

Substitution

SubtractionMultiplication

Given: m4 + m6 = 180 Prove: m5 = m6

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

Statements Reasons

Given

Angle Add. Post.Substitution

Reflexivem4 = m4

m4 + m5 = m4 + m6

m4 + m5 = 180

m4 + m6 = 180

m5 = m6 Subtraction

Given: m1 = m3 m2 = m4

Prove: mABC = mDEF

1. 1.

2. 2.

3. 3.

4. 4.

Statements Reasons

m1 = m3; m2 = m4

mABC = mDEF

m1 + m2 = m3 + m4

m1 + m2 = mABCm3 + m4 = mDEF

Given

Addition Prop.

Angle Add. Post.

Substitution

A B

C

1 2 4 3DE

F

Given: ST = RN; IT = RU Prove: SI = UN

1. ST = RN 1.

2. 2.

3. SI + IT = RU + UN 3.

4. IT = RU 4.

5. 5.

Statements Reasons

ST = SI + ITRN = RU + UN

SI = UN

Given

Segment Add. Post.

Substitution

Given

Subtraction Prop.

S I T

R U N

Postulate – A statement accepted without proof.

Theorem – A statement that can be proven using other definitions, properties, and postulates.

In this class, we will prove many of the Theorems that we will use.

If M is the midpoint of AB, then AM = ½AB and MB = ½AB.

Hypothesis: M is the midpoint of AB

Conclusion: AM = ½AB and MB = ½AB Write these pieces of the conditional

statement as your “given” and “prove” information.

Given:

Prove:

Definition of Midpoint: the point that divides a segment into two congruent segments.

If M is the midpoint of AB, then AM MB.

A BM

Midpoint Theorem: If M is the midpoint of AB, then AM = ½AB and MB = ½AB.

The theorem proves properties not given in the definition.

Proof of the Midpoint Theorem

Given: M is the midpoint of AB

Prove: AM = ½AB; MB = ½AB

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

M is the midpoint of AB

MB = ½AB

AM = MB

AM + MB = AB

AM + AM = AB 2AM = AB

AM = ½AB

Given

Def. of a Midpoint

Segment Add. Post.

Substitution

Division Property

Substitution

A M B

If BX is the bisector of ABC,

then mABX = ½mABC and

mXBC = ½mABC.

Prove: mABX = ½mABC and

mXBC = ½mABC.

Given: BX is the bisector of ABC.

A ray that divides an angle into two congruent adjacent angles.

X Y

ZW

XWYYWZ

If WY is the bisector of XWZ, then mXWY = ½mXWZ and mYWZ = ½mXWZ.

Proof of the Bisector Thm

Prove: mABX = ½mABC and

mXBC = ½mABC.

Given: BX is the bisector of ABC.

XB C

A

1. BX is the bisector of ABC.

1. Given

2. mABX = mXBC 2. Def. of an angle bisector

3. mABX + mXBC = mABC

3. Angle Addition Postulate

4. mABX + mABX = mABC

2mABX = mABC

4. Substitution

5. mABX = ½mABC 5. Division Property

6. mXBC = ½mABC 6. Substitution