2-5 proving angles congruent angle pairs vertical angles two angles whose sides form two pairs of...

2-5 Proving Angles CongruentAngle Pairs

Vertical Anglestwo angles whose sides form two pairs of opposite rays.

1

23 4

angles. vertical are 4 and 3angles. vertical are 2 and 1

Adjacent Anglestwo coplanar angles with a common side, a common vertex, and no common interior points.

5 6

angles. adjacent are 6 and 5

2-5 Proving Angles CongruentAngle Pairs

Complementary Anglestwo angles whose measures add to 90. (Not necessarily adjacent.) Each is the complement of the other.

12

angles.ary complement are B andA

angles.ary complement are 2 and 1

Supplementary Anglestwo angles whose measures add to 180. (Not necessarily adjacent.) Each is the supplement of the other.

A B

60 30

angles.ary supplement are D and C

angles.ary supplement are 6 and 5

5 6C

60

D

120

2-5 Five Angle Theorems

Vertical Angles TheoremVertical angles are congruent.

1

23 4

4 3 2 1

1

2

2 and 1 Find

497x2

942x 1

mm

m

m

12

2 and 1 Find

602

550 1

mm

xm

xm

12

2 and 1 Find

202

20100 1

mm

xm

xm

30

AOBmBOCmDOCmEOCm , , , Find

A B

E

D

C

O

35 x 92 x

542 x

angle.each of measure theand Find x

42 x

95 x

angle.each of measure theand Find x

x4

x5

angle.each of measure theand Find x

2-5 Five Angle TheoremsCongruent Supplements TheoremIf two angles are supplements of the same angle, then the two angles are congruent.

s supps :proofs in onAbbreviati

A and 1 are supplementary.B and 1 are supplementary.Therefore, A B.

3

C

4

D

C and 3 are supplementary.D and 4 are supplementary.3 4Therefore, C D.

A

1

B

s supps :proofs in onAbbreviati

Congruent Supplements TheoremIf two angles are supplements of congruent angles, then the two angles are congruent.

2-5 Five Angle Theorems, cont.

s comps :proofs in onAbbreviati s comps

:proofs in onAbbreviati

A and 1 are complementary.B and 1 are complementary.Therefore, A B.

A1

B

C and 3 are complementary.D and 4 are complementary.3 4Therefore, C D.

C

D4

3

Congruent Complements TheoremIf two angles are complements of the same angle, then the two angles are congruent.

Congruent Complements TheoremIf two angles are complements of congruent angles, then the two angles are congruent.

Right Angle TheoremAll right angles are congruent.

Congruent and Supplementary TheoremIf two angles are congruent and supplementary, then each is a right angle.

s rt. all :proofs in onAbbreviati

rt.supp. and s :proofs in onAbbreviati

2-5 Five Angle Theorems, cont.

Proving the Five Theorems

Vertical Angles Theorem

1

232 1 :Prove

angles vertical are 2 and 1 :Given

Statements Reasons

s. vertical are 2 and 1 1) Given. 1)

2

1Congruent Supplements Theorem(Same Angle)

3 1 :Prove2 supp 3 AND 2 supp 1 :Given

Statements Reasons

2 supp 3 ; 2 supp 1 1)

Given. 1)

3

s supps

21

Congruent Supplements Theorem(Congruent Angles)

3 1 :Prove42

4; supp 3 ; 2 supp 1 :Given

Statements Reasons

4 2 ;4 supp 3 ; 2 supp 1 1)

Given 1)

3

4

s supps

Given: A is a right angle.B is a right angle.

Prove: A B

Statements1. A is a right angle;

B is a right angle

Reasons1. Given

All Right Angles Congruent

A

B

s rt. All

Given: X Y ; X supp.Y Prove: X and Y are right s

Statements

1. X Y ; X supp.Y Reasons1. Given

Angles Both Congruent and Supplementary are Right Angles

X

Y

rt.supp. and s

Using the Theorems

Now you can use these five theorems as part of other proofs.

In the REASON column you can now write the short form abbreviations.

Complementary/Supplementary Proof

1. to arysupplement is 2 :Prove

1. 3

Diagram :Given

A 1B

2 3 4

C

G

E

H

F

J

D

Statements

4. Substitution POC4. 2 and 1 are supp.

1. Diagram; 3 1

2. FJD is a straight angle3. 2 and 3 are supp.

Reasons

1. Given

2. Assumed from diagram

3. If two s form a straight ,then they are supp.

More Practice

YOB. to arysupplement is AOX :Prove

YOB. XOB

Diagram :Given

Statements

X

Y

A

O

B

Reasons

4. AOX and YOB are supp.

1. Diagram; XOB YOB

2. AOB is a straight .

3. AOX and XOB are supp.

1. Given

4. Substitution POC

2. Assumed from diagram.

3. If two s form a straight, then they are supp.

Using the Theorems

4. 3 :Prove

2. 1 ABBY

ABAX Diagram; :Given

;

;

A

1

B

X Y

3 2 4

Statements ReasonsDiagram. 1.

.ABAX Given. 1.