REVIEW: VOCABULARY from Section 2-4

Complementary Angles:

Supplementary Angles:

Vertical Angles:

Two angles whose measures sum to 90.

Two angles whose measures sum to 180.

The two non-adjacent angles that are created by a pair of intersecting lines. (They are across from one another.)

Right Angle:

An angle whose measure is 90.

Straight Angle:

An angle whose measure is 180.

EXAMPLE 1Given: 1 and 2 are

complementary

Prove: ABC is a right angle.

A

B C

1 2

Statements Reasons

1. 1 and 2 are complementary

1. Given

2. m1 + m2 = 90 2. Definition of Complementary Angles

3. m1 + m2 = mABC 3. Angle Addition Postulate

4. mABC = 90 4. Substitution

5. ABC is a right angle. 5. Definition of a right angle.

Given: DEF is a straight angle.

Prove: 3 and 4 are supplementary

3 4D E F

Statements Reasons

5. 3 and 4 are supplementary.

1. Given

4. m3 + m4 = 180

2. Definition of a straight angle

3. m3 + m4 = mDEF 3. Angle Addition Postulate

2. mDEF= 180

4. Substitution

1. mDEF is a straight angle.

5. Definition of supplementary angles

EXAMPLE 2

Vertical Angle Theorem:

Vertical Angles are Congruent.

Hypothesis: Two angles are vertical angles.

Conclusion: The angles are congruent.

Conditional: If two angles are vertical angles, then the angles are congruent.

Given:

Prove:

Aside: Would the converse of this theorem work?

If two angles are congruent,

then the angles are vertical angles.

Counterexample:

FALSE

Vertical Angle Theorem Proof

Given: 1 and 2 are vertical angles.Prove: 1 2

NOTE: You cannot use the reason “Vertical Angle Theorem” or “Vertical Angles are Congruent” in this proof. That is what we are trying to prove!!

1 324

Vertical Angle Theorem Proof

Given: 1 and 2 are vertical angles.Prove: 1 2

1 324

ReasonsStatements1. 1 and 2 are vertical s.

1. Given

5. 1 2

2. m1 + m3 = 180

m3 + m2 = 1802. Angle Addition Postulate

3. m1 + m3 = m3 + m2

3. Substitution

**. m3 = m3 **. Reflexive Property

4. m1 = m2 4. Subtraction Property

5. Definition of Angles.

4. m1 = m2 and 1 2 4. Subtraction

EXAMPLE 3Given: 2 3; Prove: 1 4

132

4

ReasonsStatements

1. 2 3 1. Given

2. 2 1

3. 1 3

4. 3 4

5. 4 1

2. Vertical Angles are Congruent

4. Vertical Angles are Congruent

3. Substitution

5. Substitution

You can also say “Vertical Angle Theorem”

You can also say “Vertical Angle Theorem”

YOU CANNOT UNDER ANY CIRCUMSTANCES

USE THE REASON “DEFINITION OF

VERTICAL ANGLES” IN A PROOF!!

Given:

1 and 2 are supplementary;

3 and 4 are supplementary;

2 4

Prove: 1 3

1

3

2

4

1. 1 and 2 are supplementary3 and 4 are supplementary

1. Given

2. m1 + m2 = 180 m3 + m4 = 180

2. Definition of Supplementary Angles

3. m1 + m2 = m3 + m4

3. Substitution

4. 2 4 or m2 = m4 4. Given

5. m1 = m3 or 1 3 5. Subtraction Property

ReasonsStatements