Proving AngleRelationships

Protractor Postulate

- Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r.

Angle Congruence

Congruence of angles is reflexive, symmetric, and transitive:

1. Reflexive 1 1

2. Symmetric If 1 2, then 2 1

3. Transitive If 1 2 and 2 3,

then 1 3

Angle Addition Postulate

If R is in the interior of PQS, then mPQR + mRQS = mPQS

If mPQR + mRQS = mPQS, then R is in the interior of PQS

Q

P

R

S

Angle Addition

If and , find 44m ABD 88m ABC .m DBCA

B D

C

m ABD m DBC m ABC 44 88m DBC

44m DBC

Right Angle Theorems

List 3 - 5 facts that you observe about the perpendicular lines below:

Right Angle Theorems

Perpendicular lines intersect to form four right angles

All right angles are congruent

Perpendicular lines form congruent adjacent angles

If two angles are congruent and supplementary, then

each angle is a right angle

If two congruent angles form a linear pair, then they are

right angles

Theorems

Supplementary Theorem – if two angles form a linear pair, then they are supplementary angles.

Complementary Theorem – if the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles.

Theorems

2.6 Angles supplementary to the same angle or to congruent angles are congruent.

2.7 Angles complementary to the same angle or to congruent angles are congruent.

2.8 Vertical angles theorem: If two angles are vertical angles, then they are congruent.

Supplementary Angles

Angles supplementary to the same angle or to congruent angles are congruent

s suppl. to same or s are

Example:• m1 + m2 = 180

• m2 + m3 = 180

• Then, 1 31 3

2

Complementary Angles

Angles complementary to the same angle or to congruent angles are congruent

s compl. to same or s are

Example:• m1 + m2 = 90

• m2 + m3 = 90

• Then, 1 3

1

3

2