Proving Angle Relationships

Postulate 2.10 – Protractor Postulate

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

Proving Angle Relationships

Postulate 2.11 – Angle Addition Postulate

• If R is in the interior of PQS, then mPQR + mRQS = m PQS.

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

Answer: 50

QUILTING The diagram below shows one square for a particular quilt pattern. If and is a right angle, find

Proving Angle Relationships

Theorem 2.3 – Supplement Theorem

• If two angles form a linear pair, then they are supplementary angles.

Theorem 2.4 – Complement Theorem

• If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.

Answer: 28

are complementary angles and .

andIffind

Proving Angle Relationships

Theorem 2.5 – Angle Congruence Theorem

Congruence of angles is reflexive, symmetric, and transitive.

Reflexive: 1 1

Symmetric: If 1 2, then 2 1.

Transitive: If 1 2 and 2 3, then 1 3.

Proving Angle Relationships

Theorem 2.6• Angles supplementary to the same angle or to

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

congruent angles are congruent.Vertical Angles Theorem• If two angles are vertical angles, then they are

congruent.

In the figure, NYR and RYA form a linear pair,AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXY are congruent.

Proof:

Statements Reasons

1. Given

2. If two s form a linear pair, then they are suppl. s.

3. Given

4.

1.

2.

3.

4.

linear pairs.

Answer: mA = 52; mZ = 52

find andIf and are vertical angles and and

Proving Angle Relationships

Theorem 2.9• Perpendicular lines intersect to form four

right angles.Theorem 2.10• All right angles are congruent.Theorem 2.11• Perpendicular lines form congruent adjacent

angles.

Proving Angle Relationships

Theorem 2.12

• If two angles are congruent and supplementary, then each angle is a right angle.

Theorem 2.13

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