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
mPQR + mRQS = m PQS.
• If mPQR + mRQS = mPQS, then R is
in the interior of PQS.
QUILTING The diagram below shows one square for
a particular quilt pattern. If
and
is a right angle, find
Answer: 50
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.
If
and
find
.
Answer: 28
are complementary angles and
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.
1. Given
linear pairs.
2.
2. If two s form a
linear pair, then
they are suppl. s.
3.
3. Given
4.
4.
If
and
are vertical angles and
find
and
Answer: mA = 52; mZ = 52
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.