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Proving Angle Relationships. 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. - PowerPoint PPT Presentation
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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
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