Geometry Section 2-8 1112

Section 2-8Proving Angle Relationships

Wednesday, November 16, 2011

Essential Questions

How do you write proofs involving supplementary and complementary angles?

How do you write proofs involving congruent and right angles?


More Postulates and Theorems

Protractor Postulate:

Angle Addition Postulate:

Theorem 2.3 - Supplement Theorem:



Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180





Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180This means we can measure angles in degrees






Angle Addition Postulate: D is in the interior of ∠ABC IFF m∠ABD + m∠DBC = m∠ABC





Angle Addition Postulate: D is in the interior of ∠ABC IFF m∠ABD + m∠DBC = m∠ABC

Theorem 2.3 - Supplement Theorem: If two angles form a linear pair, then they are supplementary angles



Theorem 2.4 - Complement Theorem:

Theorem 2.5 - Properties of Angle CongruenceReflexive Property of Congruence:Symmetric Property of Congruence:

Transitive Property of Congruence:



Theorem 2.4 - Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles

Theorem 2.5 - Properties of Angle CongruenceReflexive Property of Congruence:Symmetric Property of Congruence:





Theorem 2.5 - Properties of Angle CongruenceReflexive Property of Congruence: ∠1 ≅ ∠1Symmetric Property of Congruence:






If ∠1 ≅ ∠2, then ∠2 ≅ ∠1Transitive Property of Congruence:





If ∠1 ≅ ∠2, then ∠2 ≅ ∠1Transitive Property of Congruence:

If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3



Theorem 2.6 - Congruent Supplements Theorem:

Theorem 2.7 - Congruent Complements Theorem:

Theorem 2.8 - Vertical Angles Theorem:



Theorem 2.6 - Congruent Supplements Theorem: Angles supplementary to the same angle or to congruent angles are congruent

Theorem 2.7 - Congruent Complements Theorem:





Theorem 2.7 - Congruent Complements Theorem: Angles complementary to the same angle or to congruent angles are congruent





Theorem 2.7 - Congruent Complements Theorem: Angles complementary to the same angle or to congruent angles are congruent

Theorem 2.8 - Vertical Angles Theorem: If two angles are vertical angles, then they are congruent


EVENMore Postulates and Theorems

Right Angle TheoremsTheorem 2.9:

Theorem 2.10:Theorem 2.11:

Theorem 2.12:

Theorem 2.13:



Right Angle TheoremsTheorem 2.9: Perpendicular lines intersect to form

four right anglesTheorem 2.10:Theorem 2.11:

Theorem 2.12:

Theorem 2.13:




four right anglesTheorem 2.10:All right angles are congruentTheorem 2.11:

Theorem 2.12:

Theorem 2.13:




four right anglesTheorem 2.10:All right angles are congruentTheorem 2.11: Perpendicular lines form congruent

adjacent anglesTheorem 2.12:

Theorem 2.13:





adjacent anglesTheorem 2.12: If two angles are congruent and

supplementary, then each angle is a right angleTheorem 2.13:





adjacent anglesTheorem 2.12: If two angles are congruent and

supplementary, then each angle is a right angleTheorem 2.13: If two congruent angles form a linear

pair, then they are right angles


Example 1Using a protractor, a construction worker measures that the angle a beam makes with the ceiling is 42°. What is the measure of the angle that the beam makes with the

wa%?



wa%?Ceiling

Wa!

Beam

42°



wa%?Ceiling

Wa!

Beam

42°

90°-42°



wa%?Ceiling

Wa!

Beam

42°

90°-42°

48°


Example 2At 4:00 on an analog clock, the angle between the hour and

minute hands of a clock is 120°. When the second hand bisects the angle between the hour and minute hands, what

are the measures of the angles between the minute and second hands and between the second and the hour hands?


Example 2At 4:00 on an analog clock, the angle between the hour and

minute hands of a clock is 120°. When the second hand bisects the angle between the hour and minute hands, what

are the measures of the angles between the minute and second hands and between the second and the hour hands?

Since the larger angle of 120° is bisected, two smaller angles of 60° are formed, and since those two angles add up to the larger one, both angles we are looking

for are 60°.


Example 3In the figure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent.





∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°




Given




Given

∠1 and ∠4 are supplementary




Given

∠1 and ∠4 are supplementaryLinear pairs are

supplementary




Given


supplementary

∠3 and ∠1 are supplementary




Given


supplementary

∠3 and ∠1 are supplementary Def. of supplementary




Given


supplementary


∠3 ≅ ∠4




Given


supplementary


∠3 ≅ ∠4 Angles supplementary to same ∠ are ≅


Check Your Understanding

Check out problems #1-7 on page 154 to see what you understand (or don’t) and formulate some questions on the

ideas.


Problem Set


Problem Set

p. 154 #8-20

“Compassion for others begins with kindness to ourselves.” - Pema Chodron


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Geometry Section 2-8 1112

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