Section 2-8 Proving Angle Relationships Wednesday, November 16, 2011
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?
Wednesday, November 16, 2011
More Postulates and Theorems
Protractor Postulate:
Angle Addition Postulate:
Theorem 2.3 - Supplement Theorem:
Wednesday, November 16, 2011
More Postulates and Theorems
Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180
Angle Addition Postulate:
Theorem 2.3 - Supplement Theorem:
Wednesday, November 16, 2011
More Postulates and Theorems
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:
Theorem 2.3 - Supplement Theorem:
Wednesday, November 16, 2011
More Postulates and Theorems
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
Theorem 2.3 - Supplement Theorem:
Wednesday, November 16, 2011
More Postulates and Theorems
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
Theorem 2.3 - Supplement Theorem: If two angles form a linear pair, then they are supplementary angles
Wednesday, November 16, 2011
More Postulates and Theorems
Theorem 2.4 - Complement Theorem:
Theorem 2.5 - Properties of Angle CongruenceReflexive Property of Congruence:Symmetric Property of Congruence:
Transitive Property of Congruence:
Wednesday, November 16, 2011
More Postulates and Theorems
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:
Transitive Property of Congruence:
Wednesday, November 16, 2011
More Postulates and Theorems
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: ∠1 ≅ ∠1Symmetric Property of Congruence:
Transitive Property of Congruence:
Wednesday, November 16, 2011
More Postulates and Theorems
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: ∠1 ≅ ∠1Symmetric Property of Congruence:
If ∠1 ≅ ∠2, then ∠2 ≅ ∠1Transitive Property of Congruence:
Wednesday, November 16, 2011
More Postulates and Theorems
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: ∠1 ≅ ∠1Symmetric Property of Congruence:
If ∠1 ≅ ∠2, then ∠2 ≅ ∠1Transitive Property of Congruence:
If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3
Wednesday, November 16, 2011
More Postulates and Theorems
Theorem 2.6 - Congruent Supplements Theorem:
Theorem 2.7 - Congruent Complements Theorem:
Theorem 2.8 - Vertical Angles Theorem:
Wednesday, November 16, 2011
More Postulates and Theorems
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 2.8 - Vertical Angles Theorem:
Wednesday, November 16, 2011
More Postulates and Theorems
Theorem 2.6 - Congruent Supplements Theorem: Angles supplementary 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:
Wednesday, November 16, 2011
More Postulates and Theorems
Theorem 2.6 - Congruent Supplements Theorem: Angles supplementary 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
Wednesday, November 16, 2011
EVENMore Postulates and Theorems
Right Angle TheoremsTheorem 2.9:
Theorem 2.10:Theorem 2.11:
Theorem 2.12:
Theorem 2.13:
Wednesday, November 16, 2011
EVENMore Postulates and Theorems
Right Angle TheoremsTheorem 2.9: Perpendicular lines intersect to form
four right anglesTheorem 2.10:Theorem 2.11:
Theorem 2.12:
Theorem 2.13:
Wednesday, November 16, 2011
EVENMore Postulates and Theorems
Right Angle TheoremsTheorem 2.9: Perpendicular lines intersect to form
four right anglesTheorem 2.10:All right angles are congruentTheorem 2.11:
Theorem 2.12:
Theorem 2.13:
Wednesday, November 16, 2011
EVENMore Postulates and Theorems
Right Angle TheoremsTheorem 2.9: Perpendicular lines intersect to form
four right anglesTheorem 2.10:All right angles are congruentTheorem 2.11: Perpendicular lines form congruent
adjacent anglesTheorem 2.12:
Theorem 2.13:
Wednesday, November 16, 2011
EVENMore Postulates and Theorems
Right Angle TheoremsTheorem 2.9: Perpendicular lines intersect to form
four right anglesTheorem 2.10:All right angles are congruentTheorem 2.11: Perpendicular lines form congruent
adjacent anglesTheorem 2.12: If two angles are congruent and
supplementary, then each angle is a right angleTheorem 2.13:
Wednesday, November 16, 2011
EVENMore Postulates and Theorems
Right Angle TheoremsTheorem 2.9: Perpendicular lines intersect to form
four right anglesTheorem 2.10:All right angles are congruentTheorem 2.11: Perpendicular lines form congruent
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
Wednesday, November 16, 2011
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%?
Wednesday, November 16, 2011
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%?Ceiling
Wa!
Beam
42°
Wednesday, November 16, 2011
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%?Ceiling
Wa!
Beam
42°
90°-42°
Wednesday, November 16, 2011
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%?Ceiling
Wa!
Beam
42°
90°-42°
48°
Wednesday, November 16, 2011
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?
Wednesday, November 16, 2011
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°.
Wednesday, November 16, 2011
Example 3In the figure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent.
Wednesday, November 16, 2011
Example 3In the figure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent.
Wednesday, November 16, 2011
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°
Wednesday, November 16, 2011
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
Wednesday, November 16, 2011
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
∠1 and ∠4 are supplementary
Wednesday, November 16, 2011
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
∠1 and ∠4 are supplementaryLinear pairs are
supplementary
Wednesday, November 16, 2011
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
∠1 and ∠4 are supplementaryLinear pairs are
supplementary
∠3 and ∠1 are supplementary
Wednesday, November 16, 2011
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
∠1 and ∠4 are supplementaryLinear pairs are
supplementary
∠3 and ∠1 are supplementary Def. of supplementary
Wednesday, November 16, 2011
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
∠1 and ∠4 are supplementaryLinear pairs are
supplementary
∠3 and ∠1 are supplementary Def. of supplementary
∠3 ≅ ∠4
Wednesday, November 16, 2011
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
∠1 and ∠4 are supplementaryLinear pairs are
supplementary
∠3 and ∠1 are supplementary Def. of supplementary
∠3 ≅ ∠4 Angles supplementary to same ∠ are ≅
Wednesday, November 16, 2011
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.
Wednesday, November 16, 2011
Problem Set
Wednesday, November 16, 2011
Problem Set
p. 154 #8-20
“Compassion for others begins with kindness to ourselves.” - Pema Chodron
Wednesday, November 16, 2011