Download 2-7 Proving Segment Relationships Ruler Postulate (2.8): The points

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Transcript
2-7 Proving Segment Relationships
Ruler Postulate (2.8): The points on any line or line segment can be paired with real numbers so that,
given any two points A and B on a line, A corresponds to zero and B corresponds to a positive real
number. (This postulate establishes a number line on any line)
Segment Addition Postulate (2.9):
Ex 1: Proof with Segment Addition
Given: PR = QS
P
Q
R
Prove: PQ = RS
Statement
Reason
Statement
Reason
OR
S
Segment Congruence
Reflexive Property:
Symmetric Property:
Transitive Property:
Proof of Transitive Property of Congruence (with segments)
Ė…Ė…Ė…Ė… , 𝑃𝑄
Ė…Ė…Ė…Ė… ≅ 𝑅𝑆
Ė…Ė…Ė…Ė…
Ė…Ė…Ė…Ė…Ė… ≅ 𝑃𝑄
Given: 𝑀𝑁
M
Ė…Ė…Ė…Ė…
Ė…Ė…Ė…Ė…Ė… ≅ 𝑅𝑆
Prove: 𝑀𝑁
N
Statement
Ex 2: Proof with Segment Congruence
Ė…Ė…Ė…Ė… ≅ 𝑋𝑍,
Ė…Ė…Ė…Ė…Ė… 𝑋𝑍
Ė…Ė…Ė…Ė… ≅ 𝑊𝑋
Ė…Ė…Ė…Ė…Ė…
Given: 𝑊𝑌 = 𝑌𝑍, 𝑌𝑍
Ė…Ė…Ė…Ė…Ė… ≅ 𝑊𝑌
Ė…Ė…Ė…Ė…Ė…Ė…
Prove: 𝑊𝑋
Statement
P
Reason
Y
3cm
W
Reason
Z
X
Q
R
S
2-8 Proving Angle Relationships
⃗⃗⃗⃗⃗ and a number r between 0 and 180, there is exactly one ray
Protractor Postulate (2.10): Given 𝐴đĩ
with endpoint A, extending on either side of ⃗⃗⃗⃗⃗
𝐴đĩ , such that the measure of angle formed is r.
Angle Addition Postulate (2.11):
Ex 1: Angle Addition: The time is 4 o’ clock and ten seconds. What are the measures of the angles
between the minutes and second hands and between the second and hour hands?
2.3 Supplement Theorem: If two angles from
a linear pair, then they are
__________________ angles.
2.4 Complement Theorem: If the noncommon
sides of two adjacents angles from a right angle,
then the angles are
________________________ angles.
Ex 2: Supplementary Angles
If ∠1 and ∠2 form a linear pair and 𝑚∠2 = 67, find 𝑚∠1.
Statement
Given
Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive.
Reflexive Property:
Symmetric Property:
Transitive Property:
Proof: Symmetric Property of Congruence
Statement
Reason
Theorems
2.6 Congruent Supplement Theorem: Angles supplementary to the same angle or to congruent angles
are congruent.
2.7 Congruent Complement Theorem: Angles complementary to the same angle or to congruent
angles are congruent
Ex 3: Use Supplementary Angles
Given: īƒ1 and īƒ2 form a linear pair, īƒ2 and īƒ3 form a linear pair
Prove: īƒ1 ī€ īƒ3
Statement
Reason
Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.
Ex 4: If īƒ1 and īƒ2 are vertical angles and the mīƒ1 ī€Ŋ x and mīƒ2 ī€Ŋ 228 ī€­ 3x , find mīƒ1 and mīƒ2 .
Right Angle Theorems
2.9.1
Perpendicular lines intersect to form four right angles
2.10
All right angles are congruent.
2.11
Perpendicular lines form congruent adjacent angles.
2.12
If two angles are congruent and supplementary, then each angle is a right angle.
2.13
If two congruent angles form a linear pair, then they are right angles.