Download Proving Angles Congruent

Geometric Proof
GEOMETRY (HOLT 2-6 & 2-7)
K.SANTOS
Geometric Proofs
Always will do two-column proofs (only one exception
later in year). Ignore book if it says to do flow or paragraph
proofs—always do as two-column.
First step--almost always the given (and the reason is “Given”)
Last step---the statement is always the thing you must prove
Reasons—given
properties
definitions
postulates
theorems (previously proven)
Proof—All right angles are congruent
Given: < 1 and < 2 are right angles
Prove: <1 ≅ <2
1
2
Statements
1.
2.
3.
4.
<1 and < 2 are right angles
m<1 = 90° and m<2= 90°
m<1 = m<2
<1 ≅ <2
Reasons
1.
2.
3.
4.
Given
definition of a right angle
Transitive Property (2)
Definition of congruent
angles
Common Segments Theorem (2-7-1)
Given collinear points A, B, C and D arranged as
shown, if 𝐴𝐵 ≅ 𝐶𝐷 then 𝐴𝐶 ≅ 𝐵𝐷
A
B
C
D
Proof of Common Segments Theorem
Given: 𝐴𝐵 ≅ 𝐶𝐷
Prove: 𝐴𝐶 ≅ 𝐵𝐷
Statements
1. 𝐴𝐵 ≅ 𝐶𝐷
2. AB = CD
3. BC = BC
4. AB + BC = BC + CD
5. AB + BC = AC
BC + CD = BD
6. AC = BD
7. 𝐴𝐶 ≅ 𝐵𝐷
A
B
C
D
Reasons
1.
2.
3.
4.
5.
Given
definition of congruent segments
reflexive property of equality
Addition Property of equality
Segment Addition Post
6. Substitution Property (4, 5)
7. Definition of congruent segments
Proof
Given: <A and <B are supplementary
m<A = 45°
Prove: m<B = 135°
Statements
1. <A and <B are supplementary
2. m<A + m<B = 180°
1.
2.
3. m<A = 45°
4. 45 ° + m<B = 180°
3.
4.
5. m<B = 135°
5.
Reasons
Given
definition of
supplementary angles
given
Substitution Property
(2, 3)
Subtraction Property of
equality
Proof of Part of Congruent Complements
Theorem
Given: <1 and< 2 are complements
<3 and <4 are complements
<2 ≅ <4
Then <1 ≅ <3
1
2
4
Statements
1. <1 and< 2 are complements
<3 and <4 are complements
2. m<1 +m<2 = 90
m<3 +m<4 = 90
3. m<1 + m< 2= m<3 + m<4
4. <2 ≅ <4
5. m<2 = m<4
6. m<1+m<2=m<3 +m<2
7. m<2 = m< 2
8. m< 1 = m<3
9. <1 ≅ <3
Reasons
1. given
2. def. complementary angles
3. Substitution Property (2)
4. given
5. definition of congruent angles
6. substitution property (3, 5)
7. Reflexive property
8. subtraction property
9. definition of congruent angles
3
Proof—Vertical angles are congruent
2
1
3
Given: < 1 and <3 are vertical angles
Prove: < 1 ≅ < 3
Statements
1. < 1 and <3 are vertical angles
2. < 1 and <2 are a linear pair
<2 and <3 are a linear pair
3. <1 and <2 are supplementary
<2 and <3 are supplementary
4. < 1 ≅ < 3
Reasons
1. Given
2. definition of a linear pair
3. Linear Pair Theorem
4. Congruent supplements
theorem
(supplements of the same angle are
congruent)