Download Geometry Theorems – Chapter 2

Chapters 1 & 2 Geometry Honors – Important Postulates, Definitions, and Theorems
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Commonly Used Postulates in Proof (See page 926 a complete list)
1. Segment Addition Postulate: B is between A and C iff AB + BC = AC. A
( also AB = AC – BC & BC = AC – AB )
B
C
2. Angle Addition Postulate: B is in the interior of <ACD iff m<ACB + m<BCD = m<ACD.
( also m<ACB = m<ACD – m<BCD and
A
m<BCD = m<ACD – m<ACB )
C
3. Through any two points there exists exactly one line.
B
D
4. A line contains at least two points.
5. If two lines intersect, then their intersection is exactly one point.
6. Through any three noncollinear points there exists exactly one plane.
7. A plane contains at least three noncollinear points.
8. If two points lie in a plane, then the line containing them lies in the plane.
9. If two planes intersect, then their intersection is a line.
10. Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.
Definitions
1. Point M is the midpoint of AB iff M is between A and B such that AM = MB or AM  MB
A
M
2. ⃗⃗⃗⃗⃗ bisects <ACD iff m<ACB = m<BCD or <ACB  <BCD
B
A
C
B
D
3. A point, line, plane, ray, or segment bisects a segment iff it intersects the segment at its midpoint.
4. Two lines are perpendicular iff they intersect to form a right angle.
5. An angle is a right angle iff its measure is 90.
6. An angle is an acute angle iff its measure is between 0 and 90.
7. An angle is an obtuse angle iff its measure is between 90 and 180.
8. Definition of congruent angles: Two angles are congruent iff their measures are equal.
9. Definition of congruent segments: Two segments are congruent iff their measures are equal.
Geometry Theorems – Chapter 2 – the following theorems have been, or will be, covered and proved in
class or homework. For each theorem, write the “given”, the “prove”, the diagram, and a 2-column proof.
1. Vertical Angle Theorem (vertical angles are )
Prove: <1  <3 & <2  <4
1.
2.
3.
4.
5.
6.
Statements
<1  <3; <2  <4
<1 & <2 form a linear pair
<2 & <3 form a linear pair
<3 & <4 form a linear pair
<1 & <2 are supplementary
<2 & <3 are supplementary
<3 & <4 are supplementary
m<1 + m<2 = 180
m<2 + m<3 = 180
m<3 + m<4 = 180
m<1 + m<2 = m<2 + m<3
m<2 + m<3 = m<3 + m<4
m<1 = m<3
m<2 = m<4
Reasons
1. Given
2. Def. Linear Pair
2
3. Linear Pair Postulate
4. Def. Supplementary
5. Substitution (=)
6. Subtraction (=)
Q.E.D.
2. Midpoint Theorem
Given: B is the midpoint of ̅̅̅̅
Prove: AC = 2AB; AC = 2BC;
AB = ½ AC; BC = ½ AC
1.
2.
3.
4.
5.
6.
7.
Statements
̅̅̅̅
B is the midpoint of 𝐴𝐶
AB = BC
AB + BC = AC
AB + AB = AC
2AB = AC
2BC = AC
AB= ½ AC; BC = ½ AC
Reasons
1. Given
2. Def. Midpoint
3. Segment Addition Postulate
4. Substitution (=)
5. Substitution (=)
6. Substitution (=)
7. Division (=)
Q.E.D.
3. Angle Bisector Theorem
Given: ⃗⃗⃗⃗⃗⃗ bisects <ABC
Prove: m<ABC = 2m<1; m<ABC = 2m<2
m<1 = ½ m<ABC; m<2 = ½ m<ABC
1.
2.
3.
4.
5.
6.
7.
Statements
Reasons
⃗⃗⃗⃗⃗⃗
𝐵𝐷 bisects <ABC
1. Given
m<1 = m<2
2. Def. Bisect
m<1 + m<2 = m<ABC
3. Angle Addition Postulate
m<1 + m<1 = m<ABC
4. Substitution (=)
2m<1 = m<ABC
5. Substitution (=)
2m<2 = m<ABC
6. Substitution (=)
m<1 = ½ m<ABC; m<2 = ½ m<ABC
7. Division (=)
Q.E.D.
1
4
3
4. All right angles congruent
Given: <A and <B are right angles
Prove: <A  <B
1.
2.
3.
4.
Statements
<A and <B are right angles
m<A = 90; m<B = 90
m<A = m<B
<A  <B
Reasons
1. Given
2. Def. Right Angle
3. Substitution (=)
4. Def.  Angles
Q.E.D.
5. Midpoints of congruent segments make congruent segments
Given: ̅̅̅̅ ̅̅̅̅ ; B is midpoint of ̅̅̅̅ ; E is midpoint of ̅̅̅̅
Prove: ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
1.
2.
3.
4.
5.
6.
7.
Statements
̅̅̅̅
̅̅̅̅ ;
𝐴𝐶 𝐷𝐹
B is midpoint of ̅̅̅̅
𝐴𝐶 ;
̅̅̅̅
E is midpoint of 𝐷𝐹
AC = DF
AB = BC; DE = EF
AC = 2BC; DF = 2DE
2BC = 2DE
BC = DE
AB = BC = DE = EF
Reasons
1. Given
2.
3.
4.
5.
6.
7.
Def.  Segments
Def. Midpoint
Midpoint Theorem
Substitution (=)
Division (=)
Transitive (=)
Q.E.D.
6. Bisectors of congruent angles make congruent angles
Given: <ABC  <DEF; ⃗⃗⃗⃗⃗ bisects <ABC; ⃗⃗⃗⃗⃗⃗ bisects <DEF
Prove: <ABG  <GBC  <DEH  <HEF
1.
2.
3.
4.
5.
6.
7.
Statements
<ABC  <DEF
⃗⃗⃗⃗⃗
𝐵𝐺 bisects <ABC
⃗⃗⃗⃗⃗⃗
𝐸𝐻 bisects <DEF
m<ABC = m<DEF
m<1 = m<2; m<3 = m<4
m<ABC = 2m<2; m<DEF = 2m<3
2m<2 = 2m<3
m<2 = m<3
m<1 = m<2 = m<3 = m<4
Reasons
1. Given
2.
3.
4.
5.
6.
7.
Q.E.D.
Def.  Angles
Def. Bisector
Bisector Theorem
Substitution (=)
Division (=)
Transitive (=)
7. Overlapping Segments Theorem
Case 1
Given: ̅̅̅̅ ̅̅̅̅
Prove: ̅̅̅̅ ̅̅̅̅
Case 2
Given: ̅̅̅̅
Prove: ̅̅̅̅
̅̅̅̅
̅̅̅̅
Case 1
1.
2.
3.
4.
5.
6.
Case 2
Statements
̅̅̅̅
𝐴𝐵 ̅̅̅̅
𝐶𝐷
AB = CD
AB + BC = CD + BC
AB + BC = AC
CD + BC = BD
AC = BD
̅̅̅̅
𝐴𝐶 ̅̅̅̅
𝐵𝐷
Reasons
1. Given
2. Def.  Segments
3. Addition (=)
4. Segment Addition
Postulate
5. Substitution (=)
6. Def.  Segments
Statements
̅̅̅̅
𝐴𝐶 ̅̅̅̅
𝐵𝐷
AC = BD
AB + BC = AC
CD + BC = BD
AB + BC = CD + BC
AB = CD
̅̅̅̅
𝐴𝐵 ̅̅̅̅
𝐶𝐷
1.
2.
3.
4.
5.
6.
Reasons
1. Given
2. Def.  Segments
3. Segment Addition
Postulate
4. Substitution (=)
5. Subtraction (=)
6. Def.  Segments
Q.E.D.
Q.E.D.
8. Overlapping Angles Theorem
Case 1
Given: <1  <3
Prove: <ABE  <DBC
Case 2
1
Given: <ABE  <DBC
Prove: <1  <3
2
Case 1
1.
2.
3.
4.
5.
6.
3
Case 2
Statements
Reasons
<1  <3
1. Given
m<1 = m<3
2. Def.  Angles
m<1 + m<2 = m<3 + m<2
3. Addition (=)
m<1 + m<2 = m<ABE
4. Angle Addition
m<3 + m<2 = m<DBC
Postulate
m<ABE = m<DBC
5. Substitution (=)
<ABC  <DBC
6. Def.  Angles
Q.E.D.
1.
2.
3.
4.
5.
6.
Statements
Reasons
<ABE  <DBC
1. Given
m<ABE = m<DBC
2. Def.  Angles
m<1 + m<2 = m<ABE
3. Angle Addition
m<3 + m<2 = m<DBC
Postulate
m<1 + m<2 = m<3 + m<2
4. Substitution (=)
m<1 = m<3
5. Subtraction (=)
<1  <3
6. Def.  Angles
Q.E.D.
9. Supplements of congruent (or the same) angles are congruent
Given: <1  <2; <1 and <3 are supplementary; <2 and <4 are supplementary
Prove: <3  <4
1.
2.
3.
4.
5.
6.
Statements
<1  <2
<1 & <3 are supplementary
<2 & <4 are supplementary
m<1 = m<2
m<1 + m<3 = 180
m<2 + m<4 = 180
m<1 + m<3 = m<2 + m<4
m<3 = m<4
<3  <4
Reasons
1. Given
2. Def.  Angles
3. Def. Supplementary
4. Substitution (=)
5. Subtraction (=) [4 – 2]
6. Def.  Angles
Q.E.D.
10. Complements of congruent (or the same) angles are congruent
Given: <1  <2; <1 and <3 are complementary; <2 and <4 are complementary
Prove: <3  <4
1.
2.
3.
4.
5.
6.
Statements
<1  <2
<1 & <3 are complementary
<2 & <4 are complementary
m<1 = m<2
m<1 + m<3 = 90
m<2 + m<4 = 90
m<1 + m<3 = m<2 + m<4
m<3 = m<4
<3  <4
Reasons
1. Given
2. Def.  Angles
3. Def. Complementary
4. Substitution (=)
5. Subtraction (=) [4 – 2]
6. Def.  Angles
Q.E.D.
11. If <1 and <2 are complementary and <2  <3, then <1 and <3 are complementary
(Complementary Substitution).
Given: < 1 and <2 are complementary; <2  <3
Prove: <1 and <3 are complementary
1.
2.
3.
4.
5.
Statements
Reasons
<1 & <3 are complementary ; <2  <3 1. Given
m<2 = m<3
2. Def.  Angles
m<1 + m<3 = 90
3. Def. Complementary
m<1 + m<2 = 90
4. Substitution (=)
<1 & <2 are complementary
5. Def.  Angles
Q.E.D.
12. If <1 and <2 are supplementary and <2  <3, then <1 and <3 are supplementary.
(Supplementary Substitution)
Given: < 1 and <2 are supplementary; <2  <3
Prove: <1 and <3 are supplementary
1.
2.
3.
4.
5.
Statements
Reasons
<1 & <3 are supplementary ; <2  <3
1. Given
m<2 = m<3
2. Def.  Angles
m<1 + m<3 = 90
3. Def. Supplementary
m<1 + m<2 = 90
4. Substitution (=)
<1 & <2 are complementary
5. Def.  Angles
Q.E.D.
13.  lines intersect to form four right angles
Given: line a  line b
Prove: <1, <2, <3, & <4 are right
Statements
1. a  b
2. <1 is right
3. <1 & 2 form a linear pair
4. <1 & <2 are supplementary
5. m<1 + m<2 = 180
6. m<1 = 90
7. 90 + m<2 = 180
8. m<2 = 90
9. m<2 = m<4; m<1 = m<3
10. m<4 = 90; m<3 = 90
11. <1, <2, <3, & <4 are right
Reasons
1. Given
2. Def. 
3. Def. Supplementary
4. Linear Pair Postulate
5. Def. Supplementary
6. Def. Right Angle
7. Substitution (=)
8. Subtraction (=)
9. Vertical Angle Theorem
10. Substitution (=)
11. Def. Right Angle
Q.E.D.
b
1 2
4 3
a
14. If the non-shared sides of a pair of adjacent angles are , then the adjacent angles are complementary.
⃗⃗⃗⃗⃗
Given: ⃗⃗⃗⃗⃗
Prove: <1 and <2 are complementary
B
D
1.
2.
3.
4.
5.
6.
Statements
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
𝐴𝐵
𝐴𝐶
<BAC is right
m<BAC = 90
m<BAC = m<1 + m<2
90 = m<1 + m<2
<1 & <2 are complementary
Reasons
1. Given
2. Def. 
3. Def. Right Angle
4. Angle Addition Postulate
5. Substitution (=)
6. Def. Complementary
A
1 2
C
Q.E.D.
14.5 If a pair of adjacent angles are complementary, then their non-shared sides are  (same diagram)
Given: <1 and <2 are complementary
⃗⃗⃗⃗⃗
Prove: ⃗⃗⃗⃗⃗
1.
2.
3.
4.
5.
6.
Statements
<1 & <2 are complementary
m<1 + m<2 = 90
m<BAC = m<1 + m<2
m<BAC = 90
<BAC is right
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
𝐴𝐵
𝐴𝐶
Q.E.D.
Reasons
1. Given
2. Def. Complementary
3. Angle Addition Postulate
4. Substitution (=)
5. Def. Right Angle
6. Def. 
15. “POW” = Parts of Whole Theorem
(Segments)
1.
2.
3.
4.
5.
6.
(Angles)
̅̅̅̅
Given: 𝐴𝐵
̅̅̅̅
̅̅̅̅
𝐷𝐸 &𝐵𝐶
̅̅̅̅
Prove: 𝐴𝐶
̅̅̅̅
𝐷𝐹
Statements
̅̅̅̅ 𝐷𝐸
̅̅̅̅ 𝐸𝐹
̅̅̅̅ &𝐵𝐶
̅̅̅̅
𝐴𝐵
AB = DE; BC = EF
AB + BC = DE + EF
AB + BC = AC; DE + EF = DF
AC = DF
̅̅̅̅ 𝐷𝐹
̅̅̅̅
𝐴𝐶
Q.E.D.
Reasons
1. Given
2. Def.  Segments
3. Addition (=)
4. Segment Addition Postulate
5. Substitution (=)
6. Def.  Segments
Given: <1  <3 & <2  <4
Prove: <GJI  <KMN
3
1
4
2
1.
2.
3.
4.
5.
6.
Statements
<1  <3; <2  <4
m<1 = m<3; m<2 = m<4
m<1 + m<2 = m<3 + m<4
m<1 + m<2 = m<GJI; m<3 + m<4 = m<KMN
m<GJI = m<KMN
<GJI  <KMN
Q.E.D.
Reasons
1. Given
2. Def.  Angles
3. Addition (=)
4. Angle Addition Postulate
5. Substitution (=)
6. Def.  Angless
̅̅̅̅
𝐸𝐹
16. “Con – POW” = Converse Parts of Whole Theorem
(Segments)
1.
2.
3.
4.
5.
6.
Given: ̅̅̅̅
𝐴𝐶
̅̅̅̅
̅̅̅̅
𝐷𝐹 & 𝐵𝐶
Prove: ̅̅̅̅
𝐴𝐵
̅̅̅̅
𝐷𝐸
̅̅̅̅
𝐸𝐹
Statements
Reasons
̅̅̅̅
̅̅̅̅ & ̅̅̅̅
𝐴𝐶 𝐷𝐹
𝐵𝐶 ̅̅̅̅
𝐸𝐹
1. Given
AC = DF; BC = EF
2. Def.  Segments
AB + BC = AC; DE + EF = DF
3. Segment Addition Postulate
AB + BC = DE + EF
4. Substitution (=)
AC = DF
5. Subtraction (=)
̅̅̅̅ 𝐷𝐹
̅̅̅̅
𝐴𝐶
6. Def.  Segments
Q.E.D.
(Angles)
Given: <GJI  <KMN & <2  <4
Prove: <1  <3
3
1
4
2
1.
2.
3.
4.
5.
6.
Statements
<GJI  <KMN; <2  <4
m<GJI = m<KMN; m<2 = m<4
m<GJI = m<1 + m<2; m<KMN = m<3 + m<4
m<1 + m<2 = m<3 + m<4
m<1 = m<3
<1  <3
Q.E.D.
Reasons
1. Given
2. Def.  Angles
3. Angle Addition Postulate
4. Substitution (=)
5. Subtraction (=)
6. Def.  Angles