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Perpendicularity
and
Parallelism
Theorem 5.1
Given: m is perpendicular to AB, AM = MB, P
is a point on m.
Prove: PA =PB
Proof: Statements
In right triangle PMB, right
triangle PMA
1. AM = MB
2. PM = PM
3. Right triangle PMB is
congruent to right triangle
PMA
4. PA = PB
Reasons
1. Given
2. Identity
3. LL
4. CPCTE
Theorem 5.2
Given: AB with P being any point such that PA =PB
Prove: P lies on the perpendicular bisector of AB
Construction: Let M be the midpoint of AB. Join PM
and produce it both ways.
1.
2.
3.
4.
5.
6.
7.
8.
Proof: Statements
In triangle AMP, triangle BMP
AM = MB
PA = PB
PM = PM
Triangle AMP is congruent to
triangle BMP
Angle 1 = angle 2
PM is perpendicular to AB
Since PM is perpendicular through
the midpoint of AB, it is the
perpendicular bisector of AB
Thus, P lies on the perpendicular
bisector of AB
Reasons
1.
2.
3.
4.
Construction, definition of midpoint
Given
Identity
SSS
5. CPCTE
6. Equal linear pair
7. Definition of perpendicular bisector
Theorem 5.4
Given: t cuts m at A and n at B
such that angle 1 = angle 2
Prove: m is parallel to n
1.
2.
3.
4.
5.
6.
Proof: Statements
m and n are cut by t at A and B
respectively
Suppose m is not parallel to n
m and n must then intersect at
some point P forming triangle ABP
Angle 1 > angle 2
But angle 1 > angle 2 is impossible
Therefore, m is parallel to n
Reasons
1. Given
2. Assumption
3. Nonparallel coplanar lines
intersect
4. Exterior angle theorem1
5. It is given angle 1 = angle 2
6. As the assumption leads to a
contradiction, it must be false and
its negation is true
Corollary 5.5
Given: t intersects m and n such that angle 1 =
angle 2
Prove: m is parallel n
Proof: Statements
1. angle 1 = angle 2
2. angle 1 = angle 3
3. angle 3 = angle 2
4. m is parallel to n
Reasons
1. Given
2. Vertical angles are equal
3. Substitution
4. Alternate interior angles
are equal or AIP
Theorem 5.9
Given: n is perpendicular to m, s is perpendicular to m
Prove: n is parallel to s
Proof: Statements
1. n is perpendicular to
m, s is perpendicular
to m
2. h, g are right angles
3. h = g
4. n is parallel to s
Reasons
1. Given
2. Definition of perpendicular
lines
3. All right angles are equal
4. Corresponding angles are
equal
Theorem 5.10
Given: m is parallel to n, t cuts m at A and n
at B
Prove: angle 1 = angle 2
1.
2.
3.
4.
5.
Proof: Statements
Suppose m is parallel to n but
angle 1 is not equal to angle 2.
through A draw a line AR such
that angle RAB = angle 2
Angle RAB and angle 2 are
equal alternate interior angles
n is parallel to AR
But this contradicts the parallel
postulate if AR is distinct from
m
Therefore, the assumption in
no. 1 is false and angle 1 =
angle 2
Reasons
1. Assumption
2. Construction
3. AIP
4. m is parallel to n, given
Corollary 5.11
Given: m is parallel to n with t a transversal cutting both of
them
Prove: a. Angle 1 = angle 2
b. Angle 1 = angle 6
Proof: Statements
a. 1. Angle 2 = angle 4
2. Angle 1 = angle 2
3. Angle 1 = angle 4
b. 1. Angle 1 = angle 4
2. Angle 4 = angle 6
3. Angle 1 = angle 6
Reasons
a. 1. Alternate interior angles are
equal, m is parallel to n
2. Vertical angles theorem
3. Substitution
b. 1. Proved in a
2. Vertical angles theorem
3. Substitution
Theorem 5.12
Given: Triangle ABC
Prove: Angle A + angle B + angle C = 180°
Construction: Through A draw YAX is parallel to BC
Proof: Statements
1. YAX is parallel to BC
2. Angle 1 = angle B, angle
3 = angle C
3. Angle 1 + angle 2 +
Angle 3 = 180°
4. Angle A + angle B +
Angle C = 180°
Reasons
1. Construction
2. Alternate interior angles
are equal, YAX is
parallel to BC
3. Angles about a point on
one side of a line
4. Substitution
Theorem 5.13
Given: Triangle ABC with exterior angle ACD
Prove: Angle AGC = angle A + angle B
Construction: draw CX is parallel to BA
Proof: Statements
1. CX is parallel to BA
2. Angle A = angle 2
3. Angle B = angle 3
4. Angle A + angle B = angle 2 +
angle 3
5. But, angle 2 + angle 3 = angle
ACD
6. Angle A + angle B = angle ACD
Reasons
1. Construction
2. Alternate interior angles are equal,
CX is parallel to BA
3. Corresponding angles are equal,
CX is parallel to BA
4. Addition of no. 2 and no. 3
5. Angle Addition Postulate
6. Substitution
Theorem 5.14
Given: Triangle ABC, triangle XYZ with BC = YZ,
angle B = angle Y, angle A = angle X
Prove: Triangle ABC is congruent triangle XYZ
Proof: Statements
1. Angle A + angle B + angle C = 180 =
angle X + angle Y + angle Z
2. Angle A = angle X, angle B = angle Y
3. Angle A + angle B, angle X + angle Y
4. Angle C = angle Z
In triangle ABC, triangle XYZ
5. Angle B = angle Y, BC = YZ
6. Angle C = angle Z
7. Triangle ABC is congruent triangle
XYZ
Reasons
1. Angle sum of a triangle
2. Given
3. APE
4. Subtract no. 3 from no. 1
5. Given
6. Proved
7. ASA
Corollary 5.15
Given: right triangle ACB, right triangle XZY
AB = XY, angle B = angle Y
Prove: right triangle ACB is congruent to right
triangle XZY
Reasons
Proof: Statements
1. AB = XY, angle B =
1. Given
angle Y
2. Angle C = angle Z
2. All right angles are
equal
3. Right triangle ACB is
congruent to right
triangle XZY
3. SAA
Theorem 5.17
Given: angle AOB, angle 1 = angele2, PD is
perpendicular to OB, PC is perpendicular to OA
Prove: PD = PC
Proof: Statements
In right triangle OCP, right
triangle ODP
1. OP = OP
2. Angle 1 = angle 2
3. Right triangle OCP is
congruent to right
triangle ODP
4. PC = PD
Reasons
1. Identity
2. Given
3. Hypotenuse angle or Hy
A
4. CPCTE
Theorem 5.21
Given: triangle ABC, angle B > angle C
Prove: AC > AB
1.
2.
3.
4.
5.
6.
7.
8.
9.
Proof: Statements
Suppose AC is not greater than
AB
Then, AC = AB or AC < AB
Suppose AC = AB
Then, angle C = angle B
This contradicts the given,
therefore, AC is not equal to AB
Suppose AC < AB
Then angle C > angle B
This contradicts the given,
therefore, AC is not less than
AB
If AC is not equal to AB and AC
is not less than AB, then AC >
AB
Reasons
1. Assumption
2. Trichotomy law
3. Assumption
4. Angles opposite equal sides
6. Assumption
7. Angle opposite greater side
Theorem 5.22
Given: triangle ABC
Prove: b + c > a, a + b > c
Construction: Produce CA to X so that AX =
AB. Join BX
1.
2.
3.
4.
5.
6.
7.
8.
Proof: Statements
In triangle AXB, AX = AB
Angle 1 = angle 2
Angle XBC > angle 2
Angle XBC > angle 1
CX > a
But, CX = CA + AX
CX = b + c
b+c>a
1.
2.
3.
4.
5.
6.
7.
8.
Reasons
Construction
Isosceles triangle theorem
Whole-part postulate
Substitution
Side opposite greater angle
Construction
Substitution
Substitution