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Mth 97
Fall 2013
Chapter 4
4.1 Reasoning and Proof in Geometry
Direct reasoning or ________________________reasoning is used to draw a conclusion from a series of
statements. Conditional statements, “if p, then q,” play a central role in deductive reasoning.
“If p, the q” is written ______________, and can also be read “p ___________ q” or” p only if q.”
p is the _____________________ and q is the___________________
What can you deduce from the following statements?
If ∆ABC is an equilateral triangle, then it is also an equiangular triangle.
∆ABC has 3 congruent sides.
Therefore (symbol
, a 3 dot triangle), ∆ABC is an _______________________________.
Not all statements are written in conditional form (if…then), but we can write most statements in this
form. Rewrite the following statements as conditional statements.
The only polyhedron with four vertices is a triangular pyramid.
_____________________________________________________________________________________
A cylinder is not a polyhedron.
_____________________________________________________________________________________
Law of Detachment
Example:
If p → q is a true conditional statement and p is true, then q is true.
1. If an angle is obtuse, its measure is greater than 90° and less than 180°.
2. ‫ﮮ‬B is obtuse.
3. __________________________________________
Law of Syllogism
Example:
If p → q and q → r are true conditional statements, then p → r is also true.
1. If M is the midpoint of AB , then AM = MB.
2. If the measures of two segments are equal, then they are congruent.
3. _____________________________________
Other forms of the conditional “If p, then q.”
Converse of p → q
q→p
Inverse of p → q
not p →not q
Contrapositive of p → q
not q →not p
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Mth 97
Fall 2013
Chapter 4
Write the converse, inverse and contrapositive of “If ABCD is a square, it has four right angles.”
Converse ____________________________________________________________________________
Inverse ______________________________________________________________________________
Contrapositive _________________________________________________________________________
A biconditional statement exists only when the conditional statement and its converse are both _________
If p → q and q →p are true, then_____________, read “p if and only if q.” (Shorthand for this is p iff q.)
Example: Write the converse of the statement below. If both the statement and its converse are true,
write the biconditional statement.
If three sides of a triangle are congruent, then the three angles of the triangle are congruent.
Converse: _____________________________________________________________________________
Biconditional: ___________________________________________________________________________
Do ICA 6
Vertical Angles are formed by a pair of intersecting lines and are opposite each other. In the sketch below,
the pair ‫ﮮ‬1 and ‫ﮮ‬3 and the pair ‫ﮮ‬2 and ‫ﮮ‬4 are vertical angles.
Theorem 4.1 – If two angles are a pair of vertical angles,
2
1
3
4
.
their measures are equal. m‫ﮮ‬1 = m‫ﮮ‬3 and m‫ﮮ‬2 = m‫ﮮ‬4
See proof on top of page 191
Proof – A proof is a convincing mathematical argument. The two kinds of proof used extensively in
Geometry are paragraph proof and statement-reason proof (2 column proof).
If two angles are congruent and supplementary, then they are right angles.
Given: 1  2 and 1 and 2 are supplementary
1 2
Prove: 1 and 2 are right angles
Statement
1.
2.
3.
4.
5.
6.
1  2 ; m 1 + m 2 = 180
m 1 = m 2
m 2 + m 2 =180; 2(m 2 ) = 180
m 2 = 90
m 1 = 90
1 and 2 are right angles
Reason (Other samples of this proof are on page 191.)
1.
2.
3.
4.
5.
6.
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Mth 97
Fall 2013
Chapter 4
4.2 Triangular Congruence Conditions
Definition of Congruent Triangles: ∆ABC  ∆DEF, if and only if
1. All 3 pairs of corresponding angles are congruent.
2. All 3 pairs of corresponding sides are congruent.
Reflexive Property – Something is congruent or equal to ________________
Examples:
∆DEF  ∆DEF or 15 = 15
Symmetric Property – It doesn’t matter which side of the = or  you are on.
Examples:
If ∆ABC  ∆DEF, then ∆DEF  ∆ABC. If 5 + 2 = 7, then 7 = ______________
Transitive Property – If two thing are  or = to the same thing, then they are  or = to each other.
Examples: If ∆ABC  ∆DEF and ∆DEF  ∆GHI, then If ∆ABC  ∆GHI.
or If 4 + 4 = 8 and 8 = (2)(4), then 4 + 4 = ______________
SAS Postulate
LL Theorem
Postulates and Theorems used to prove Two Triangles are Congruent
If two sides and the included angle of one
triangle are congruent respectively to two
sides and the included angle of another
triangle, then the two triangles are
congruent.
If two legs of one right triangle are
congruent respectively to two legs of
another right triangle, then the two
triangles are congruent.
HL Theorem
If the hypotenuse and a leg of one right
triangle are congruent respectively to the
hypotenuse and the leg of another right
triangle, then the two triangles are
congruent.
ASA Postulate If two angles and the included side of one
triangle are congruent respectively to two
angles and the included side of another
triangle, then the two triangles are
congruent.
SSS Postulate If three sides of one triangle are congruent
respectively to three sides of another
triangle, then the two triangles are
congruent.
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Mth 97
Fall 2013
Chapter 4
Theorem 4.4 – Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of two sides of a triangle equals
the square of the third side, then the triangle is a right triangle.
For each pair of triangles decide whether they are necessarily congruent. If so, write and appropriate
congruence statement and specify which congruence principle applies.
F
a)
A
B
b)
D
c)
C
I
H
d)
T
H
G
U
O
M
P
A
Y
X
W
V
N
Proof using triangular congruence
Given: RA  TA, PA  CA, RP  CT
R
A
C
Prove : R  T
P
Subgoal:
T
Statement
RA  TA, PA  CA, RP  CT
2. ∆RAP  ________________
3. R  T
1.
Reason
1.
2.
3.
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Mth 97
Fall 2013
Chapter 4
Section 4.3 – Problem Solving Using Triangle Congruence
Isosceles Triangles
Theorem 4.5 – In an isosceles triangle, the angles opposite the congruent sides are congruent.
Given: ∆ABC with AB  AC
A
Prove: B  C
C
B
Subgoal to be prove first: ∆ABC  ∆ACB
Statements
1.
Reasons
AB  AC
1.
2. AC  AB
3. A  A
4. ∆ABC  ∆ACB
5. B  C
2.
3.
4.
5.
Corallary 4.6 – Every equilateral triangle is equiangular.
If
then
Theorem 4.7 – If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Given: ∆ABC with B  C
A
Prove: AB  AC
C
B
Subgoal to be prove first: ∆ABC  ∆ACB
Statements
Reasons
B  C
C  B
1.
2.
3. BC  BC
4. ∆ABC  ∆ACB
3.
4.
AB  AC
5.
1.
2.
5.
Corallary 4.8 – Every equiangular triangle is equilateral.
If
then
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Mth 97
Fall 2013
Chapter 4
Theorem 4.7 – If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
A
A
If
then
If
, then
B
C
B
C
Corollary 4.8 – Every equiangular triangle is equilateral.
If
, then
Perpendicular Bisector of a Segment
Definitions:
1) To _____________ means to divide in half.
2) An __________ ________________ of a line segment is a line, line segment or ray that divides the angle
into two congruent angles.
3) The ______________________ __________________ of a line segment is a line, line segment or ray that
passes through the midpoint of the segment and is perpendicular to the line segment.
4) A _________________ of a triangle is a segment whose endpoints are the vertex of an angle of the
triangle and the midpoint of the side opposite that angle.
Theorem 4.9 – In an isosceles triangle, the ray that bisects the vertex angle bisects the base and is
perpendicular to it.
A
A
If
then
B
C
B
D C
See proof on page 210.
Theorem 4.10 – Perpendicular Bisector Theorem
A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the
endpoints of the segment.
P
P
See proof on page 211.
A
B
A
B
P is on the perpendicular bisector of AB if and only if ______________________
An _______________________angle of a triangle is formed by on side of a triangle and an extension of an
adjacent side.
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Mth 97
Fall 2013
Chapter 4
Theorem 4.11 – Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of either of the
nonadjacent interior angles.
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___________ and __________
Proof is on pages 212 – 213.
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Section 4.4 – Basic Geometric Constructions with a Compass and Strait Edge
Construction 1: Copy a line segment
a) Draw a line segment longer than the one you wish to copy.
b) Open your compass to match the length of you wish to copy.
c) Without changing the compass setting, place the tip of the compass at one end of the
new line segment and swing an arc to intersect the segment.
Copy each line segment.
A
B
C
D
Constuct a line segment twice as long as CD .
Construction 2: Copy an Angle
a) Draw a line segment or a ray. This will be one side of the new angle.
b) Place the tip of the compass at the vertex of the angle you wish to copy and swing an arc that
intersects the sides of the angle.
c) Without changing the compass setting, place the tip of your compass on an endpoint of your
copy and swing a similar arc that intersects your copy.
d) Open your compass to match the distance between the intersection points of the arc and the
sides of the angle you are copying.
e) Without changing the compass setting, copy this length on the arc beginning where the arc
intersected the side or your new angle.
f) Draw a segment from the endpoint of your copy (the new vertex) through the intersection of
your arcs. This is the second side of your angle.
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Mth 97
Fall 2013
Chapter 4
Copy the angle.
Construction 3: Bisect an Angle
a. Place the compass point on the vertex of the angle and swing an arc that intersects both sides of the
angle.
b. Set your compass for any opening that over half the distance between the points where your arc
intersected the sides of your angle. Your will use this setting to draw the next two arcs. Place your
compass point on each of these intersections and draw arcs that interest in the interior of your
angle.
c. Draw a segment or ray from the vertex of your angle through the intersection point of the last two
rays you drew.
Constructing Perpendiculars
Construction 4: Construct a perpendicular to a point on a line by bisecting the straight angle.
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Mth 97
Fall 2013
Chapter 4
Construction 5: Construct a perpendicular to a line from a point not on the line
a. Place the compass on the point not on the line and swing an arc that intersects the line in two
places.
b. Use the same compass setting to draw the next two arcs. Using each intersection point of the arc
with your line from the first step as a center draw arcs that intersect either above or below your
line.
c. Draw a segment from the point not on your line through the intersection point of your last two arcs
that intersects your line.
Construct a 45°angle by bisecting one of the right angles above.
Construction 6: Construct the Perpendicular Bisector of a Segment
a. Open your compass to more than half the length of the segment to be bisected. Use this compass
setting for this steps a and b. Place your compass on one endpoint and swing an arc above and
below this segment.
b. Place the compass point on the other end of the segment and swing arcs above and below the
segment.
c. Draw a line through the intersection points of the arc.
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Mth 97
Fall 2013
Chapter 4
Construction 7: Construct and equilateral triangle give the length of one side
a. Open the compass to the length of the segment to be copied. Use this compass setting for both
arcs in the next step.
b. Place the compass on each endpoint and swing an arc either, both arcs above or both below.
c. The intersection point of the two arcs is the third point of your triangle. Use your straight edge to
help you draw the triangle.
Construction 8: Methods of constructing a 30 – 60 right triangle
1. First construct an equilateral triangle.
2. Bisect one of the angles.
See pages 221-222.
1. First construct the perpendicular bisector of a
segment
2. Copy the length of your line segment using one
endpoint as a center so that the second endpoint
of your copy lies on the perpendicular bisector.
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Mth 97
Fall 2013
Chapter 4
Construction 9: Construct a triangle given the lengths of two sides and the included angle
a. Copy the angle.
b. Copy the length of one side of the triangle to a side of the angle.
c. Copy the length of the other side of the triangle to the second side of the angle.
d. Draw a segment connecting the endpoints of the segments you just drew.
Construction 10: Construct a triangle give two angles and the included side
a. Copy the segment.
b. Using one endpoint as a center, copy one of the angles.
c. Using the other endpoint as a center copy the other angle. The intersection point of the two new
sides of the angles you copied is the third vertex of your triangle.
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