Transcript Geometry 2_1 Conditional Statements

GEOMETRY: CHAPTER 2
Ch. 2.1 Conditional
Statements
A conditional statement is a logical statement that
has two parts, a hypothesis and a conclusion.
When a conditional statement is written in if-then
form , the “if” part contains the hypothesis and the
“then” part contains the conclusion.
Ex.1
If it is raining, then there are clouds in the sky.
Hypothesis
Conclusion
Ex. 2
Rewrite the statements in if-then form.
a. All birds have feathers.
b. Two angles are supplementary if they are
a linear pair.
c. The measure of two angles added
together is 90 degrees, and the angles
are complementary.
a. If an animal is a bird, then it has feathers.
b. If two angles are a linear pair, then they
are supplementary.
Ex. 2. (cont.)
Rewrite the statements in if-then form.
c. The measure of two complementary
angles added together is 90 degrees.
If two angles are complementary, then their
measures add up to 90 degrees.
Ex. 3. Rewrite the conditional statement in ifthen form.
3x + 2 = 8, because x= 2
Ex. 3. (cont.)Rewrite the conditional statement in
if-then form.
3x + 2 = 8, because x= 2
If x= 2, then 3x + 2 = 8
The negation of a statement is the opposite of the
original statement. Notice that Statement 2 is
already negative, so its negation is positive.
Statement 1 —The sky is overcast.
Negation — The sky is not overcast.
Statement 2 —The ball is not Abby’s.
Negation — The ball is Abby’s.
Conditional Statements can be true or
false. To show that a conditional
statement is true, you must prove that the
conclusion is true every time the
hypothesis is true.
To show that a conditional statement is
false, you need to give only one
counterexample.
To write the converse of a conditional statement,
exchange the hypothesis and conclusion.
To write the inverse of a conditional statement,
negate both the hypothesis and the conclusion.
To write the contrapositive, first write the converse
and then negate both the hypothesis and
conclusion.
Ex. 4 Conditional, Converse, Inverse, Contrapositive
Conditional
Statement
Converse
If mA  78 , then A is acute.
TRUE
If A is acute, then mA  78 .
FALSE
Inverse
If mA  78 , then A is not acute. FALSE
0
0
Contrapositive If
0
A is not acute, then mA  78 . TRUE
0
Ex. 5.
Write the if-then form, the converse, the inverse,
and the contrapositive of the conditional
statement “Mission students are female.” Decide
whether each statement is true or false.
Ex. 5. (cont.)
If-then form: If a student attends Mission, then she
is female. TRUE
Converse: If a student is female, then she attends
Mission. FALSE.
Inverse: If a student does not attend Mission, then
the student is not female. FALSE
Contrapositive: If a student is not female, the
student does not attend Mission. TRUE.
A conditional statement and its contrapositive
are either both true or both false.
Similarly, a converse inverse of a conditional
statement are either both true or both false.
When two statements are both true or both
false, they are called equivalent statements.
Taken from: http://z.about.com/d/math/1/0/M/2/perpendicular.jpg
KEY CONCEPT—PERPENDICULAR LINES
Definition: If two lines intersect to form a right
angle, then they are perpendicular lines.
The definition can also be written using the
converse: If two lines are perpendicular lines, then
they intersect to form a right angle.
You can write “line l is perpendicular to line m” as
lm
Taken from:
http://education.yahoo.com/homework_help/math_help/s
olutionimages/minigeogt/2/1/1/minigeogt_2_1_1_25_10/f
-10-24-1.gif
Ex. 6. Decide whether each
statement about the diagram
is true.
a. 1 and 2 are a linear pair.
b. 1 and 3 are vertical angles.
c. 2 and 4 are vertical angles.
d. The lines forming 1, 2, and 3
and 4 are perpendicular to one another.
Point, Line, and Plane Postulates:
Postulate 5—Through any two points there
exists exactly one line.
Postulate 6—A line contains at least two points.
Postulate 7—If two lines intersect, then their
intersection is exactly one point.
Postulate 8—Through any three noncollinear
points there exists exactly one
plane.
Postulate 9—A plane contains at least three
noncollinear points.
Postulate 10—If two points lie in a plane, then
the line containing them lies in
the plane.
Postulate 11—If two planes intersect, then
their intersection is a line.
Ex.7.
Solution:
a. Postulate 7—If two lines intersect, then
their intersection is exactly one point.
b. Postulate 11—If two planes intersect, then
their intersection is a line.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.
Ex. 8: Use the diagram to
write examples of
Postulates 9 and 10.
Postulate 9—Plane P contains at least three
noncollinear points A, B, and C.
Postulate 10—Point A and point B lie in the
same plane P, so line n containing A and B
also lies in plane P.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.
Extra Ex. 9: Use the
diagram to write examples
of postulates 6 and 8.
Postulate 6: Line l contains at least two points R
and S.
Postulate 8: Through noncollinear points R, S,
and W, there exists exactly one plane M.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.
Concept Summary
Interpreting a Diagram
When you interpret a diagram, you can assume information
about size or measure only if it is marked.
You can assume
All points shown are coplanar
That angle AHB and angle BHD are a linear pair.
That angle AHF and BHD are vertical angles.
A, H, J and D are collinear.
Line AD and line BF intersect at H.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.
Concept Summary
(cont.)
You cannot assume:
G, F , and E are collinear.
BF and CE intersect.
BF and CE do not intersect.
BHA  CJA
AD  BF or mAHB  90 .
0
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.
Perpendicular Figures—A line is a line
perpendicular to a plane if and only if the line
intersects the plane in a point and is perpendicular
to every line in the plane that intersects it at that point.
In a diagram, a line
perpendicular to a plane
must be marked with a
right angle symbol.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 98.
Ex. 10
Sketch a diagram showing
FH  segment EG at its midpoint M .
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 98.
Images used for this presentation came from
the following websites:
http://z.about.com/d/math/1/0/M/2/perpendic
ular.jpg
http://education.yahoo.com/homework_help/
math_help/solutionimages/minigeogt/2/1/1
/minigeogt_2_1_1_25_10/f-10-24-1.gif