unit 6 because idk shit Flashcards
Properties of inequalities 1
If a > b, and c ≥ d, then a + c > b + d (LIKE ADD. PROP.)
Properties of inequalities 1 EX
a = 5
b = 2
c = 3
d = 1
5 >2
+3 +1
8 > 3
Stays true
Properties of inequalities 2
If a > b and C > 0, then ac > bc and a/c > b/c
Properties of inequalities 2 EX
2
OR FOR NEGATIVES,
-2
Remember, sign FLIPS when dividing or multiplying negative numbers
Properties of inequalities 3
If a > b and c < 0, then ac < bc and a/c < b/c
Properties of inequalities 4
If a > b and b > c, then a > c (LIKE TRANS. PROP.)
Properties of Inequalities 5
If a = b + c and c > 0, then a > b (SUM IS GREATER THAN ITS PARTS”
PRACTICE: If XY = YZ + 5, then XY > YZ (T or F)
True
PRACTICE: If m<A = m<B +m<C, then m<b> m<C (T or F)</b>
False
PRACTICE: If m<H = m<J + m<K, then m<K> m<H</K>
False
PRACTICE: If 10 = y + 2, then y > 10
False
The Exterior Angle Inequality Theorem
The measure of exterior angles of a triangle is greater than the measures of either remote interior angle.
-Statement: If p, then q.
-What is the Inverse?
If not p, then not q
-Statement: If p, then q.
-What is the Contrapositive?
If not q, then not p
If the statement is true (or false),
Then the contrapositive is true (or false)
Statements and contrapositives of those statements are called what?
Logically equivalent statements
What does a conditional venn diagram look like?
Circle p is inside circle q
PRACTICE: (If p, then q.) All runners are athletes. Leroy is a runner. What’s the conclusion?
Leroy is an athlete.
PRACTICE: (If p, then q.) All runners are athletes. Lucia is not an athlete. What’s the conclusion?
Lucia is not a runner.
PRACTICE: (If p, then q.) All runners are athletes. Linda is an athlete. What’s the conclusion
No conclusion.
PRACTICE: (If p, then q.) All runners are athletes. Larry is not a runner. What’s the conclusion?
No conclusion.
What do indirect proofs start with?
“Assume that (opp. of what your proving is true)”
How to write an Indirect proof?
- Assume temporarily that the conclusion is not true
- Reason logically until you reach a contradiction of a known fact
- Point out that the temporary assumption is false and the conclusion is true.
PRACTICE:
Given: n is an integer and n^2 is even
Prove: n is even
SOMETHING LIKE THIS:
-Assume temporarily that n is not even
-Then n is odd and
n^2 = n x n
=odd x odd
=odd
This however contradicts the original given information that n^2 is even, meaning the assumption was false and that n is even
If one side of a triangle is longer than the second side,
Then the angle opposite to the first side is larger than the angle opposite the second side
If one angle of a triangle is larger than a second angle,
Then the side opposite to the first angle is longer than the side opposite to the second angle.
The perpendicular segment from a point to a line is-
-the shortest segment from the point to the plane.
The perpendicular segment from a point to a plane is-
-the shortest segment from the point to the plane.
The sum of the lengths of any two sides of a triangle is-
-greater than the lengths of the third side.
Is it possible for a triangle to have sides with the following lengths?
6, 8, 10
Yes
Is it possible for a triangle to have sides with the following lengths?
3, 4, 8
No
Is it possible for a triangle to have sides with the following lengths?
2.5, 4.1, 5.0
Yes
Is it possible for a triangle to have sides with the following lengths?
6, 6, 5
Yes queen
SAS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
SSS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
