Respuesta :
Write a conjecture that describes the pattern in
each sequence. Then use your conjecture to
find the next item in the sequence.
1. 15, 30, 45, 60
SOLUTION:
Here, 15 is added to each term to get the next term.
So, the next term is 60 + 15 = 75.
2.
SOLUTION:
Each figure is rotated by an angle of 90° and then the
mirror image of the rotated figure is taken. So, the
next figure will be as shown.
Use the following statements to write a
compound statement for each conjunction or
disjunction. Then find its truth value.
p : 5 < –3
q: All vertical angles are congruent.
r: If 4x = 36, then x =9.
3. p and q
SOLUTION:
Find the conjunction p and q. A conjunction is true
only when both statements that form it are true.
The statement p is "5 < –3" which is false. The
statement q is "all vertical angles are congruent." is
true. Since p is false, the conjunction is false.
4.
SOLUTION:
Find the disjunction . A disjunction is true if at
least one of the statements is true. Then find the
conjunction . A conjunction is true only
when both statements that form it are true.
The statement is 5 < –3 or all vertical
angles are congruent, and if 4x = 36, then x = 9. The
statement is true because it is enough for
either of the statements p or q to be true for the
statement to be true and it is true that all
vertical angles are congruent. Here, r is a true
statement. Therefore, is a true
statement.
5. PROOF Write a paragraph proof.
Given:
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given two sets of congruent
segments. Use the properties that you have learned
about congruent segments and equivalent expressions
in algebra to walk through the proof.
Proof: Since ,
JK = BC and KL = AB by the definition of congruent
segments. By the Addition Property, JK + KL = CB
+ AB. Using the Segment Addition Postulate, JL =
JK + KL and AC = AB + BC. By substitution, JL =
AC. Because the measures are equal, by
the definition of congruent segments.
6. SPORTS Refer to the Venn diagram that represents
the sports students chose to play at South High
School last year.
a. Describe the sports that the students in the nonintersecting
portion of the tennis region chose.
b. How many students played soccer and tennis?
SOLUTION:
a. .The students in the non-intersecting portion of the
tennis region chose only to play tennis.
b. Find (soccer and tennis). The number of students
in the intersecting region is 23. So, 23 students played
soccer and tennis.
7. Determine whether the stated conclusion is valid
based on the given information. If not, write invalid.
Explain your reasoning.
Given: If a lawyer passes the bar exam, then he or
she can practice law. Candice passed the bar exam.
Conclusion: Candice can practice law.
SOLUTION:
By the Law of Detachment if p → q is a true
statement and p is true, then q. Here, the statement
“if a lawyer passes the bar exam, then he or she can
practice law” is a true statement and Candice passed
the bar exam. So, Candice can practice law is a valid
statement.
8. PROOF Copy and complete the following proof.
Given: 3(x – 4) = 2x + 7
Prove: x = 19
Proof:
SOLUTION:
The 2nd row is uses the Distributive Property to
simplify the left side of the equation.
The 3rd row changes the subtraction property to
subtract 2x from each side.
The 4th row uses the Addition Property to add 12 to
each side
Determine whether each statement is always,
sometimes, or never true.
9. Two angles that are supplementary form a linear
pair.
SOLUTION:
Two angles that are supplementary form a linear pair
only if they share one of their legs. So, the statement
is sometimes true.
or
10. If B is between A and C, then AC + AB = BC.
SOLUTION:
If B is between A and C, then AB + BC = AC. So,
the statement is never true.
11. If two lines intersect to form congruent adjacent
angles, then the lines are perpendicular.
SOLUTION:
If two lines intersect the two adjacent angles have to
be supplementary and since they are congruent each
one of the angles measure 90 degrees. So, the lines
are perpendicular. Therefore, the statement is
always true.
Find the measure of each numbered angle, and
name the theorems that justify your work.
12.
SOLUTION:
Angles 1, 2, and 3 form a linear pair. Since we are
given that angle 3 is right, angles 1 and 2 are
complementary. By the Complementary Angles
Theorem, the sum of their measures is 90.
Solve for x.
2x = 96
x = 48
Therefore,
13.
SOLUTION:
The angles 7 and 8 are supplementary. So, by the
Supplementary Angles Theorem the sum of their
measures is 180.
Solve for x.
5x = 165
x = 33
Therefore, = 2(33) + 15 or 81 and = 3(33)
or 99.
