Question

1. (124) given that $overline{st}paralleloverline{uv}$ and $overline{su}paralleloverline{tv}$, find $mangle stu$.

a. 62
b. 78
c. 34
d. 46
e. none of the above

1. (123) given that $overline{st}paralleloverline{uv}$ and $overline{su}paralleloverline{tv}$, find $mangle tvw$.

a. 116
b. 145
c. 96
d. 110
e. none of the above

1. (126) identify the converse of the following definition. then determine whether this converse is true and whether the definition passes the reversibility test.

if two angles are complementary, then the sum of their measures is 90.
a. if the sum of the measures of two angles is 90, then the angles are complementary. the converse is not true, and the definition does not pass.
b. if each of the measures of two angles is 90, then the angles are complementary. the converse is not true, and the definition does not pass.
c. if the sum of the measures of two angles is 90, then the angles are complementary. the converse is true, and the definition passes.
d. if the measure of an angle is 90, then the angle is complementary. the converse is not true, and the definition does not pass.
e. if each of the measures of two angles is 90, then the angles are complementary. the converse is true, and the definition passes.

1. (126) identify the converse of the following definition. then determine whether this converse is true and whether the definition passes the reversibility test.

if amy travels from new york to los angeles, then she has crossed the country.
a. if amy travels the country, then she goes from new york to los angeles. the converse is not true, and the definition does not pass.
b. if amy crosses the country, then she has traveled from new york to los angeles. the converse is true, and the definition passes.
c. if amy travels the country, then she goes from new york to los angeles. the converse is true, and the definition passes.
d. if amy crosses the country, then she has traveled from new york to los angeles. the converse is not true, and the definition does not pass.
e. if amy is a traveler, then she travels from new york to los angeles. the converse is true, and the definition passes.

Explanation:

Step1: Use properties of parallelogram

Since $\overline{ST}\parallel\overline{UV}$ and $\overline{SU}\parallel\overline{TV}$, $STVU$ is a parallelogram. In a parallelogram, opposite - angles are equal and consecutive angles are supplementary.

Step2: Find the value of $x$ for question 5

In parallelogram $STVU$, $\angle SUT$ and $\angle STV$ are opposite angles. Also, $\angle SUT = 82^{\circ}$. So, $3x + 4=82$.
Subtract 4 from both sides: $3x=82 - 4=78$.
Divide both sides by 3: $x = 26$.

Step3: Calculate $\angle STU$ for question 5

$\angle STU$ and $\angle SUT$ are consecutive angles in the parallelogram, so $\angle STU+\angle SUT = 180^{\circ}$.
$\angle STU=180 - 82=98^{\circ}$, but this is not in the options, so the answer for question 5 is E. None of the above.

Step4: Calculate $\angle TVW$ for question 6

First, find the value of $x$ from the angle - relationship in the parallelogram.
Since $\angle SUT$ and $\angle STV$ are opposite angles, $3x + 4=82$, $x = 26$.
$\angle TUV$ and $\angle TVW$ are supplementary (linear - pair).
In the parallelogram, $\angle TUV = 82^{\circ}$, so $\angle TVW=180 - 82 = 98^{\circ}$, but this is not in the options, so the answer for question 6 is E. None of the above.

Step5: Find converse and check reversibility for question 7

The converse of the statement "If two angles are complementary, then the sum of their measures is 90" is "If the sum of the measures of two angles is 90, then the angles are complementary".
This converse is true, and the definition passes the reversibility test. So the answer for question 7 is C.

Step6: Find converse and check reversibility for question 8

The original statement is "If Amy travels from New York to Los Angeles, then she has crossed the country".
The converse is "If Amy crosses the country, then she has traveled from New York to Los Angeles".
There are many ways to cross the country other than traveling from New York to Los Angeles, so the converse is not true, and the definition does not pass. So the answer for question 8 is D.

Answer:

1. E. None of the above
2. E. None of the above
3. C. If the sum of the measures of two angles is 90, then the angles are complementary. The converse is true, and the definition passes.
4. D. If Amy crosses the country, then she has traveled from New York to Los Angeles. The converse is not true, and the definition does not pass.