Question

1. to remember your converses, inverses, and contrapositives, learn this chart: conditional if p, then q. converse inverse contrapositive 2. given (mangle12 = 121^{circ}) and (mangle6 = 75^{circ}), find the measure of each missing angle. which of the following is the contrapositive of the conditional (\if...then\) statement if the radius of a circle is perpendicular to a line at a point on the circle, then the line is tangent to that circle. a if a line is not tangent to a circle, then the radius is not perpendicular to the line at a point on that circle. b if a line is tangent to a circle, then the radius of the circle is perpendicular to the line at a point on that circle. c if the radius of a circle is perpendicular to a line at a point on the circle, then the line is not tangent to that circle. d if the radius of a circle is not perpendicular to a line at a point on the circle, then the line is not tangent to that circle.

Explanation:

Step1: Recall angle - relationship rules

Vertical angles are equal. Adjacent angles on a straight - line sum to 180°.

Step2: Find angle measures using vertical - angle property

Since vertical angles are equal:
$m\angle1=m\angle5$, $m\angle2 = m\angle6=75^{\circ}$, $m\angle3=m\angle7$, $m\angle4=m\angle8$, $m\angle9=m\angle13$, $m\angle10=m\angle14$, $m\angle11=m\angle15$ (not in the figure but for general understanding), $m\angle12=m\angle16$ (not in the figure but for general understanding).
We know $m\angle12 = 121^{\circ}$, so $m\angle11=180 - 121=59^{\circ}$ (adjacent angles on a straight - line).
$m\angle1=m\angle5 = 59^{\circ}$ (vertical angles with $\angle11$ and $\angle12$ respectively).
$m\angle3=m\angle7 = 121^{\circ}$ (vertical angles with $\angle12$).
$m\angle4=m\angle8 = 59^{\circ}$ (vertical angles with $\angle11$).
$m\angle9=m\angle13 = 75^{\circ}$ (vertical angles with $\angle6$).
$m\angle10=m\angle14 = 105^{\circ}$ (adjacent to $\angle9$ on a straight - line, $180 - 75$).

For the contrapositive of the conditional statement:
The conditional statement is "If the radius of a circle is perpendicular to a line at a point on the circle, then the line is tangent to that circle".
The contrapositive of a conditional statement "If P, then Q" is "If not Q, then not P".
Let P be "the radius of a circle is perpendicular to a line at a point on the circle" and Q be "the line is tangent to that circle".
The contrapositive is "If a line is not tangent to a circle, then the radius is not perpendicular to the line at a point on that circle".

Answer:

a. $m\angle1 = 59^{\circ}$
b. $m\angle2 = 75^{\circ}$
c. $m\angle3 = 121^{\circ}$
d. $m\angle4 = 59^{\circ}$
e. $m\angle5 = 59^{\circ}$
f. $m\angle7 = 121^{\circ}$
g. $m\angle8 = 59^{\circ}$
h. $m\angle9 = 75^{\circ}$
i. $m\angle10 = 105^{\circ}$
j. $m\angle11 = 59^{\circ}$
k. (not clear which $\angle1$ you mean again, assume it's the first one we calculated, so) $m\angle1 = 59^{\circ}$
l. $m\angle14 = 105^{\circ}$
The contrapositive of the conditional statement is A. If a line is not tangent to a circle, then the radius is not perpendicular to the line at a point on that circle.