Question

1. solve for x.

(there is a circle with a central angle of 50° in the image)
(a) if x = 100° draw a scar on one arm.
(b) if x = 50° draw a scar on the face.

1. solve for x.

(there is a circle with a 45° angle and an angle x in the image)
(a) if x = 135° draw 2 bats in the background.
(b) if x = 67.5° draw 2 birds in the background.

Explanation:

Step1: Recall the total angle around a point

A full circle (around a point) has a total angle of \( 360^\circ \). In the given diagram, there is a straight line (which is a \( 180^\circ \) angle) and another angle of \( 45^\circ \), and we need to find \( x \). Wait, actually, looking at the circle, the angles around the center: the straight line is \( 180^\circ \), and we have a \( 45^\circ \) angle and \( x \), but actually, the three angles (the straight line is split into two, plus the \( 45^\circ \))? Wait, no, the diagram shows a circle with a diameter (so \( 180^\circ \)) and a radius making an angle of \( 45^\circ \) with one part of the diameter, and \( x \) with the other. Wait, actually, the sum of angles around the center: the total around a point is \( 360^\circ \), but here we have a semicircle? No, the circle is divided into three parts? Wait, no, the diagram has a straight line (diameter) so \( 180^\circ \), and a radius making \( 45^\circ \) with one side, and \( x \) with the other. Wait, actually, the angle on one side of the diameter: the \( 45^\circ \) and \( x \) should add up to \( 180^\circ \)? No, that's not right. Wait, no, the total angle around the center is \( 360^\circ \), but the straight line is \( 180^\circ \), so the other semicircle is also \( 180^\circ \). Wait, the diagram shows a circle with a diameter (so two semicircles, each \( 180^\circ \)) and a radius that splits one semicircle into \( 45^\circ \) and \( x \). Wait, no, the three angles: the straight line is \( 180^\circ \), and then the \( 45^\circ \) and \( x \) are in the other semicircle? No, I think I made a mistake. Wait, actually, the correct approach: the angle along a straight line is \( 180^\circ \). In the diagram, we have a straight line (so \( 180^\circ \)) and a \( 45^\circ \) angle, and we need to find \( x \) such that the sum of angles around the center? Wait, no, the circle's center: the total angle is \( 360^\circ \), but the diameter is \( 180^\circ \), so the remaining angle is also \( 180^\circ \). Wait, the diagram shows a diameter (horizontal line) and a radius going down to make \( 45^\circ \) with the diameter, and \( x \) is the angle between the other radius and the diameter. Wait, no, actually, the three angles: the straight line (diameter) is \( 180^\circ \), and the two angles \( 45^\circ \) and \( x \) are in the other semicircle? No, that's not. Wait, let's think again. The sum of angles around a point is \( 360^\circ \). The straight line (diameter) is \( 180^\circ \), so the other two angles ( \( 45^\circ \) and \( x \)) and the straight line? No, the diagram has a circle with center, a horizontal diameter, a radius going down to the left making \( 45^\circ \) with the left half of the diameter, and a radius going down to the right making \( x \) with the right half of the diameter. Wait, then the sum of \( 45^\circ \), \( x \), and the \( 180^\circ \) (the straight line) would be \( 360^\circ \)? No, that's not. Wait, no, the horizontal diameter is \( 180^\circ \), so the angles below the diameter: \( 45^\circ \) and \( x \) should add up to \( 180^\circ \)? No, that would be if they are on a straight line. Wait, I think I messed up. Let's look at the total angle around the center. The center has angles: the straight line (diameter) is \( 180^\circ \), and the other two angles ( \( 45^\circ \) and \( x \)) are in the other semicircle? No, the circle is \( 360^\circ \), so \( 180^\circ \) (diameter) + \( 45^\circ \) + \( x \) = \( 360^\circ \)? No, that would mean \( 45 + x = 180 \), so \( x = 135^\circ \). Yes! Because the total around the center is \( 360^\circ \), and the diameter is \( 180^\circ \), so the remaining two angles ( \( 45^\circ \) and \( x \)) must add up to \( 180^\circ \)? Wait, no, \( 360 - 180 = 180 \), so \( 45 + x = 180 \), so \( x = 180 - 45 = 135^\circ \). So \( x = 135^\circ \), which matches option (a).

Answer:

(for problem 10):
\( x = 135^\circ \) (so option (a) is correct: (a) If \( x = 135^\circ \) draw 2 bats in the background)