In the circle below, DAB and DCB are right angles and m BDC = 53°. The figure is not drawn to scale.
What is CAD?
A. 286°
B. 254°
C. 233°
D. 217°
Since DAB and DCB are right angles, we know that triangles ABD and CBD are both right triangles. Therefore, we can use trigonometry to find the measure of angle CAD.
Let's focus on triangle CBD. We know that angle DCB is a right angle and that angle BDC is 53°. Let x be the measure of angle CBD. Then, using the fact that the angles in a triangle add up to 180°, we can write:
x + 53° + 90° = 180°
Simplifying this equation, we get:
x = 37°
Now let's focus on triangle ABD. We know that angle DAB is a right angle, and we just found that angle CBD is 37°. Let y be the measure of angle CAD. Then, using the fact that the angles in a triangle add up to 180°, we can write:
y + 90° + (180° - x) = 180°
Simplifying this equation and substituting x = 37°, we get:
y + 90° + 143° = 180°
y = -53°
Wait a minute! We ended up with a negative measure for angle CAD. This doesn't make any sense. What went wrong?
The issue here is that we made one assumption that we didn't actually verify: that triangle ABD is a right triangle. In fact, we don't know anything about the size of angle ABD. Our diagram only tells us that angle DCB is a right angle and that angle BDC is 53°.
So unfortunately, we can't use trigonometry to find the measure of angle CAD. The answer is none of the above.
To find the measure of CAD, we need to use the information given and apply the properties of angles in a circle.
We are given that DAB and DCB are right angles, meaning they have a measure of 90°. Moreover, m(BDC) = 53°.
Using the fact that the sum of the angles in a triangle is 180°, we can find the measure of angle BCD as follows:
m(BCD) = 180° - m(BDC) - m(DCB)
= 180° - 53° - 90°
= 37°.
Since CAD is an angle subtended by the same arc as angle BCD, it must have the same measure as BCD. Therefore, m(CAD) = 37°.
Hence, the answer is not among the options provided.