⦁ This sketch shows the flight path of a plane. These are the bearings: AB 022° BC 105° CA 262° Calculate the angles inside triangle ABC.( Hint: North lines are parallel lines.) Give reasons if you use angle rules.
So, draw the triangle ABC. Where do yo get stuck?
To get started, note that ∡A = 90-22-8 = 60°
To calculate the angles inside triangle ABC, we can use the properties of triangles and angles. The given bearings AB 022°, BC 105°, and CA 262° represent the directions of each line segment.
First, let's assign labels to the angles inside triangle ABC. Let angle A be ∠ABC, angle B be ∠BCA, and angle C be ∠CAB.
To find angle A (∠ABC), we use the angle sum property of a triangle, which states that the sum of the angles inside a triangle is always 180 degrees. Therefore, we have:
∠ABC + ∠BCA + ∠CAB = 180°
Now let's substitute the known bearings into the equation:
∠ABC + 105° + 262° = 180°
Simplifying the equation:
∠ABC = 180° - 105° - 262°
∠ABC = -187°
Since the value of ∠ABC is -187°, something is off because angles cannot have negative values.
To correct this issue, we need to consider the concept of bearings. Bearings are measured clockwise from the north direction. In this case, the bearing between AB is 022°, which means angle A (∠ABC) is 022° clockwise from the north direction.
To convert this bearing to the standard angle measurement (where angles are measured counterclockwise from the positive x-axis), we subtract the bearing from 360°:
∠ABC = 360° - 022°
∠ABC = 338°
Now we can find angle B (∠BCA) using a similar approach. The bearing between BC is 105°, which means angle B (∠BCA) is 105° clockwise from the north direction. Converting it to the standard angle measurement, we have:
∠BCA = 360° - 105°
∠BCA = 255°
Finally, we can find angle C (∠CAB) by subtracting angles A (∠ABC) and B (∠BCA) from 180°:
∠CAB = 180° - ∠ABC - ∠BCA
∠CAB = 180° - 338° - 255°
∠CAB = -413°
Similarly to angle A, the value of ∠CAB is incorrectly negative, indicating an error. Applying the concept of bearings once again, the bearing between CA is 262°, meaning angle C (∠CAB) is 262° clockwise from the north direction.
Converting it to the standard angle measurement, we have:
∠CAB = 360° - 262°
∠CAB = 98°
After correcting for the bearing conversion, we find:
Angle A (∠ABC) = 338°
Angle B (∠BCA) = 255°
Angle C (∠CAB) = 98°
These are the measures of the angles inside triangle ABC.