A side of the triangle below has been extended to form an exterior angle of 127°. Find the value of x
Since the exterior angle of a triangle is equal to the sum of its adjacent interior angles, we can subtract the given 127° from 180° to find the measure of the corresponding interior angle:
180° - 127° = 53°
This angle is opposite to the side of length x, so we can use the sine ratio to find x:
sin 53° = x / (10 + x)
Cross-multiplying, we get:
x = (10 + x) sin 53°
Expanding:
x = 10 sin 53° + x sin 53°
Subtracting x sin 53° from both sides:
x - x sin 53° = 10 sin 53°
Factoring out x:
x (1 - sin 53°) = 10 sin 53°
Dividing both sides by (1 - sin 53°):
x = 10 sin 53° / (1 - sin 53°)
Using a calculator:
x ≈ 17.6
Therefore, the value of x is approximately 17.6 units.
To find the value of x, we need to use the fact that the measure of an exterior angle of a triangle is equal to the sum of the two opposite interior angles.
In this case, we know that the exterior angle is 127°. Let's call the two opposite interior angles a and b.
So, we have the equation a + b = 127°.
Since the triangle has three interior angles, we know that the sum of all interior angles of a triangle is 180°.
Therefore, we have another equation a + b + x = 180°.
Combining both equations, we get:
(a + b) + x = 180°
127° + x = 180°
To solve for x, we subtract 127 from both sides:
x = 180° - 127°
x = 53°
So, the value of x is 53°.