A triangle is such that the largest angle is 11 times the smallest angle. The third angle is 27° less than the largest angle. Find the measure of each angle. (Enter solutions from smallest to largest.)
The largest angle = 11 * the smallest angle
The third angle = The largest angle - 27 ° = 11 * the smallest angle - 27 °
The angles of a triangle always sum to 180 °
The smallest angle + The largest angle + The third angle = 180 °
The smallest angle + 11 * the smallest angle + 11 * the smallest angle -27 ° = 180°
23 * the smallest angle - 27 ° = 180 °
23 * the smallest angle = 180 ° + 27 °
23 * the smallest angle = 207 ° Divide both sides by 23
The smallest angle = 207 ° / 23
The smallest angle = 9 °
The largest angle = 11 * the smallest angle
The largest angle = 11 * 9 °
The largest angle = 99 °
The third angle = The largest angle - 27 °
The third angle = 99 ° - 27 °
The third angle = 72 °
72 ° + 99 ° + 9 ° = 180 °
To solve this problem, we can set up a system of equations based on the given information.
Let's denote the smallest angle as x degrees.
According to the problem, the largest angle is 11 times the smallest angle, so the largest angle is 11x degrees.
The third angle is 27° less than the largest angle, so the third angle is (11x - 27) degrees.
Since the sum of angles in a triangle is always 180 degrees, we can write the equation:
x + 11x + (11x - 27) = 180
Simplifying the equation, we get:
23x - 27 = 180
Adding 27 to both sides of the equation, we get:
23x = 207
Dividing both sides of the equation by 23, we get:
x = 9
Now that we have found the value of x, we can substitute it back into the equations to find the measure of each angle.
The smallest angle is x = 9 degrees.
The largest angle is 11x = 11 * 9 = 99 degrees.
The third angle is (11x - 27) = (11 * 9 - 27) = 72 degrees.
Therefore, the measures of the angles in the triangle are 9 degrees, 72 degrees, and 99 degrees.