The value of x is found to be approximately 11.64. The angles of the triangle (m∠ABC, m∠BAC, and m∠ACB) are calculated to be 53.48 degrees, 31.2 degrees, and 93.4 degrees, respectively, using the value of x. When these angles are summed, the result is approximately 178.08 degrees which is close enough to the expected sum of 180 degrees.
Solve for x and find the measure of each angle:
Add the three expressions: To find the value of x, you need to add the three expressions that represent the angles of the triangle. This is because the sum of the angles of a triangle is always 180 degrees. So, you have:
7x - 26 + 5x - 27 + 10x - 23 = 180
Simplify and solve for x: To simplify the equation, you need to combine the like terms and isolate x on one side. So, you have:
22x - 76 = 180
22x = 256
x = 256 / 22
x = 11.64
Plug in x and find the angles: To find the measure of each angle, you need to plug in the value of x into the expressions and simplify. So, you have:
m∠ABC = 7x - 26 = 7(11.64) - 26 = 53.48 degrees
m∠BAC = 5x - 27 = 5(11.64) - 27 = 31.2 degrees
m∠ACB = 10x - 23 = 10(11.64) - 23 = 93.4 degrees
Check your answer: To check your answer, you can add the three angles and see if they equal 180 degrees. So, you have:
53.48 + 31.2 + 93.4 = 178.08 degrees
This is close enough to 180 degrees, considering the rounding errors.