In /_\ABC, bar(AC) is extended through point C to point D,m/_CAB=(x+14)^(@), m/_ABC=(x-3)^(@), and m/_BCD=(4x-11)^(@). What is the value of x? Answer:

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In  /_\ABC, bar(AC) is extended through point  C to point  D,m/_CAB=(x+14)^(@),  m/_ABC=(x-3)^(@), and  m/_BCD=(4x-11)^(@). What is the value of  x? Answer:

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Write Equation for Triangle ABC: We know that the sum of the interior angles of a triangle is 180 degrees. In triangle ABC, we can write an equation using the given angle measures. m/_CAB + m/_ABC + m/_BCA = 180^(@) Substitute the given expressions for m/_CAB and m/_ABC. (x + 14)^(@) + (x - 3)^(@) + m/_BCA = 180^(@)Use Exterior Angle Theorem: We also know that the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. In this case, m/_BCD is the exterior angle to triangle ABC at vertex C. m/_BCD = m/_CAB + m/_ABC Substitute the given expressions for m/_BCD, m/_CAB, and m/_ABC. (4x - 11)^(@) = (x + 14)^(@) + (x - 3)^(@)Solve for x: Now we solve the equation from the previous step to find the value of x. (4x - 11) = (x + 14) + (x - 3) Combine like terms. 4x - 11 = 2x + 11 Subtract 2x from both sides. 2x - 11 = 11 Add 11 to both sides. 2x = 22 Divide both sides by 2. x = 11Check Solution: We should check our solution by substituting x back into the original angle expressions to ensure they sum to 180 degrees for the triangle and that the exterior angle equals the sum of the two opposite interior angles. m/_CAB = (11 + 14)^(@) = 25^(@) m/_ABC = (11 - 3)^(@) = 8^(@) m/_BCD = (4 * 11 - 11)^(@) = 33^(@) Check if m/_CAB + m/_ABC + m/_BCD = 180^(@) 25^(@) + 8^(@) + 33^(@) = 66^(@), which is not equal to 180^(@). This indicates a mistake in our calculations.

In $\triangle \mathrm{ABC}, \overline{A C}$ is extended through point $\mathrm{C}$ to point $\mathrm{D}, \mathrm{m} \angle C A B=(x+14)^{\circ}$ , $\mathrm{m} \angle A B C=(x-3)^{\circ}$ , and $\mathrm{m} \angle B C D=(4 x-11)^{\circ}$ . What is the value of $x ?$ $\newline$ Answer:

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