Question

question
in \\( \triangle a b c, \overline{a c} \\) is extended through point \\( c \\) to point \\( d, m \angle b c d=(9 x+2)^{circ}, m \angle c a b=(3 x+20)^{circ} \\), and \\( m \angle a b c=(x+17)^{circ} \\). find \\( m \angle b c d \\).
answer
\\( m \angle b c d= \\)

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. So, \(m\angle BCD=m\angle CAB + m\angle ABC\).
Substitute the given expressions: \((9x + 2)=(3x + 20)+(x + 17)\).

Step2: Simplify the right - hand side

First, simplify the right - hand side: \((3x + 20)+(x + 17)=3x+x+20 + 17=4x+37\).
So the equation becomes \(9x + 2=4x+37\).

Step3: Solve for \(x\)

Subtract \(4x\) from both sides: \(9x-4x + 2=4x-4x+37\), which gives \(5x+2=37\).
Then subtract 2 from both sides: \(5x+2 - 2=37 - 2\), so \(5x=35\).
Divide both sides by 5: \(x=\frac{35}{5}=7\).

Step4: Find \(m\angle BCD\)

Substitute \(x = 7\) into the expression for \(m\angle BCD\).
\(m\angle BCD=(9x + 2)\).
\(m\angle BCD=9\times7+2=63 + 2=65\).

Answer:

\(65\)