Question

definition of an angle bisector
26 = 5x + 11
subtract 11 on both sides
15 = 5x
divide by 5 on both sides
x = 3
information:
m∠dbc = m∠abd = 65°
m∠abc = m∠abd + m∠dbc
solve:
m∠abc = 65 + 65
m∠abc = 130°
your turn

1. if m∠fcd = x + 48, m∠bcf = x + 17, and m∠bcd = 95°, then find the value of x.
2. if m∠abd = 39° and m∠dbc = 22°, what is m∠abc? draw two angle pairs that will justify your answer.

Explanation:

Step1: Use angle - addition property

We know that \(\angle FCD=\angle FCB+\angle BCD\). Given \(\angle FCD = x + 48\) and \(\angle BCD=x + 17\), and \(\angle FCB\) is an angle. By the angle - addition formula \(x + 48=(x + 17)+\angle FCB\), but we can also use the fact that \(\angle FCD-\angle BCD=\angle FCB\).

Another way: Since \(\angle FCD=x + 48\) and \(\angle BCD=x + 17\), and we know that \(\angle FCD-\angle BCD=\angle FCB\) (angle - subtraction property in terms of the given expressions).

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Wait, correct approach: \(\angle FCD+\angle BCD = 180^{\circ}\) (linear - pair of angles, since \(B\), \(C\), \(F\) and \(D\) are related as in a straight - line like formation for the sum of these two adjacent angles). So \(x + 48+x + 17=180\)

Step2: Simplify the equation

Combine like terms: \((x+x)+(48 + 17)=180\), so \(2x+65 = 180\)

Subtract 65 from both sides: \(2x=180 - 65\), so \(2x=115\)

Divide both sides by 2: \(x=\frac{115}{2}=57.5\)

Answer:

\(x = 57.5\)