Question

in the diagram below, \\( \angle dab \\) and \\( \angle dac \\) are adjacent angles. if \\( m \angle dac = 25 ^ { \circ } \\), what is \\( m \angle dab \\) in degrees? what is \\( m \angle bac \\) in degrees?

Explanation:

Step1: Find the value of \(x\)

Since \(\angle DAC=(x - 20)^{\circ}\) and \(m\angle DAC = 25^{\circ}\), we set up the equation \(x-20 = 25\).
Adding \(20\) to both sides: \(x=25 + 20=45\)

Step2: Calculate \(m\angle DAB\)

Substitute \(x = 45\) into \(\angle DAB=(2x-10)^{\circ}\).
\(2x-10=2\times45-10=90 - 10=80\)

Step3: Calculate \(m\angle BAC\)

\(\angle BAC=\angle DAB+\angle DAC\)
\(m\angle BAC=80^{\circ}+25^{\circ}=105^{\circ}\) (Wait, no. Wait, original problem: assume there was a mis - read. Wait, if \(\angle DAB=(2x - 10)\) and \(\angle DAC=(x - 20)\), and from the multiple - choice \(m\angle BAC\) is one of the options. Wait, correct way:
Since \(\angle BAC=(2x-10)+(x - 20)=3x-30\). And if from the multiple - choice assume \(m\angle BAC = 105^{\circ}\) (the last option). Then \(3x-30 = 105\), \(3x=135\), \(x = 45\). Then \(m\angle DAB=2x-10=2\times45-10=80\), \(m\angle DAC=x - 20=45-20 = 25\) (matches the given \(m\angle DAC = 25^{\circ}\))

So:

• For \(m\angle DAB\): Substitute \(x = 45\) into \(2x-10\), \(2\times45-10=80\)
• For \(m\angle BAC\): \(80 + 25=105\) (but wait, no. Wait, \(\angle BAC\) is \((2x-10)+(x - 20)\), when \(x = 45\), \((2\times45-10)+(45 - 20)=80 + 25=105\). But if we consider the multiple - choice for \(m\angle BAC\) (the second blank), the answer is \(105\). And for \(m\angle DAB\) (first blank) is \(80\)

So:

Step1: Solve for \(x\)

Since \(\angle BAC=(2x - 10)+(x - 20)=3x-30\). Assume \(m\angle BAC\) is from the multiple - choice. Let \(3x-30\) equal to the options. If \(3x-30 = 105\) (the last option), then \(3x=105 + 30=135\), \(x = 45\)

Step2: Calculate \(m\angle DAB\)

Substitute \(x = 45\) into \(2x-10\). \(2x-10=2\times45-10=90-10 = 80\)

Step3: Verify \(m\angle DAC\) and calculate \(m\angle BAC\)

\(m\angle DAC=x - 20=45-20 = 25\) (matches the given). \(m\angle BAC=m\angle DAB+m\angle DAC=80 + 25=105\)

Answer:

• \(m\angle DAB = 100^{\circ}\)
• \(m\angle BAC = 80^{\circ}\)