Question

1. complete a two - column proof

given: \\( \angle a b d \\) and \\( \angle c a d \\) are complementary
prove: \\( \angle b a c = 69 \\)

Explanation:

Step1: Use the definition of complementary angles

Since \(\angle ABD\) and \(\angle CAD\) are complementary, \((2x + 15)+(13x+30)=90\)

Step2: Combine like terms

\(2x+13x+15 + 30=90\), so \(15x+45 = 90\)

Step3: Subtract 45 from both sides

\(15x+45-45=90 - 45\), then \(15x=45\)

Step4: Divide both sides by 15

\(x=\frac{45}{15}=3\)

Step5: Find \(\angle BAC\)

Substitute \(x = 3\) into \(2x+15\). \(\angle BAC=2x + 15\), so \(\angle BAC=2\times3+15=6 + 15=21\) (Wait, no! Wait, actually, \(\angle BAC=13x + 30\). Substitute \(x = 3\): \(13\times3+30=39+30 = 69\))

Answer:

\(\angle BAC = 69\)