Question

1. right - angle abc is shown. bd is inside ∠abc. m∠abd = 43°. a. use the definition of complementary angles to find the measure of complementary angles. b. if m∠abd=x + 3, and m∠dbc = 4x+2, find the value of x.

Explanation:

Step1: Recall complementary - angle definition

Complementary angles add up to 90°. So, \(m\angle ABD + m\angle DBC=90^{\circ}\).

Step2: Substitute angle - measures

Given \(m\angle ABD=x + 3\) and \(m\angle DBC = 4x+2\), we substitute into the equation: \((x + 3)+(4x+2)=90\).

Step3: Simplify the left - hand side

Combine like terms: \(x+4x+3 + 2=90\), which simplifies to \(5x+5 = 90\).

Step4: Isolate the variable term

Subtract 5 from both sides: \(5x+5−5=90−5\), resulting in \(5x=85\).

Step5: Solve for x

Divide both sides by 5: \(\frac{5x}{5}=\frac{85}{5}\), so \(x = 17\).

Answer:

\(x = 17\)