Question

determine the missing angle measure(s) in each problem at right using the geometric relationships shown in the diagram at right. be sure to write down the conjecture that justifies each calculation. each part is a separate problem.
a. if ( d = 110 ^ { circ } ) and ( k = 5 x - 20 ^ { circ } ), write an equation and solve for ( x ).
b. if ( b = 4 x - 11 ^ { circ } ) and ( n = x + 26 ^ { circ } ), write an equation and solve for ( x ). then determine the measure of ( n ).

Explanation:

Part a

Step1: Identify the relationship

Since \(d\) and \(k\) are corresponding angles (conjecture: Corresponding Angles Postulate), \(d = k\).

Step2: Set up the equation

Given \(d = 110^{\circ}\) and \(k=5x - 20^{\circ}\), the equation is \(5x-20 = 110\).

Step3: Solve for \(x\)

Add \(20\) to both sides: \(5x=110 + 20\), so \(5x=130\).
Divide both sides by \(5\): \(x=\frac{130}{5}=26\).

Part b

Step1: Identify the relationship

\(b\) and \(n\) are supplementary angles (conjecture: Consecutive Interior Angles Theorem). So \(b + n=180^{\circ}\).

Step2: Set up the equation

Given \(b = 4x-11^{\circ}\) and \(n=x + 26^{\circ}\), the equation is \((4x-11)+(x + 26)=180\).

Step3: Simplify the equation

Combine like - terms: \(4x+x-11 + 26=180\), \(5x+15 = 180\).

Step4: Solve for \(x\)

Subtract \(15\) from both sides: \(5x=180 - 15\), \(5x=165\).
Divide both sides by \(5\): \(x=\frac{165}{5}=33\).

Step5: Find the measure of \(n\)

Substitute \(x = 33\) into \(n=x + 26\). So \(n=33+26=59^{\circ}\).

Answer:

a. \(x = 26\)
b. \(x = 33\), \(n = 59^{\circ}\)