Question

1. lines pq and rs are parallel, and ad is a transversal. a. determine the value of n. n = 25 b. what is m∠abq? m∠abq = 139° c. what is m∠bcr? m∠bcr = 41°

Explanation:

Step1: Use corresponding - angles property

Since lines PQ and RS are parallel and AD is a transversal, the corresponding angles are equal. So, \(2n - 9=5n + 14\).

Step2: Solve the equation for n

First, move the terms with n to one - side: \(2n-5n=14 + 9\). Then, \(-3n=23\), and \(n=-\frac{23}{3}\). But it seems there is a mistake in the setup. If we consider the supplementary - angles property (since \((2n - 9)+(5n + 14)=180\) for same - side interior angles).

Step3: Solve the correct equation

Expand the left - hand side: \(2n-9+5n + 14=180\). Combine like terms: \(7n+5 = 180\). Subtract 5 from both sides: \(7n=180 - 5=175\). Divide both sides by 7: \(n = 25\).

Step4: Find \(m\angle ABQ\)

Substitute \(n = 25\) into the expression for the angle \(\angle ABQ\) which is \(2n-9\). So, \(m\angle ABQ=2\times25-9=50 - 9=41^{\circ}\).

Step5: Find \(m\angle BCR\)

Since \(\angle BCR\) and \(\angle ABQ\) are corresponding angles (because PQ\(\parallel\)RS and AD is a transversal), \(m\angle BCR=m\angle ABQ = 41^{\circ}\).

Answer:

a. \(n = 25\)
b. \(m\angle ABQ=41^{\circ}\)
c. \(m\angle BCR=41^{\circ}\)