Question

mn||ph, where rv is a transversal with m∠rsn=(3x - 17)° and m∠stp=(7x - 3)° as shown.

1. determine the value of x.
2. what is m∠rsn?
3. what is m∠stp?
4. what is m∠rsm?
5. what is m∠hts?

Explanation:

Step1: Use angle - relationship for parallel lines

Since \(MN\parallel PH\) and \(RV\) is a transversal, \(\angle RSN\) and \(\angle STP\) are same - side interior angles. Same - side interior angles are supplementary, so \((3x - 17)+(7x - 3)=180\).

Step2: Combine like - terms

Combine the \(x\) terms and the constant terms: \((3x+7x)+(-17 - 3)=180\), which simplifies to \(10x-20 = 180\).

Step3: Isolate the variable term

Add 20 to both sides of the equation: \(10x-20 + 20=180 + 20\), resulting in \(10x=200\).

Step4: Solve for \(x\)

Divide both sides of the equation by 10: \(\frac{10x}{10}=\frac{200}{10}\), so \(x = 20\).

Step5: Find \(m\angle RSN\)

Substitute \(x = 20\) into the expression for \(m\angle RSN\): \(m\angle RSN=3x - 17=3\times20-17=60 - 17 = 43^{\circ}\).

Step6: Find \(m\angle STP\)

Substitute \(x = 20\) into the expression for \(m\angle STP\): \(m\angle STP=7x - 3=7\times20-3=140 - 3 = 137^{\circ}\).

Step7: Find \(m\angle RSM\)

Since \(\angle RSN\) and \(\angle RSM\) are a linear - pair (they are adjacent and supplementary), \(m\angle RSM=180 - m\angle RSN\). So \(m\angle RSM=180 - 43=137^{\circ}\).

Step8: Find \(m\angle HTS\)

\(\angle HTS\) and \(\angle RSN\) are vertical angles. Vertical angles are congruent. So \(m\angle HTS=m\angle RSN = 43^{\circ}\).

Answer:

1. \(x = 20\)
2. \(m\angle RSN=43^{\circ}\)
3. \(m\angle STP=137^{\circ}\)
4. \(m\angle RSM=137^{\circ}\)
5. \(m\angle HTS=43^{\circ}\)