Question

b. if ( p = 130^{circ} ), can you now determine the measures of any of the angles from part (a) that you could not before? calculate the measures for all that you can. be sure to justify your reasoning.

Explanation:

Step1: Find angle \( j \)

Since \( j \) and \( p \) are supplementary (they form a linear pair), \( j + p=180^{\circ} \). Given \( p = 130^{\circ} \), then \( j=180^{\circ}-p \).
\( j = 180^{\circ}-130^{\circ}=50^{\circ} \)

Step2: Find angle \( k \)

\( k \) and \( p \) are vertical angles. Vertical angles are equal. So \( k=p \).
\( k = 130^{\circ} \)

Step3: Find angle \( q \)

\( q \) and \( j \) are vertical angles. Vertical angles are equal. So \( q = j \).
\( q=50^{\circ} \)

Step4: Find angle \( s \)

\( s \) and \( p \) are corresponding angles (because of the parallel lines). Corresponding angles are equal. So \( s = p \).
\( s=130^{\circ} \)

Step5: Find angle \( r \)

\( r \) and \( j \) are corresponding angles (because of the parallel lines). Corresponding angles are equal. So \( r = j \).
\( r=50^{\circ} \)

Step6: Find angle \( x \)

\( x \) and \( p \) are vertical angles (for the lower - part intersection). Vertical angles are equal. So \( x = p \).
\( x=130^{\circ} \)

Step7: Find angle \( y \)

\( y \) and \( j \) are vertical angles (for the lower - part intersection). Vertical angles are equal. So \( y = j \).
\( y=50^{\circ} \)

Answer:

\( j = 50^{\circ}, k = 130^{\circ}, q = 50^{\circ}, r = 50^{\circ}, s = 130^{\circ}, x = 130^{\circ}, y = 50^{\circ} \)