Question

in part (a), prove that h || k. in parts (b) and (c), solve for x given that q || p and s || r, respectively. a. diagram with h, k, angles x, y, ( x + y = 180^circ ) b. diagram with a, b, c, angles ( 31^circ ), ( 26^circ ), ( x ) c. diagram with a, b, c, angles ( 142^circ ), ( 141^circ ), ( x ) since ( x + y = 180^circ ) and if ( x ) is the corresponding angle to ( x ), then ( x ) is supplementary to ( y ) and ( x + y = 180^circ ) therefore ( x ) and ( x ) both have the same measure, ( 180 - y ) since ( x + y = 180^circ ) and if ( x ) is the corresponding angle to ( x ), then ( x ) is complementary to ( y ) and ( x + y = 180^circ ) therefore ( x ) and ( x ) both have the same measure, ( 90 - y ) since ( x ) and ( y ) are supplementary angles, the lines are parallel. b. solve for ( x ) ( x = square^circ )

Explanation:

Part (a) Proof that \( h \parallel k \)

To prove \( h \parallel k \), we use the Converse of the Same - Side Interior Angles Theorem. The Same - Side Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the same - side interior angles are supplementary. The converse of this theorem states that if two lines are cut by a transversal and the same - side interior angles are supplementary, then the two lines are parallel.

Let the transversal be the line that intersects \( h \) and \( k \), forming angles \( x \) and \( y \). We are given that \( x + y=180^{\circ} \). Angles \( x \) and \( y \) are same - side interior angles formed by the transversal intersecting lines \( h \) and \( k \). By the Converse of the Same - Side Interior Angles Theorem, since \( x + y = 180^{\circ} \) (the same - side interior angles are supplementary), we can conclude that \( h\parallel k \).

Part (b) Solve for \( x \) when \( q\parallel p \)

When dealing with parallel lines \( q\parallel p \) and a transversal (or a figure with alternate - interior or corresponding angles), we can use the property of alternate - interior angles or the sum of angles in a triangle - like figure.

We can draw a line through point \( B \) parallel to \( q \) and \( p \). Let's call this line \( l \). Then, the angle of \( 31^{\circ} \) and one part of \( x \) are alternate - interior angles, and the angle of \( 25^{\circ} \) and the other part of \( x \) are alternate - interior angles.

So, \( x=31^{\circ}+ 25^{\circ}\)

Step 1: Identify the angle addition

We know that when we have two parallel lines and a transversal, and we draw a line parallel to them through the vertex of the angle \( x \), we can split \( x \) into two angles that are equal to the given angles \( 31^{\circ} \) and \( 25^{\circ} \) (by alternate - interior angles theorem).
So \( x=31 + 25\)

Step 2: Calculate the sum

\(x = 56^{\circ}\)

Part (c) Solve for \( x \) when \( s\parallel r \)

We know that the sum of the interior angles on the same side of a transversal for parallel lines and the angle \( x \) can be related to the given angles. The sum of the angles around a point or in a polygon - like figure formed by the parallel lines and the transversal can be used.

The sum of the angles \( 142^{\circ} \), \( x \), and \( 141^{\circ} \) should be related to the sum of angles on a straight line (since \( s\parallel r \)). The sum of angles on a straight line is \( 180^{\circ}\times2 = 360^{\circ}\) (because we have two parallel lines and the angles formed around the transversal).

So, \(142 + x+141=360\)

Step 1: Set up the equation

The sum of the three angles \( 142^{\circ} \), \( x \), and \( 141^{\circ} \) is equal to \( 360^{\circ} \) (since the total angle around the intersection of the transversal with the two parallel lines is \( 360^{\circ} \)). So the equation is \(142 + x+141 = 360\)

Step 2: Simplify the left - hand side

\(142+141+x=283 + x\)

Step 3: Solve for \( x \)

Subtract \( 283 \) from both sides of the equation \(283 + x=360\)
\(x=360 - 283\)
\(x = 77^{\circ}\)

Final Answers

a. By the Converse of the Same - Side Interior Angles Theorem, since \( x + y=180^{\circ} \), \( h\parallel k \).

b. \( x=\boxed{56}\)

c. \( x=\boxed{77}\)

Answer:

We know that the sum of the interior angles on the same side of a transversal for parallel lines and the angle \( x \) can be related to the given angles. The sum of the angles around a point or in a polygon - like figure formed by the parallel lines and the transversal can be used.

The sum of the angles \( 142^{\circ} \), \( x \), and \( 141^{\circ} \) should be related to the sum of angles on a straight line (since \( s\parallel r \)). The sum of angles on a straight line is \( 180^{\circ}\times2 = 360^{\circ}\) (because we have two parallel lines and the angles formed around the transversal).

So, \(142 + x+141=360\)

Step 1: Set up the equation

The sum of the three angles \( 142^{\circ} \), \( x \), and \( 141^{\circ} \) is equal to \( 360^{\circ} \) (since the total angle around the intersection of the transversal with the two parallel lines is \( 360^{\circ} \)). So the equation is \(142 + x+141 = 360\)

Step 2: Simplify the left - hand side

\(142+141+x=283 + x\)

Step 3: Solve for \( x \)

Subtract \( 283 \) from both sides of the equation \(283 + x=360\)
\(x=360 - 283\)
\(x = 77^{\circ}\)

Final Answers

a. By the Converse of the Same - Side Interior Angles Theorem, since \( x + y=180^{\circ} \), \( h\parallel k \).

b. \( x=\boxed{56}\)

c. \( x=\boxed{77}\)