1y2 11. Algebra If the slope of \( \overline{X Y} \) is 0 , and \( X(2,3) \) and \( Y(-1, k) \), find \( k \). 12. Angles \( A \) and \( B \) are a pair of alternate interior angles on a pair of parallel

To solve the problem, we start with the first part regarding the slope of the line segment \( \overline{XY} \). ### Part 1: Finding \( k \) The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \] In this case, the points are \( X(2, 3) \) and \( Y(-1, k) \). The slope of \( \overline{XY} \) is given to be 0, which means the line is horizontal. Therefore, the change in \( y \) must be 0: \[ \frac{k - 3}{-1 - 2} = 0 \] This simplifies to: \[ \frac{k - 3}{-3} = 0 \] For the fraction to equal 0, the numerator must be 0: \[ k - 3 = 0 \] Solving for \( k \): \[ k = 3 \] ### Part 2: Angles \( A \) and \( B \) Since angles \( A \) and \( B \) are described as alternate interior angles formed by a transversal intersecting two parallel lines, we can conclude that: - Alternate interior angles are equal when the lines are parallel. Thus, if \( A \) and \( B \) are alternate interior angles, we have: \[ A = B \] If you have specific values or relationships for angles \( A \) and \( B \), please provide them for further analysis. ### Summary 1. The value of \( k \) is \( 3 \). 2. Angles \( A \) and \( B \) are equal if they are alternate interior angles formed by a transversal intersecting parallel lines.