Question

the diagram shows two parallel lines cut by a transversal. one angle measure is shown. explain how you determined the measure of angle y.

Explanation:

1. First, identify the relationship between the given angle and angle \( y \). If the given angle and \( y \) are vertical angles, they are equal. If they are supplementary (form a linear pair), their sum is \( 180^\circ \). If the lines are parallel, corresponding angles, alternate interior angles, or alternate exterior angles can be used. For example, if the given angle and \( y \) are vertical angles, \( y \) equals the given angle. If they are supplementary, \( y = 180^\circ - \text{given angle} \). If using parallel lines and transversal, use angle - relationship theorems (corresponding angles are equal, alternate interior angles are equal, etc.) to find \( y \) based on the given angle's position.
2. Let's assume the given angle (let's say its measure is \( x \)) and angle \( y \) are vertical angles. Vertical angles are always equal because when two lines intersect, the opposite angles formed are congruent. So if we know the measure of the given angle, angle \( y \) will have the same measure as the given angle because of the vertical angles theorem. If the given angle and \( y \) are supplementary (they form a linear pair, meaning they are adjacent and their non - common sides form a straight line), then we use the fact that the sum of supplementary angles is \( 180^\circ \). So \( y=180^\circ - x \), where \( x \) is the measure of the given angle. If the lines are parallel and the given angle and \( y \) are corresponding angles, then by the corresponding angles postulate (if two parallel lines are cut by a transversal, then corresponding angles are congruent), \( y \) will be equal to the given angle.

Answer:

To determine the measure of angle \( y \) when two parallel lines are cut by a transversal:

1. Identify the angle relationship:

• If angle \( y \) and the given angle are vertical angles: By the Vertical Angles Theorem, vertical angles are congruent. So \( m\angle y=\) measure of the given angle.
• If angle \( y \) and the given angle are supplementary (form a linear pair): Since supplementary angles sum to \( 180^\circ \), \( m\angle y = 180^\circ-\) measure of the given angle.
• If angle \( y \) and the given angle are corresponding/alternate interior/alternate exterior angles (due to parallel lines and transversal): By the Corresponding Angles Postulate (or Alternate Interior/Exterior Angles Theorems), these angles are congruent, so \( m\angle y=\) measure of the given angle.