Question

1. fencing a diagonal brace strengthens the wire fence and prevents it from sagging. the brace makes a 50° angle with the wire as shown. find the value of the variable y.

Explanation:

Step1: Identify the angle relationship

The two angles (50° and \( y \)) are alternate interior angles or consecutive interior angles? Wait, actually, looking at the diagram (a fence with a diagonal brace, so the horizontal rails are parallel, and the brace is a transversal). So the 50° angle and \( y \) are supplementary? Wait, no, maybe they are same - side interior angles? Wait, no, actually, if the horizontal parts are parallel, and the brace is a transversal, then the angle of 50° and \( y \) are complementary? Wait, no, let's think again. Wait, the fence has vertical posts, so the horizontal rails are parallel, and the diagonal brace forms a transversal. So the angle between the brace and the top rail is 50°, and the angle between the brace and the bottom rail ( \( y \)): since the vertical posts are perpendicular to the horizontal rails (assuming the fence is rectangular - like), so the two angles (50° and \( y \)) should be complementary? Wait, no, if the horizontal rails are parallel, and the brace is a transversal, then the consecutive interior angles are supplementary. Wait, maybe the diagram is such that the two angles (50° and \( y \)) are complementary? Wait, no, let's consider that the vertical posts make right angles with the horizontal rails. So the triangle formed? Wait, no, the fence structure: horizontal top and bottom, vertical posts, and a diagonal brace. So the angle between the brace and the vertical post? No, the problem says "the brace makes a 50° angle with the wire as shown". Wait, maybe the two angles (50° and \( y \)) are complementary because they are in a right - angled situation? Wait, no, if the horizontal rails are parallel, and the brace is a transversal, and the vertical posts are perpendicular to the horizontal rails, then the angle between the brace and the top rail is 50°, and the angle between the brace and the bottom rail ( \( y \)): since the sum of angles on a straight line (the vertical post) is 90°? Wait, no, maybe it's a case of alternate interior angles or same - side interior angles. Wait, actually, in a fence with horizontal rails (parallel) and a diagonal brace (transversal), the angle of 50° and \( y \) are supplementary? No, wait, if the horizontal rails are parallel, and the brace is a transversal, then the consecutive interior angles are supplementary. But if the vertical posts are perpendicular to the horizontal rails, then the angle between the brace and the vertical post is 90° - 50° = 40°, but no. Wait, maybe the two angles (50° and \( y \)) are complementary. Wait, no, let's think again. The problem is about a diagonal brace, so the two angles (50° and \( y \)) are adjacent to a right angle? Wait, maybe the fence has vertical posts, so the horizontal rails are horizontal, vertical posts are vertical. The diagonal brace forms a triangle with the vertical post and the horizontal rail. So the angle between the brace and the horizontal rail is 50°, and the angle between the brace and the vertical post is \( y \). Since the horizontal and vertical are perpendicular (90°), then 50°+ \( y \)= 90°, so \( y = 90° - 50°=40° \)? Wait, no, that might not be right. Wait, maybe the two angles (50° and \( y \)) are alternate interior angles. Wait, the horizontal rails are parallel, so the angle of 50° and \( y \) are alternate interior angles, so they are equal? No, that doesn't make sense. Wait, maybe the diagram is such that the two angles (50° and \( y \)) are supplementary. Wait, no, let's look at the standard fence - brace problem. In a fence, the diagonal brace and the horizontal rails: if the top rail and bottom rail are parallel, and the brace is a transversal, then the angle between the brace and the top rail (50°) and the angle between the brace and the bottom rail ( \( y \)) are same - side interior angles, so they are supplementary? No, that would be 180° - 50° = 130°, which doesn't seem right. Wait, maybe the problem is that the two angles are complementary. Wait, maybe the vertical post is perpendicular to the horizontal rail, so the triangle formed by the vertical post, horizontal rail, and diagonal brace is a right triangle. So the angle between the brace and the horizontal rail is 50°, so the angle between the brace and the vertical post ( \( y \)) is 90° - 50° = 40°? Wait, but maybe the diagram is different. Wait, the problem says "the brace makes a 50° angle with the wire as shown". Wait, maybe the two angles (50° and \( y \)) are vertical angles? No. Wait, maybe the fence has horizontal rails (parallel) and the diagonal brace, so the angle of 50° and \( y \) are alternate interior angles, so they are equal? No, that would be 50°, but that seems wrong. Wait, maybe I made a mistake. Let's re - examine. If the horizontal rails are parallel, and the brace is a transversal, then the consecutive interior angles are supplementary. But if the vertical posts are perpendicular to the horizontal rails, then the angle between the brace and the vertical post is 90° - 50° = 40°, but maybe \( y \) is that angle. Wait, maybe the answer is 40°, but let's think again. Wait, the problem is about a diagonal brace strengthening the fence, so the two angles (50° and \( y \)) are complementary because the vertical post is perpendicular to the horizontal rail, so 50°+ \( y \)= 90°, so \( y = 40° \).

Step1: Determine the angle relationship

Assume the horizontal rails are parallel and the vertical posts are perpendicular to the horizontal rails (forming right angles). The diagonal brace acts as a transversal. The angle of 50° and \( y \) are complementary (since they are part of a right - angled triangle formed by the vertical post, horizontal rail, and diagonal brace).

Step2: Calculate the value of \( y \)

We know that for complementary angles, the sum of the two angles is \( 90^{\circ} \). So, if one angle is \( 50^{\circ} \), then:
\[ y=90^{\circ}- 50^{\circ} \]
\[ y = 40^{\circ} \]

Answer:

\( y = 40^{\circ} \)