What is the degree measure of the smaller of the 2 angles formed by the line and the ray shown in the figure below? A. \(14^{\circ}\) B. \(28^{\circ}\) C. \(29^{\circ}\) D. \(58^{\circ}\) E. Cannot be determined from the given information

Geometry is a branch of mathematics focused on questions of shape, size, relative position of figures, and the properties of space. Angles are a central concept in geometry; they are formed by two rays (or line segments) diverging from a common point called the vertex. One of the fundamental ideas in the study of angles is the sum of angles in various configurations.

For example, when two rays form a straight line, the angle they create is called a straight angle, which is always equal to 180 degrees. This principle is crucial in solving many geometrical problems and is often used as a starting point for deducing other properties of a geometric figure. Understanding the properties of angles allows us to calculate measures, construct geometric figures accurately, and solve problems involving angles in polygons, circles, and beyond.

In geometry, the straight angle sum refers to the total measure of the angles that can fit on a straight line, which is always 180 degrees. This concept is derived from the fact that a straight angle itself is one that measures 180 degrees. Thus, when a straight line is divided into two angles by a ray, as in the aforementioned exercise, the sum of the two resulting angles must still equal 180 degrees, due to the unchanging nature of a straight angle.

Example Application

In the given exercise, a ray divides a straight angle, and the problem asks for the degree measure of the smaller angle formed. This smaller angle, when combined with its adjacent angle, must sum up to 180 degrees because the two angles together recreate the straight angle. This relationship serves as a pillar for determining possible measures of angles when given limited information and allows for the elimination of incorrect answers.

Problem solving in mathematics involves a step-by-step approach to break down complex problems into manageable parts. In the realm of geometry, problem solving often utilizes known properties and theorems, like the straight angle sum property, to work towards a solution.

Understanding the Process

The exercise presented underlines the problem-solving strategy of elimination based on geometric principles. To find the smaller angle formed by a line and a ray, one must understand that the sum of this angle with its supplementary angle on the line will always be 180 degrees. By subtracting each possible option for the smaller angle from 180 degrees, one can check if the complementary angle is valid and in accordance with the given choices.

The solution illustrates that, given various choices and using geometric rules, one can eliminate possibilities until the correct answer is found or, as demonstrated, we can conclude that the available information is insufficient. This method showcases the logical reasoning skills in geometrical problem solving, emphasizing a systematic approach to arriving at a valid conclusion.