Solving for Unknown Angles
Solving for unknown angles is a fundamental aspect of geometry that focuses on determining the measures of angles in various two-dimensional shapes. One of the key principles involved is that the sum of the interior angles in a triangle is always 180 degrees. When tasked with finding an unknown angle in a triangle, one can easily calculate it by subtracting the sum of the known angles from 180 degrees. Additionally, problems may involve intersecting lines, forming pairs of angles that can be classified as either adjacent or opposite. Adjacent angles, which share a side, always sum to 180 degrees, while opposite angles are equal in measure. For quadrilaterals, the sum of the interior angles is 360 degrees, expanding the complexity of angle calculations. Understanding these properties not only aids in solving for unknown angles but also enhances one’s overall comprehension of geometric relationships. This knowledge can be applied across various mathematical contexts, making it a valuable skill for students and enthusiasts alike.
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Solving for Unknown Angles
Solving for unknown angles is a category of math problems that usually concentrates on the properties of two-dimensional shapes. For example, solving for unknown angles with triangles relies on the fact that all triangles have three sides, which intersect to form three inside angles, and the sum of these three inside angles is always 180 degrees. In a typical problem asking one to solve for the unknown angle in a triangle, one will be given the values of two of the inside angles. The task is to understand that by adding together the quantities of the two known angles, and subtracting this sum from 180, one can determine the value of the unknown angle.
A similar category of problem involves the use of a horizontal line that is intersected by a non-perpendicular line, creating a pair of angles, one obtuse and the other acute. As in the previous example, the sum of the two angles must equal 180 degrees, so if one is given the value of either the obtuse or the acute angle, one can calculate the value of the missing angle. Solving for unknown angles may also be performed for the inside angles of a quadrilateral, with the main difference being that the sum of any quadrilateral’s inside angles will always be 360 degrees.
Overview
Some problems that require solving for unknown angles do not use a closed figure, but are instead more akin to the example above, in which a horizontal line is intersected. In these cases, the typical diagram shows two lines intersecting to form an X. The point of intersection creates four angles around it, and the student will usually be given the value of one of these angles and asked to determine the values of the other three. This is a straightforward task, but it is one that requires an understanding that the intersection of the two lines creates two different types of angles: adjacent angles and opposite angles.
Adjacent angles share one side, whereas opposite angles do not share sides but are instead located diagonally across from each another, with the point of intersection between them. Adjacent angles and opposite angles have special relationships with one another. Adjacent angles, as in the example above, always add up to 180 degrees when they are combined. Opposite angles, on the other hand, are always equal to one another. Given one of the four angles created by two intersecting lines, then, one can use these properties of adjacent and opposite angles to find the other three angles of the intersection. For example, if the four angles are A, B, C and D, with A and C opposite to one another, and A equals 30 degrees, C must also equal 30 degrees. It is then possible to show that B, because it is adjacent to A, must equal 150 degrees. The final unknown angle, D, must also equal 150 degrees, because it is opposite to B.
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