About Finding Unknown Angle Measures
What is an Angle?
An angle is a figure formed by two rays or line segments. We can observe angles on objects all around us. The two rays that form an angle have a common intersection point known as the vertex of the angle. The two rays that make an angle are known as the arms of the angle.
We use the symbol ∠ to represent an angle. Angles are measured using an instrument known as a protractor. The extent to which the two rays are separated, that is, how wide the arms are from each other, is measured in degrees (°).
Angles are classified into different types based on their measurement: acute angle, right angle, obtuse angle, straight angle, reflex angle, zero angle and a complete angle.
All closed shapes that have vertices have angles. The number of angles present in a closed shape depends on the number of sides present in the shape. Here are some interesting relationships between angles found on different shapes:
- The angle around a point is \(360^\circ\).
- The angles on a straight line add up to \(180^\circ\).
Here, \(a + b = 180^\circ\).
- The angles inside a triangle add up to \(180^\circ\).
Here, \(a + b + c = 180^\circ\).
- An exterior angle of a triangle is equal to the sum of the opposite interior angles.
Here, \(x=y+z\).
- The angles inside a quadrilateral add up to \(360^\circ\).
Here, \(a + b + c + d = 360^\circ\).
These angle relationships can be used to solve many problems in everyday life.
Adjacent Angles
Two angles that share a common side and a vertex are known as adjacent angles.
Here, \(\angle~AOB\) and \(\angle~BOC\) are adjacent angles, as they share a common vertex O, and a common arm OB.
Here, \(\angle~AOB\) and \(\angle~BOC\) and \(\angle~BOC\) and \(\angle~COD\) are adjacent angles. But \(\angle~AOB\) and \(\angle~COD\) are not adjacent angles as they do not share a common arm.
Complementary Angles
Any two angles whose measures add up to \(90^\circ\) are known as complementary angles. Complementary angles need not be always adjacent angles; they can be two separate angles having a sum of \(90^\circ\).
Here, \(\angle~AOB \) and \(\angle~BOC \) and \(\angle~PQR \) and \(\angle~XYZ \)are complementary angles.
Supplementary Angles
Two angles are said to be supplementary angles when they add up to \(180^\circ\). Supplementary angles need not be always adjacent angles; they can be two separate angles having a sum of \(180^\circ\).
Here, the pair \(\angle~AOB\) and \(\angle~BOC\) and the pair \(\angle~PQR\) and \(\angle~XYZ\) are supplementary angles as they add up to 180 degrees.
Vertical Angles
Vertical angles are the opposite angles formed when two lines intersect. Vertical angles are always equal.
In this figure, \(a=b\) and \(c=d\).
Solved Examples on Finding Missing Angles
Example 1: Name a pair of adjacent angles, complementary angles, supplementary angles, and vertical angles from the following figure:
Solution:
In this figure, all pairs of angles that share a common arm and a common vertex are adjacent angles. For example, \(\angle~AXB\) and \(\angle~BXC\) are adjacent angles.
All pairs of angles that add up to \(90^\circ\) are complementary angles. \(\angle~CXD\) is a right angle, so \(\angle~CXF\) and \(\angle~FXD\) will add up to \(90^\circ\) degrees and hence form a pair of complementary angles.
Similarly, all pairs of angles that add up to \(180^\circ\) are supplementary angles. \(\angle~AXD\) is an angle in a straight line, therefore it is a straight angle.
Hence, \(\angle~AXB\) and \(\angle~BXD\) will add up to \(180^\circ\) degrees and form a pair of supplementary angles.
\(\angle~CXF\) and \(\angle~GXE\) are two angles formed by the intersection of two lines CE and GF. Hence, they are vertical angles.
Example 2: Classify the pair of angles \(\angle~AOB\) and \(\angle~COD\) and find the value of x.
Solution:
Since \(\angle~AOB\) and \(\angle~COD\) are two angles formed by the intersection of two lines AD and BC, they are vertical angles. All vertical angles are equal.
So, \(\angle~AOB = COD\)
Since \(\angle~AOB = 45^\circ\), \(\angle~COD = 45^\circ\)
Example 3: Find the values of x, y, and z.
Solution:
In this figure, \(x+68^\circ = 180^\circ\). This is because they are supplementary angles. In other words, the angles on a straight line add up to \( 180^\circ\).
\(x+68^\circ = 180^\circ\)
So, \(x = 180^\circ- 68^\circ\)
\(x = 112^\circ \)
Now, \(y = 68^\circ \) as y and the 68 angle are vertical angles. They are formed by the intersection of two lines.
Now, x and z are also vertical angles as they are formed by the intersection of two lines.
Hence, \(x = z = 112^\circ\)
Therefore, \(x = z = 112^\circ\) and \(y = 68^\circ \)
Example 4: A ladder is resting on a wall as shown in the figure.
Find the value of \(x \).
Solution:
We can consider the wall to be a straight line. Then, \(x \) and the \(15^\circ \) angles are angles on a straight line. That means they are supplementary angles.
So, \(x + 15^\circ = 180^\circ \)
\(x = 180^\circ – 15^\circ \)
\(x = 165^\circ \)
Congruent angles are the angles that are of equal measure. Two congruent angles coincide when they are superimposed.
The sum of two complementary angles is \(90^\circ\). On the other hand, the sum of two supplementary angles is \(180^\circ\).