Angles and angle pairs
An angle is formed by two rays sharing an endpoint. Recognising special pairs of angles is the key to finding unknown measures throughout geometry.
Naming and measuring angles
An angle consists of two rays, its sides, sharing an endpoint called the vertex. Angle ABC has vertex B. Angles are measured in degrees with a protractor; a full turn is 360 degrees. Angles with equal measures are congruent.
Types of angles
An acute angle measures less than 90 degrees; a right angle exactly 90, marked with a small square; an obtuse angle between 90 and 180; and a straight angle exactly 180, forming a line. Perpendicular lines meet at right angles.
Angle addition and bisectors
The angle addition postulate says that if point D lies in the interior of angle ABC, then \(m\angle ABD + m\angle DBC = m\angle ABC\). An angle bisector divides an angle into two congruent angles: bisecting a 70-degree angle gives two 35-degree angles.
Complementary and supplementary
Two angles are complementary if their measures add to 90 degrees and supplementary if they add to 180. The complement of 28 degrees is 62 degrees; its supplement is 152 degrees. The angles need not be adjacent.
Adjacent angles and linear pairs
Adjacent angles share a vertex and a side but no interior points. A linear pair is two adjacent angles whose outer sides form a straight line, so they are supplementary. If one angle of a linear pair is 115 degrees, the other is 65.
Vertical angles
When two lines cross, the angles opposite each other are vertical angles, and vertical angles are always congruent. The proof is short: each vertical angle forms a linear pair with the same neighbouring angle, so both equal 180 degrees minus that angle.
Solving with algebra
Angle relationships give equations. If two vertical angles measure 3x + 10 and 5x - 30 degrees, they are equal, so 3x + 10 = 5x - 30, giving x = 20 and an angle of 70 degrees. The two angles forming linear pairs with them each measure 110 degrees.
Worked example
The hands of a clock at 3:00 form a right angle, 90 degrees, since each hour mark is 360 / 12 = 30 degrees apart. At 2:00, they are two hour marks apart, 60 degrees. At 4:00 the angle is 120 degrees, whose supplement is 60 degrees.
Key takeaways
An angle is two rays sharing a vertex, measured in degrees. Acute angles are under 90, right angles exactly 90, obtuse angles between 90 and 180, and straight angles 180. Angle measures add, and a bisector splits an angle in half. Complementary angles sum to 90 and supplementary angles to 180. A linear pair is supplementary. Vertical angles, opposite each other where two lines cross, are always congruent. These facts turn angle diagrams into equations.
Flashcard
Two rays sharing a vertex
Check your understanding
Two vertical angles measure 3x + 10 and 5x - 30 degrees. What is each angle?