Points, Lines and Angles
Learning objectives
- Describe points, lines, rays and segments.
- Classify angles and find complements and supplements.
- Prove that vertical angles are equal.
- Use parallel lines and a transversal to find equal and supplementary angles.
Basic objects
Geometry starts from a few basic ideas. A point has a position but no size. A line is straight and extends forever in both directions. A plane is a flat surface that extends forever; in this course all figures lie in one plane.
A line segment \(AB\) is the part of a line between the points \(A\) and \(B\); its length is written \(|AB|\). A ray \(AB\) starts at \(A\) and passes through \(B\). Two lines in a plane are parallel if they never meet.
An angle is formed by two rays with a common endpoint, called the vertex. We measure angles in degrees; a full turn is \(360^\circ\).
Measuring and classifying angles
Drag the point or use the slider to turn the second ray. The page shows the angle, its type, its complement and its supplement.
An angle of measure \(\theta\) is acute if \(0^\circ<\theta<90^\circ\), right if \(\theta=90^\circ\), obtuse if \(90^\circ<\theta<180^\circ\), straight if \(\theta=180^\circ\), and reflex if \(180^\circ<\theta<360^\circ\).
Two angles are complementary if their measures add up to \(90^\circ\), and supplementary if they add up to \(180^\circ\).
Find an angle that equals twice its complement.
Let the angle be \(\theta\). Then \(\theta=2(90^\circ-\theta)\), so \(3\theta=180^\circ\) and \(\theta=60^\circ\).
Vertical angles
When two lines cross, the opposite (vertical) angles are equal.
Show proof
Call two adjacent angles \(\alpha\) and \(\beta\); together they form a straight angle, so \(\alpha+\beta=180^\circ\). The angle \(\gamma\) vertical to \(\alpha\) is also adjacent to \(\beta\), so \(\beta+\gamma=180^\circ\). Hence \(\alpha=180^\circ-\beta=\gamma\). ∎
Parallel lines and a transversal
A line that crosses two other lines is a transversal. Change the angle, then choose a pair type to see which angles are highlighted.
If two parallel lines are cut by a transversal, then corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add up to \(180^\circ\).
In the system of Euclid, the equality of corresponding angles for parallel lines is equivalent to the parallel postulate, so we accept it as our starting point. The other two statements follow from it and from vertical angles.
Exercises
Find the complement and the supplement of \(52^\circ\).
Show solution
The complement is \(90^\circ-52^\circ=38^\circ\) and the supplement is \(180^\circ-52^\circ=128^\circ\).
Two vertical angles measure \((3x+10)^\circ\) and \((5x-30)^\circ\). Find \(x\) and the angles.
Show solution
Vertical angles are equal: \(3x+10=5x-30\), so \(x=20\) and each angle is \(70^\circ\).
Two parallel lines are cut by a transversal and one of the eight angles is \(75^\circ\). Find the other seven.
Show solution
Four of the angles measure \(75^\circ\) and the other four measure \(180^\circ-75^\circ=105^\circ\).
Two supplementary angles differ by \(20^\circ\). Find them.
Show solution
Let \(x=y+20^\circ\) and \(x+y=180^\circ\). Then \(2y=160^\circ\), so \(y=80^\circ\) and \(x=100^\circ\).
Find the angle that equals three times its supplement.
Show solution
Let the angle be \(\theta\): \(\theta=3(180^\circ-\theta)\), so \(4\theta=540^\circ\) and \(\theta=135^\circ\).
Two adjacent angles are supplementary. Prove that their bisectors are perpendicular.
Show solution
Let the angles be \(2\alpha\) and \(2\beta\) with \(2\alpha+2\beta=180^\circ\). The bisectors make angles \(\alpha\) and \(\beta\) with the common ray, on its two sides, so the angle between them is \(\alpha+\beta=90^\circ\).
Summary
- Angles are measured in degrees and classified as acute, right, obtuse, straight or reflex.
- Complementary angles add up to \(90^\circ\); supplementary angles add up to \(180^\circ\).
- Vertical angles are equal.
- Parallel lines cut by a transversal give equal corresponding and alternate interior angles.