Lines and Angles

The One Thing

Lines and angles are solved by identifying the geometric relationship shown in a diagram and then applying the corresponding equality or sum property. The principal tools are straight-line and full-turn angle sums, vertically opposite and linear-pair relationships, and the angle relationships produced when a transversal intersects parallel lines.

Definitions and Results

• Line: A straight path extending infinitely in both directions.
• Line segment: A part of a line with two fixed endpoints.
• Ray: A part of a line that starts at one endpoint and extends infinitely in one direction.
• Angle: The figure formed by two rays having a common endpoint, called the vertex.
• Adjacent angles: Two angles with a common vertex and a common arm, where their interiors do not overlap.
• Linear pair: A pair of adjacent angles whose non-common arms form a straight line.
• Vertically opposite angles: The opposite angles formed when two lines intersect; each pair is equal.
• Complementary angles: Two angles whose sum is 90 degrees.
• Supplementary angles: Two angles whose sum is 180 degrees.
• Transversal: A line that intersects two or more lines at distinct points.
• Parallel lines: Lines in the same plane that never meet, however far they are extended.
• Corresponding angles: Angles lying in the same relative position at the two intersections of a transversal with two lines.
• Alternate interior angles: Interior angles on opposite sides of a transversal.
• Alternate exterior angles: Exterior angles on opposite sides of a transversal.
• Co-interior angles: Interior angles on the same side of a transversal, also called consecutive interior angles.
• Complete angle: An angle measuring 360 degrees.
• Straight angle: An angle measuring 180 degrees.
• Right angle: An angle measuring 90 degrees.
• Acute angle: An angle measuring less than 90 degrees.
• Obtuse angle: An angle greater than 90 degrees but less than 180 degrees.
• Angles on a straight line: Their measures add up to 180 degrees.
• Angles around a point: Their measures add up to 360 degrees.
• Intersecting lines: Vertically opposite angles are equal, and angles in a linear pair are supplementary.
• Parallel lines and a transversal: Corresponding angles, alternate interior angles, and alternate exterior angles are equal; co-interior angles are supplementary and sum to 180 degrees.
• Converse results: If corresponding angles are equal, alternate interior angles are equal, or co-interior angles are supplementary, the two lines are parallel.
• Parallelism through a common parallel: If two lines are parallel to the same line, they are parallel to each other.
• Perpendicularity and parallel lines: If a line is perpendicular to one of two parallel lines, it is perpendicular to the other as well.
• Degree notation: The measure of an angle is commonly written with the degree symbol, such as 45 degrees.

Worked Methods

Finding an unknown angle on a straight line

• Identify the adjacent angles whose non-common arms form a straight line.
• Use the fact that angles on a straight line add up to 180 degrees.
• Subtract the known angle from 180 degrees.

Thus, if one angle is 45 degrees, the adjacent angle is:

$$180^{\circ }-45^{\circ }=135^{\circ }$$

Finding an unknown angle around a point

• Identify all angles meeting at the point.
• Use the fact that angles around a point add up to 360 degrees.
• Add the known angles and subtract their total from 360 degrees.

Using vertically opposite angles

• Confirm that two straight lines intersect.
• Identify the pair of opposite angles.
• Set the vertically opposite angles equal.

For example, if one angle at the intersection is 45 degrees, the angle directly opposite it is also 45 degrees.

Using a linear pair

• Confirm that the two angles are adjacent.
• Check that their non-common arms form a straight line.
• Use the supplementary relationship:

$$\text{first angle}+\text{second angle}=180^{\circ }$$

Using complementary or supplementary angles

• Determine whether the angles are complementary or supplementary.
• For complementary angles, use:

$$\text{first angle}+\text{second angle}=90^{\circ }$$

• For supplementary angles, use:

$$\text{first angle}+\text{second angle}=180^{\circ }$$

• Rearrange the equation to find the unknown angle.

Using corresponding angles with parallel lines

• Identify the two parallel lines and the transversal.
• Locate the angles in the same relative position at the two intersections.
• Set the corresponding angles equal.
• Solve for the unknown angle.

Using alternate interior angles

• Identify the interior region between the two parallel lines.
• Locate the angles on opposite sides of the transversal.
• Use the equality of alternate interior angles.
• Solve for the unknown angle.

Using alternate exterior angles

• Identify the angles outside the two parallel lines.
• Confirm that they lie on opposite sides of the transversal.
• Set the alternate exterior angles equal.
• Solve for the unknown angle.

Using co-interior angles

• Identify the interior angles on the same side of the transversal.
• Use the supplementary relationship:

$$\text{first angle}+\text{second angle}=180^{\circ }$$

• Rearrange the equation to find the unknown angle.

Testing whether two lines are parallel

• Identify the angle relationship shown by the transversal.
• Check whether corresponding angles are equal, alternate interior angles are equal, or co-interior angles are supplementary.
• Apply the appropriate converse theorem.
• Conclude that the two lines are parallel.

Using parallel lines and perpendicularity

• Identify the two parallel lines.
• Determine whether a third line is perpendicular to one of them.
• Apply the result that a line perpendicular to one of two parallel lines is perpendicular to the other.

Where It Goes Wrong

• Treating adjacent angles as automatically equal; adjacent angles share a vertex and arm, but only a linear pair is necessarily supplementary.
• Forgetting that vertically opposite angles are equal only when they are formed by two intersecting lines.
• Using 360 degrees for angles on a straight line instead of 180 degrees; 360 degrees applies to angles around a point.
• Confusing corresponding, alternate interior, alternate exterior, and co-interior positions relative to the transversal.
• Applying equal-angle results without confirming that the two lines are parallel; co-interior angles are supplementary, not equal.
• Omitting the converse condition when using angle measurements to prove that two lines are parallel.

What Gets Asked

• Define a line, line segment, ray, angle, adjacent angles, a linear pair, vertically opposite angles, complementary angles, supplementary angles, a transversal, parallel lines, and the principal angle pairs.
• Classify angles as complete, straight, right, acute, or obtuse.
• Calculate an unknown angle using angles on a straight line or angles around a point.
• Find an unknown angle using vertically opposite angles or a linear pair.
• Apply complementary and supplementary angle relationships.
• Identify corresponding, alternate interior, alternate exterior, and co-interior angles in a diagram.
• Calculate unknown angles formed when a transversal intersects parallel lines.
• Use converse theorems to determine whether two lines are parallel.
• Apply the facts that lines parallel to the same line are parallel to each other and that a line perpendicular to one of two parallel lines is perpendicular to the other.