Angles, Parallel Lines and Triangles, SAT math lesson

Two straight roads cross, and a railway line cuts across two parallel streets. Every crossing like that creates angles, and a handful of facts tie them together. Almost every SAT angle question comes down to picking the right fact, saying why it applies and writing one equation. Angles in this lesson are measured in degrees.

Key terms

Vertical angles

Definition When two lines intersect, they form two pairs of vertical angles, the pairs of angles that are opposite each other at the intersection point and share only the vertex. Vertical angles are congruent, meaning they have equal measures. The reason is that each of two vertical angles forms a linear pair with the same angle next to them, so both equal $180^\circ$ minus that angle.

In plain words Where two lines cross, the angles across from each other are equal.

Example Two lines intersect, and two vertical angles measure $(3x + 10)^\circ$ and $(5x - 30)^\circ$ . Setting them equal gives

3x + 10 = 5x - 30, \quad \text{so} \quad x = 20.

Each of these angles measures $70^\circ$ . Before substituting, notice which angle the question actually asks for. If it asks for an angle next to one of them, the answer is $110^\circ$ , not $70^\circ$ .

Supplementary angles

Definition Two angles are supplementary when their measures add up to $180^\circ$ . Two adjacent angles whose outer sides form a straight line are called a linear pair, and the angles of a linear pair are always supplementary. Two angles are complementary when their measures add up to $90^\circ$ .

In plain words Angles that sit side by side along a straight line fill a half turn, so together they make $180^\circ$ .

Example Two angles form a linear pair and measure $(2x)^\circ$ and $(3x + 20)^\circ$ . Then

2x + (3x + 20) = 180,

so $5x = 160$ and $x = 32$ . The angles measure $64^\circ$ and $116^\circ$ , and indeed $64 + 116 = 180$ .

Corresponding angles

Definition A transversal is a line that crosses two other lines at two different points. At each of the two intersection points it forms four angles. Two angles are corresponding when they are in the same position at the two intersection points, for example both above the crossed line and to the right of the transversal. If the two lines are parallel, corresponding angles are congruent. Conversely, if a pair of corresponding angles is congruent, the two lines are parallel. If the lines are not parallel, corresponding angles are not equal, so this fact must never be used without knowing the lines are parallel.

In plain words A transversal crossing two parallel lines makes the same picture of angles at both crossings. Slide one crossing along the transversal and it lands exactly on the other.

Example Lines $\ell$ and $m$ are parallel, and a transversal $t$ crosses them. The angle above $\ell$ and to the right of $t$ measures $65^\circ$ . The corresponding angle above $m$ and to the right of $t$ also measures $65^\circ$ , and the angle next to it along $m$ measures $180^\circ - 65^\circ = 115^\circ$ .

Alternate angles

Definition Two angles at the two intersection points of a transversal are alternate interior angles when both lie between the two crossed lines and on opposite sides of the transversal. They are alternate exterior angles when both lie outside the two lines and on opposite sides of the transversal. If the two lines are parallel, alternate interior angles are congruent and alternate exterior angles are congruent. Also, the two interior angles on the same side of the transversal are supplementary. Each of these statements also works in reverse. If a pair of alternate angles is congruent, or two same side interior angles are supplementary, the lines are parallel.

In plain words With parallel lines, the angles that make a Z shape are equal, and the two inside angles on the same side of the transversal add up to $180^\circ$ .

Example Parallel lines are cut by a transversal, and two alternate interior angles measure $(4x - 12)^\circ$ and $(2x + 30)^\circ$ . Because the lines are parallel,

4x - 12 = 2x + 30, \quad \text{so} \quad x = 21.

Each angle measures $72^\circ$ . The interior angle on the same side of the transversal as either of them measures $180^\circ - 72^\circ = 108^\circ$ .

Triangle angle sum

Definition In every triangle in a plane, the measures of the three interior angles add up to $180^\circ$ ,

m\angle A + m\angle B + m\angle C = 180^\circ.

An exterior angle of a triangle is formed by one side and the extension of an adjacent side. Its measure equals the sum of the measures of the two interior angles that are not next to it, called the remote interior angles. To see why the sum is $180^\circ$ , draw through one vertex the line parallel to the opposite side. The two angles on either side of the triangle’s angle at that vertex form alternate interior angles with the triangle’s other two angles, so they equal those angles, and all three angles fit together along a straight line.

In plain words Tear off the three corners of a paper triangle and put them side by side. They make a straight line.

Example The angles of a triangle measure $x^\circ$ , $(2x)^\circ$ and $(3x)^\circ$ . Then $6x = 180$ , so $x = 30$ and the angles are $30^\circ$ , $60^\circ$ and $90^\circ$ . In another triangle two angles measure $48^\circ$ and $67^\circ$ . The exterior angle at the third vertex is

48^\circ + 67^\circ = 115^\circ,

and the third interior angle is $180^\circ - 115^\circ = 65^\circ$ .

Figures and scale

Definition A diagram in general shows how points, lines and angles are arranged, and it need not be drawn to scale. Facts such as two lines being parallel, an angle being a right angle or two lengths being equal can be used only if they are stated or marked. On the SAT, figures are drawn to scale unless the question says that a figure is not drawn to scale. Even in a figure drawn to scale, an exact answer has to come from the given numbers and from angle facts that apply, not from measuring the picture.

In plain words Trust what the question tells you, not what the picture seems to show. If the question says the figure is not to scale, the picture may be badly off.

Example A figure shows two lines that look parallel, crossed by a transversal, with a note that it is not drawn to scale. The angles marked $70^\circ$ and $(x + 15)^\circ$ are in corresponding positions. Since nothing says the lines are parallel, the value of $x$ cannot be found. If the question adds that the lines are parallel, then $x + 15 = 70$ and $x = 55$ .

Check yourself

Two lines intersect at a point. Two vertical angles formed by the lines measure $(4x + 12)^\circ$ and $(6x - 20)^\circ$ . What is the measure of an angle adjacent to either of them? (A) $16^\circ$ (B) $76^\circ$ (C) $104^\circ$ (D) $164^\circ$

Show the answer

The correct choice is (C). Vertical angles are equal, so $4x + 12 = 6x - 20$ and $x = 16$ . Each vertical angle measures $76^\circ$ , and an adjacent angle forms a linear pair with it, so it measures $180^\circ - 76^\circ = 104^\circ$ . Choice (A) is the value of $x$ , choice (B) is the vertical angle itself, not the angle asked for, and choice (D) subtracts $x$ rather than the angle from $180$ .

Parallel lines $\ell$ and $m$ are cut by a transversal. Two interior angles on the same side of the transversal measure $(3x + 15)^\circ$ and $(5x + 5)^\circ$ . What is the value of $x$ ?

Show the answer

The answer is $20$ . Interior angles on the same side of a transversal of parallel lines are supplementary, so

(3x + 15) + (5x + 5) = 180,

which gives $8x = 160$ and $x = 20$ . The angles are $75^\circ$ and $105^\circ$ . Setting the two angles equal would give $x = 5$ , which is wrong because these angles are not alternate or corresponding.

In triangle $ABC$ , the exterior angle at vertex $C$ measures $130^\circ$ , and $m\angle A = 57^\circ$ . What is $m\angle B$ ? (A) $50^\circ$ (B) $57^\circ$ (C) $73^\circ$ (D) $123^\circ$

Show the answer

The correct choice is (C). The exterior angle at $C$ equals the sum of the remote interior angles, so $57^\circ + m\angle B = 130^\circ$ and $m\angle B = 73^\circ$ . Choice (A) is the interior angle at $C$ , since $180^\circ - 130^\circ = 50^\circ$ . Choice (B) repeats $m\angle A$ , as if the triangle were isosceles. Choice (D) is $180^\circ - 57^\circ$ , which is not an angle of this triangle.

Angles, Parallel Lines and Triangles

Theory, part 93

Practice, part 94

Key terms

Vertical angles

Supplementary angles

Corresponding angles

Alternate angles

Triangle angle sum

Figures and scale

Check yourself

How this lesson connects

Builds on

Used later in