How do I find missing angles with parallel lines and a transversal?

Decide whether the two angles are equal or supplementary. If both look acute or both look obtuse they are equal (corresponding or alternate interior); if one is acute and one is obtuse they add to $180^\circ$ (co-interior). This quick check resolves most transversal questions.

What is the difference between similar and congruent triangles?

Congruent triangles are identical in both shape and size, so every corresponding side and angle matches exactly. Similar triangles have the same shape but can differ in size — their angles are equal and their corresponding sides are proportional, which lets you solve for missing lengths with a ratio.

What is the exterior angle theorem?

An exterior angle of a triangle — formed by extending one side — equals the sum of the two interior angles not adjacent to it. It lets you find an unknown angle by adding the two "far" angles directly, instead of finding the adjacent angle and subtracting from $180^\circ$.

Do the interior angles of every triangle add up to 180 degrees?

Yes. The three interior angles of any triangle in a plane always sum to exactly $180^\circ$, regardless of the triangle's shape. That lets you find the third angle whenever you know the other two.

CramClub • Angles, Lines, and Triangles

Interior angles sum to $180^\circ$

Vertical angles equal; a straight line = $180^\circ$

Corresponding & alternate interior angles are equal; co-interior sum to $180^\circ$

Exterior angle $=$ sum of the two remote interior angles

Equal angles; corresponding sides in a constant ratio

The Rules That Govern Angles and Triangles

Many SAT geometry problems never require a formula from the reference sheet — they hinge on a small set of angle relationships you must have memorized. Master these and you can chase an unknown angle across a diagram in seconds. Because SAT figures are often "not drawn to scale," you have to reason from the rules, not from how the picture looks.

Angles Formed by Intersecting Lines

When two lines cross, they create four angles with two dependable properties. Vertical angles — the pair directly opposite each other — are always equal. Adjacent angles along a straight line are supplementary: they sum to $180^\circ$ . Angles that together fill a right angle are complementary, summing to $90^\circ$ , and angles filling a full turn around a point sum to $360^\circ$ .

If two intersecting lines form one angle of $50^\circ$ , its vertical partner is also $50^\circ$ , and each of its two neighbors is $180^\circ - 50^\circ = 130^\circ$ . Notice how one given angle instantly determines all four.

Parallel Lines Cut by a Transversal

This is the single richest source of SAT angle problems. When a transversal crosses two parallel lines, eight angles appear — but they take only two distinct values. Angles in matching positions (corresponding angles) are equal. Alternate interior angles, on opposite sides of the transversal and between the parallels, are equal. Co-interior (same-side interior) angles are supplementary, summing to $180^\circ$ .

The practical shortcut: any two of the eight angles are either equal or add to $180^\circ$ , and you can tell which by a glance. If both angles are acute or both are obtuse, they are equal; if one is acute and one is obtuse, they are supplementary. That single test resolves most transversal questions without naming which formal category applies.

The Triangle Angle Sum and Exterior Angles

The interior angles of every triangle add to $180^\circ$ . Know two angles and the third is forced. An exterior angle — formed by extending one side — equals the sum of the two remote (non-adjacent) interior angles. This exterior-angle shortcut skips a step: instead of finding an interior angle and subtracting from $180^\circ$ , you add the two far angles directly.

Special triangles carry built-in angle facts. An isosceles triangle has two equal sides and, opposite them, two equal base angles. An equilateral triangle has three $60^\circ$ angles. So if an isosceles triangle has a vertex angle of $40^\circ$ , each base angle is $\tfrac{180^\circ - 40^\circ}{2} = 70^\circ$ .

Similar and Congruent Triangles

Two triangles are congruent if they have identical size and shape — every corresponding side and angle matches. They are similar if they have the same shape but possibly different size: corresponding angles are equal and corresponding sides are proportional.

Similarity is the workhorse. Once you establish that two triangles are similar — often because they share an angle and both contain a right angle, or because a line is drawn parallel to one side — you set up a proportion and solve for a missing length:

$\frac{\text{side in triangle 1}}{\text{matching side in triangle 2}} = \frac{\text{another side in triangle 1}}{\text{its match in triangle 2}}.$

A drawn line parallel to one side of a triangle creates a smaller, similar triangle (the "side-splitter" setup), and shadow problems create similar triangles from the sun's parallel rays. In every case, the key discipline is matching each side to its true corresponding side before you cross-multiply.

A Fully Worked Example

Lines $m$ and $n$ are parallel, and a transversal crosses them. The angle just above line $m$ measures $(3x + 10)^\circ$ , and the corresponding angle just above line $n$ measures $(5x - 30)^\circ$ . Find the measure of that angle.

Step 1 — Identify the relationship. The two angles sit in matching positions on parallel lines, so they are corresponding and therefore equal.

Step 2 — Set the expressions equal. $3x + 10 = 5x - 30$ .

Step 3 — Solve. $10 + 30 = 5x - 3x \Rightarrow 40 = 2x \Rightarrow x = 20$ .

Step 4 — Evaluate. $3(20) + 10 = 70^\circ$ , and $5(20) - 30 = 70^\circ$ agrees. The angle is $70^\circ$ .

Connecting to Right Triangles

When a triangle contains a right angle, its sides are tied together by the Pythagorean theorem and its angles unlock trigonometry — but the angle-chasing skills here come first, and they alone solve a large share of SAT geometry questions.

Two angles of a triangle measure $50^\circ$ and $60^\circ$ . What is the third angle?

Two lines intersect, forming one angle of $130^\circ$ . Find the measures of the other three angles.

Lines $m$ and $n$ are parallel. A transversal makes an angle of $(3x + 10)^\circ$ above line $m$ and a corresponding angle of $(5x - 30)^\circ$ above line $n$ . What is the measure of that angle?

An isosceles triangle has a vertex angle of $40^\circ$ . What is the measure of each base angle?

An exterior angle of a triangle measures $110^\circ$ . One of the two remote interior angles is $45^\circ$ . What is the other remote interior angle?

In triangle $ABC$ , point $D$ is on side $AB$ and point $E$ is on side $AC$ , with segment $DE$ parallel to $BC$ . Given $AD = 6$ , $DB = 4$ , and $BC = 15$ , find the length of $DE$ .

Applying the equal-angle rules when the lines are not actually parallel. Without parallelism, corresponding and alternate angles need not be equal.

Use the parallel-line relationships only when the problem states the lines are parallel (or marks them with matching arrows). Otherwise fall back on the triangle sum and vertical angles.

Confusing equal angles with supplementary ones at a transversal.

Do the acute/obtuse check: if the two angles are both acute or both obtuse they are equal; if one is acute and one is obtuse they add to $180^\circ$ .

Setting up a similarity proportion with sides in mismatched order, or using $AD/DB$ where the full side $AD/AB$ is needed.

Match each side to its true corresponding side, and when a parallel line splits a triangle, compare the small triangle side to the whole side of the big triangle, e.g. $DE/BC = AD/AB$ .

Using the exterior-angle rule as though the exterior angle equals just one remote interior angle.

An exterior angle equals the sum of both remote interior angles. If $110^\circ$ is exterior and one remote angle is $45^\circ$ , the other is $110 - 45 = 65^\circ$ , not $110^\circ$ .

In a triangle, one angle measures $42^\circ$ and a second measures $73^\circ$ . What is the third angle?

A $6$ -foot vertical pole casts an $8$ -foot shadow. At the same time, a nearby flagpole casts a $20$ -foot shadow. How tall is the flagpole?

Two parallel lines are cut by a transversal. One interior angle measures $65^\circ$ . What is the measure of the co-interior (same-side interior) angle?

An isosceles triangle has a vertex angle of $40^\circ$ . What is the measure of each base angle?

An exterior angle of a triangle measures $110^\circ$ , and the two remote interior angles are equal. What is the measure of each remote interior angle?

In triangle $ABC$ , $D$ is on $AB$ and $E$ is on $AC$ with $DE$ parallel to $BC$ . If $AD = 6$ , $DB = 4$ , and $BC = 15$ , what is the length of $DE$ ?