Math · Geometry and Trigonometry

SAT Lines, Angles, and Triangles: The Complete Guide

Q: Are the triangle angle sum and the exterior angle theorem on the SAT reference sheet?

The reference sheet states that the angles of a triangle sum to 180 degrees. The exterior angle theorem, the parallel-line angle rules, and the similarity tests are not on the sheet, so you need to know them.

Q: Can I assume lines in a figure are parallel?

Only if the question says so or the figure's marks show it, such as matching arrowheads. Digital SAT figures are usually not drawn to scale, so judge from the given information.

Q: Does SSA prove that two triangles are congruent?

No. Two sides and an angle that is not between them (SSA) do not guarantee congruence, because they can describe two different triangles. Use SSS, SAS, ASA, AAS, or HL for right triangles.

Q: How do I find the possible length of the third side of a triangle?

Use the triangle inequality. The third side must be strictly greater than the difference of the other two sides and strictly less than their sum.

Q: How do I know two triangles are similar?

Two pairs of equal angles (AA) is the most common test. You can also use sides in proportion (SSS) or two proportional sides with an equal included angle (SAS).

Lines, angles, and triangles questions test the rules that connect angles and sides in straight-edged figures: parallel lines, the triangle angle sum, isosceles and equilateral triangles, and similar and congruent triangles. The rules are short. Most points come from knowing them well and chaining them together on one figure. This guide lists the rules the SAT uses, shows how to recognize each one, and works three examples from easy to hard.

Section: Math, Geometry and Trigonometry
Calculator: Usually not needed; Desmos is available if you want it
Figures: Often not drawn to scale, so use only the given information
Answer types: Angles in degrees, lengths, and ratios, in multiple choice or student-produced response

What this skill tests

This skill covers the relationships among the angles and sides of lines, polygons, and triangles. A question gives you some measures or lengths and asks for an unknown that follows from a rule. You will recognize it by its stems, which are often phrased like these:

Two lines are parallel and are cut by a transversal. Find the measure of one of the angles.
In triangle $ABC$ , one angle is given. Find another angle or an exterior angle.
Triangles $ABC$ and $DEF$ are similar. Find a side length of one of them.
Which of the following could be the length of the third side of a triangle whose other sides are $7$ and $11$ ?

Figures on the Digital SAT are usually not drawn to scale, and they show only what the problem gives you. Do not assume that two lines are parallel, that two sides are equal, or that an angle is a right angle unless the problem says so or the marks on the figure show it.

The core idea at a glance

Most questions in this skill need a chain of two or three rules. The typical sequence is to find a straight-line or parallel-line angle, use the triangle angle sum to find a third angle, and then use an exterior angle or similarity to reach the answer. The key habit is to write each new angle on the figure as soon as you find it.

For lengths, the strongest tool is similarity. If two triangles have matching angles, their corresponding sides are in the same ratio. A segment drawn parallel to one side of a triangle creates a pair of similar triangles automatically, which is why parallel-line questions so often lead to proportions.

Core rules: angles, triangles, and similarity

Learn the angle relationships first. Each one is short, and most lines-and-angles questions use two or three of them together. Remember that the parallel-line rules apply only when the two lines are parallel.

Relationship	Rule	Memory cue
Angles on a straight line	Sum to $180^\circ$	A straight line
Angles around a point	Sum to $360^\circ$	A full turn
Vertical angles	Equal, whether or not the lines are parallel	Opposite angles at an X
Complementary angles	Sum to $90^\circ$	A square corner
Supplementary angles	Sum to $180^\circ$	A straight line
Corresponding angles	Equal (parallel lines)	F shape
Alternate interior angles	Equal (parallel lines)	Z shape
Same-side interior angles	Sum to $180^\circ$ (parallel lines)	C shape

The triangle rules come next. The most used one is the angle sum:

a^\circ+b^\circ+c^\circ=180^\circ

The three interior angles of every triangle sum to

180^\circ

\text{exterior angle at } C=\angle A+\angle B

|a-b|<c<a+b

Possible length

c

of the third side, given sides

a

and

b

Triangles are congruent when they match in size and shape, and similar when they match in shape only. The congruence tests and their limits are summarized below.

Test	What it proves	Note
SSS	Congruence	All three sides are equal
SAS	Congruence	Two sides and the angle between them
ASA	Congruence	Two angles and the side between them
AAS	Congruence	Two angles and a side that is not between them
HL	Congruence (right triangles only)	Hypotenuse and one leg
SSA	Not a valid test	Can describe two different triangles
AAA	Similarity only	Same angles, possibly different sizes

Similarity uses AA (two pairs of equal angles), SSS with proportional sides, or SAS with a proportional pair of sides and an equal included angle. Similar triangles have corresponding sides in a constant ratio $k$ , perimeters in ratio $k$ , and areas in ratio $k^2$ .

\frac{AD}{AB}=\frac{DE}{BC}

DE\parallel BC

, with

D

\overline{AB}

and

E

\overline{AC}

, then triangles

ADE

and

ABC

are similar.

For polygons with $n$ sides, the interior angles sum to $(n-2)\cdot180^\circ$ . A regular polygon has equal interior angles, so each one measures $\frac{(n-2)\cdot180^\circ}{n}$ .

Step-by-step method

Mark every given value on the figure, and write the target of the question in the margin.
Look for parallel lines and a transversal. Name each angle pair using the table, but only if the lines are actually parallel.
Find a triangle and apply the angle sum. Write the third angle on the figure.
For a length question, search for similar triangles: a parallel segment, a shared angle, or a pair of right angles.
Set up the proportion with corresponding vertices in the same order. Match $AD$ with $AB$ , not with $BC$ .
Solve, check that the size of the result fits the figure, and confirm that the question asks for what you found. It may ask for a supplement or an exterior angle rather than the interior angle.

Worked examples

The three examples below move from a one-step angle sum to a multi-step isosceles problem and finally to a similar-triangle proportion. Try each one before reading the steps.

Worked example

Example 1: The third angle of a triangle (easy)

In triangle $RST$ , $\angle R=47^\circ$ and $\angle S=68^\circ$ . What is the measure of $\angle T$ , in degrees?

A $65^\circ$
B $113^\circ$
C $115^\circ$
D $133^\circ$

Show the solution

Use the triangle angle sum: $\angle R+\angle S+\angle T=180^\circ$ .
Substitute: $47+68+\angle T=180$ .
Since $47+68=115$ , we get $\angle T=180-115=65^\circ$ . The answer is A.
Why not C ( $115^\circ$ )? That is the sum of the two given angles, not the third angle.
Why not D ( $133^\circ$ )? That subtracts only one given angle from $180^\circ$ , since $180-47=133$ .
Why not B ( $113^\circ$ )? It is an arithmetic slip, since $180-67=113$ . It would appear if $68$ were misread as $67$ .

Answer: A

Worked example

Example 2: An isosceles triangle with an exterior angle (medium)

In triangle $PQR$ , $PQ=PR$ . The exterior angle at $Q$ , formed by extending side $RQ$ past $Q$ , measures $110^\circ$ . What is the measure of $\angle P$ , in degrees?

A $40^\circ$
B $70^\circ$
C $110^\circ$
D $140^\circ$

Show the solution

The interior angle at $Q$ and the exterior angle at $Q$ form a straight line, so $\angle Q=180-110=70^\circ$ .
Since $PQ=PR$ , the sides opposite the base angles are equal. Side $PQ$ faces $\angle R$ , and side $PR$ faces $\angle Q$ . Therefore $\angle R=\angle Q=70^\circ$ .
The angle sum gives $\angle P=180-70-70=40^\circ$ . The answer is A.
Why not B ( $70^\circ$ )? That is a base angle. The question asks for the angle at $P$ , which is the vertex angle.
Why not C ( $110^\circ$ )? That is the exterior angle at $Q$ , which the problem gives you. It is not the angle at $P$ .
Why not D ( $140^\circ$ )? That is the sum of the two base angles, $70+70$ , which equals the exterior angle at $P$ . It is not the interior angle at $P$ .

Answer: A

Worked example

Example 3: A parallel segment and a proportion (hard, student-produced response)

Note: Figure not drawn to scale.

In triangle $ABC$ , point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ , with $\overline{DE}\parallel\overline{BC}$ . If $AD=4$ , $DB=6$ , and $DE=5$ , what is the length of $\overline{BC}$ ?

Show the solution

Because $DE\parallel BC$ , the corresponding angles at $D$ and $B$ are equal, and the corresponding angles at $E$ and $C$ are equal. Triangle $ADE$ is similar to triangle $ABC$ . The two triangles share the angle at $A$ .
The full side is $AB=AD+DB=4+6=10$ .
Match corresponding sides in the same order. $AD$ corresponds to $AB$ , and $DE$ corresponds to $BC$ . So $\frac{AD}{AB}=\frac{DE}{BC}$ , which gives $\frac{4}{10}=\frac{5}{BC}$ .
Cross-multiply: $4\cdot BC=50$ , so $BC=12.5$ , which is the same as $\frac{25}{2}$ .
Why not $7.5$ ? That comes from $DE\cdot DB/AD=5\cdot6/4$ , which pairs the segments $DE$ and $DB$ with the wrong partners. A proportion must match each side with the side that corresponds to it.

Answer: 12.5

Common traps

The hardest questions

The hardest questions in this skill use the rules in an unusual order, count possibilities, or nest one similar pair inside another. Four patterns cover most of them.

Pattern 1: counting possible integer lengths. With sides $7$ and $11$ , the third side $c$ satisfies $11-7<c<11+7$ , so $4<c<18$ . The integer values of $c$ run from $5$ to $17$ , which gives $13$ values. Remember that the endpoints are excluded.

Pattern 2: nested similar triangles. A parallel segment that cuts off a small triangle at the top with scale factor $k$ makes that small triangle similar to the whole triangle. Its area is $k^2$ times the whole triangle's area. For $k=\frac23$ , the area is $\left(\frac23\right)^2=\frac49$ of the whole.

Pattern 3: polygon angles. A regular hexagon has interior angles of $\frac{(6-2)\cdot180^\circ}{6}=120^\circ$ . Its exterior angles are $60^\circ$ each, and six of them sum to $360^\circ$ , which is a useful check.

Pattern 4: an unmarked isosceles triangle. If the angle between two equal sides is $40^\circ$ , the other two angles are equal, and each is $\frac{180-40}{2}=70^\circ$ . Always look for equal sides that the figure may not label.

Timing and practice plan

Aim for about a minute on a one-step angle question and two to three minutes on a similarity question with several parallel lines. Speed comes from recognizing the pattern quickly, so drill the recognition as well as the arithmetic.

Draw the angle table from memory. Practice until you can name each pair of angles formed by a transversal in under five seconds.
Solve ten angle-sum problems, writing each new angle on the figure before moving on.
Solve five similarity problems in which you list the corresponding vertices before writing any proportion.
Practice the triangle inequality with five pairs of sides, and state the range of the third side before looking at the choices.
Review every error and label it as a missed rule, a mismatched pair, or an arithmetic slip.

In Solar, repeat these drills with this skill's practice questions. Move on when you can solve a set of mixed angle and similarity questions without writing a wrong proportion.

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Core rules: angles, triangles, and similarity
Step-by-step method
Worked examples
Common traps
The hardest questions

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Lines, angles, and triangles: FAQ

Are the triangle angle sum and the exterior angle theorem on the SAT reference sheet?

The reference sheet states that the angles of a triangle sum to 180 degrees. The exterior angle theorem, the parallel-line angle rules, and the similarity tests are not on the sheet, so you need to know them.

Can I assume lines in a figure are parallel?

Only if the question says so or the figure's marks show it, such as matching arrowheads. Digital SAT figures are usually not drawn to scale, so judge from the given information.

Does SSA prove that two triangles are congruent?

No. Two sides and an angle that is not between them (SSA) do not guarantee congruence, because they can describe two different triangles. Use SSS, SAS, ASA, AAS, or HL for right triangles.

How do I find the possible length of the third side of a triangle?

Use the triangle inequality. The third side must be strictly greater than the difference of the other two sides and strictly less than their sum.

How do I know two triangles are similar?

Two pairs of equal angles (AA) is the most common test. You can also use sides in proportion (SSS) or two proportional sides with an equal included angle (SAS).