📚SAT (Digital) Unit 4 Review

This topic covers the geometry of triangles, angles, and lines as tested on the Digital SAT. You'll need to apply the triangle angle sum theorem, use properties of vertical angles and parallel lines cut by a transversal, and solve problems involving congruence and similarity of triangles. You should also know how a scale factor affects side lengths versus angle measures, and be ready to identify which statements are needed to complete a geometric proof or satisfy a theorem. Expect roughly 2–4 questions on these concepts across the math section, sometimes as straightforward angle calculations and sometimes as multi-step problems involving similar triangle proportions.

Triangle Angle Sum and Exterior Angles

The triangle angle sum theorem states that the three interior angles of any triangle add up to $180°$ . This is one of the most directly tested facts on the SAT.

Example 1 (Straightforward): In triangle $ABC$ , angle $A = 47°$ and angle $B = 68°$ . What is the measure of angle $C$ ?

$\angle C = 180° - 47° - 68° = 65°$

The exterior angle theorem is a useful extension: an exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Example 2 (Exterior Angle): In triangle $PQR$ , side $QR$ is extended to point $S$ . If $\angle P = 40°$ and $\angle Q = 75°$ , what is the measure of exterior angle $PRS$ ?

$\angle PRS = \angle P + \angle Q = 40° + 75° = 115°$

You could also find $\angle R = 180° - 40° - 75° = 65°$ , then note that $\angle PRS = 180° - 65° = 115°$ . Both approaches give the same answer, which is a good way to check your work.

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Vertical Angles and Parallel Lines with a Transversal

Vertical angles are the pairs of opposite angles formed when two lines intersect. They are always congruent. If two lines cross and one angle measures $130°$ , the angle directly across from it is also $130°$ , and each adjacent angle is $180° - 130° = 50°$ .

When a transversal crosses two parallel lines, eight angles are formed. The relationships you need to know:

Corresponding angles (same position at each intersection): congruent
Alternate interior angles (opposite sides of the transversal, between the parallel lines): congruent
Alternate exterior angles (opposite sides of the transversal, outside the parallel lines): congruent
Same-side interior angles (same side of the transversal, between the parallel lines): supplementary (sum to $180°$ )

A practical shortcut: when a transversal cuts parallel lines at a non-right angle, every acute angle in the figure is equal, and every obtuse angle is equal. Any acute angle plus any obtuse angle equals $180°$ .

Example 3: Lines $l$ and $m$ are parallel, and a transversal crosses both. One of the angles at line $l$ measures $125°$ . What is the measure of the alternate interior angle at line $m$ ?

The alternate interior angle is congruent to the $125°$ angle, so it also measures $125°$ .

Example 4 (Combined): Two parallel lines are cut by a transversal. At the first intersection, one angle is $3x + 10$ . At the second intersection, the same-side interior angle is $2x + 20$ . Find $x$ .

Same-side interior angles are supplementary:

$(3x + 10) + (2x + 20) = 180$

$5x + 30 = 180$

$5x = 150$

$x = 30$

Congruence and Similarity of Triangles

Congruence means two triangles have exactly the same size and shape: all corresponding sides are equal and all corresponding angles are equal. The standard criteria for proving triangles are congruent are:

SSS (three pairs of sides equal)
SAS (two sides and the included angle equal)
ASA (two angles and the included side equal)
AAS (two angles and a non-included side equal)

Note that SSA is not a valid congruence criterion. The SAT may test whether you can distinguish valid from invalid criteria.

Similarity means two triangles have the same shape but possibly different sizes. Corresponding angles are equal, and corresponding sides are proportional. The most commonly tested criterion is:

AA (two pairs of angles equal). Since the triangle angle sum theorem guarantees the third angle matches, two angles are enough.
SAS Similarity (two pairs of sides in proportion with the included angle equal)
SSS Similarity (all three pairs of sides in proportion)

Example 5: Triangles $ABC$ and $DEF$ are similar. $AB = 8$ , $DE = 12$ , and $BC = 6$ . What is $EF$ ?

Set up a proportion using corresponding sides:

$\frac{AB}{DE} = \frac{BC}{EF}$

$\frac{8}{12} = \frac{6}{EF}$

Cross-multiply:

$8 \cdot EF = 12 \cdot 6$

$8 \cdot EF = 72$

$EF = 9$

Scale Factor and Its Effects

When two figures are similar with a scale factor of $k$ , all lengths in the larger figure are $k$ times the corresponding lengths in the smaller figure. Critically, angle measures remain unchanged regardless of the scale factor. A triangle scaled up by a factor of 5 still has the exact same three angle measures.

This distinction matters because the SAT will sometimes ask what happens to angles when a figure is scaled, and the answer is always: nothing.

For other measurements:

All linear measurements (sides, perimeters, heights, medians) scale by $k$
Areas scale by $k^2$
Volumes (for 3D figures) scale by $k^3$

Example 6: Two similar triangles have a scale factor of $k = 3$ . The smaller triangle has sides of length 4, 5, and 7. What is the perimeter of the larger triangle?

Each side of the larger triangle is 3 times the corresponding side:

$\text{Sides of larger triangle: } 4 \times 3 = 12, \quad 5 \times 3 = 15, \quad 7 \times 3 = 21$

$\text{Perimeter} = 12 + 15 + 21 = 48$

Alternatively, the smaller triangle's perimeter is $4 + 5 + 7 = 16$ , and perimeter scales by $k$ :

$16 \times 3 = 48$

Example 7 (Tricky): Two similar triangles have corresponding sides of 5 and 20. If the area of the smaller triangle is 30, what is the area of the larger triangle?

The scale factor is $k = \frac{20}{5} = 4$ . Area scales by $k^2$ :

$\text{Area of larger triangle} = 30 \times 4^2 = 30 \times 16 = 480$

A common mistake is multiplying the area by $k$ instead of $k^2$ , which would give the wrong answer of 120.

Identifying Statements Needed for Proofs

Some SAT questions won't ask you to solve for a number. Instead, they'll describe a geometric situation and ask which additional statement is needed to prove that two triangles are congruent or similar, or to satisfy a given theorem. These are essentially reasoning questions about proofs.

How these questions work: You're given some information about a figure and a conclusion (like "triangle $ABC$ is congruent to triangle $DEF$ "). You need to pick the answer choice that provides the missing piece.

Example 8: In the figure, $\overline{BD}$ bisects $\angle ABC$ , and $\overline{BD}$ is perpendicular to $\overline{AC}$ . Which of the following, along with the given information, is sufficient to prove that triangle $ABD$ is congruent to triangle $CBD$ ?

Think about what you already know:

$\angle ABD = \angle CBD$ (because $BD$ bisects $\angle ABC$ )
$\angle BDA = \angle BDC = 90°$ (because $BD \perp AC$ )
$BD = BD$ (shared side)

You already have ASA, so you might not need anything else. But if the question removes one of these givens, the correct answer would supply the missing criterion. The strategy is: list what you know, figure out which congruence criterion you're closest to completing (SSS, SAS, ASA, AAS), and pick the answer that fills the gap.

For similarity proofs, remember that establishing two pairs of equal angles (AA) is usually the fastest path. If the problem involves parallel lines cut by a transversal, the angle relationships from that setup often provide the two equal angle pairs you need.

What to Watch For on Test Day

Don't confuse congruence criteria. SSA is not valid for congruence. AA is sufficient for similarity but not for congruence. Know which criteria apply to which situation.
Scale factor applies to lengths, not angles. When a triangle is scaled by factor $k$ , sides multiply by $k$ and areas multiply by $k^2$ , but angles stay exactly the same.
Set up proportions carefully for similar triangles. Make sure you're matching corresponding sides. If triangle $ABC \sim$ triangle $DEF$ , then $AB$ corresponds to $DE$ , $BC$ to $EF$ , and $AC$ to $DF$ . Mixing up the correspondence is a common source of wrong answers.
Use parallel lines to establish similarity. When a transversal cuts parallel lines, the equal angles created (alternate interior, corresponding) often give you the two angle pairs needed for AA similarity. Look for this pattern in figures with parallel lines and triangles.
For proof-style questions, inventory what you know first. Write down every angle or side relationship the problem gives you, including vertical angles and shared sides that aren't explicitly stated. Then see which congruence or similarity criterion is one step away from being complete.

📚SAT (Digital) Unit 4 Review