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Digital SAT · Lines, Angles & Triangles

How to Solve Similarity and Congruence on the Digital SAT

Similar figures are the same shape: equal corresponding angles and proportional corresponding sides. Congruent figures go further, matching in both angles and lengths. Most SAT questions use similarity to find a missing side through a proportion of corresponding sides. Match the parts in order first, then write the ratio and solve. One more fact earns points: the ratio of areas is the square of the ratio of sides.

Written from Perfect1600’s analysis of every similarity and congruence question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
per test
typical difficulty
Mostly easy
typical difficulty
practice questions
22
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

similarity
Equal angles, proportional sides

Match corresponding parts in order, then set up a proportion.

proportion
No calculator

A correspondence-and-proportion task, not a graphing one.

side ratio²
Area ratio

Areas scale by the square of the side ratio, a common oversight.

How to recognize similarity and congruence questions

  • Two triangles or figures are said to be similar or congruent.
  • Corresponding sides or angles are given, with one unknown.
  • The words similar, congruent, corresponding, or proportional appear.
  • It may give a side ratio and ask for an area ratio, or the reverse.

Why students miss these

The rule is short, but pairing corresponding parts is where people slip. Naming the triangles in an order that does not match the correspondence yields a proportion with mismatched sides. The area relationship trips others: they scale an area by the side ratio rather than its square. Writing the vertices in corresponding order before any ratio heads off both.

The step-by-step method

  1. 1

    Confirm the correspondence

    List the vertices of each figure in the order that matches, so corresponding parts line up.

  2. 2

    Use equal angles or proportional sides

    Similar figures share equal angles; corresponding sides are in a constant ratio.

  3. 3

    Set up the proportion

    Write a ratio of corresponding sides equal to another, then solve for the unknown.

  4. 4

    Square the ratio for areas

    The ratio of areas of similar figures is the square of the ratio of sides.

Worked examples

Easy example
Triangles ABCABC and DEFDEF are similar, with ABAB corresponding to DEDE. If AB=4AB = 4 and DE=12DE = 12, what is the scale factor from triangle ABCABC to triangle DEFDEF?
A
13\frac{1}{3}
B
32\frac{3}{2}
C
2
3

A: Incorrect. This is the factor from DEFDEF to ABCABC, the reciprocal.

B: Incorrect. This does not result from dividing the corresponding lengths.

C: Incorrect. This does not result from 124\frac{12}{4}.

D: Correct. The scale factor is DEAB=124=3\frac{DE}{AB} = \frac{12}{4} = 3.

Explanation

Scale factor =124=3= \frac{12}{4} = 3.

Medium example
Triangles ABCABC and DEFDEF are similar, and the ratio of ABAB to DEDE is 3 to 5. If AB=9AB = 9, what is the length of DEDE?
A
12
15
C
18
D
21

A: Incorrect. This does not result from the proportion 35=9DE\frac{3}{5} = \frac{9}{DE}.

B: Correct. From 35=9DE\frac{3}{5} = \frac{9}{DE}, DE=15DE = 15.

C: Incorrect. This doubles ABAB instead of using the given ratio.

D: Incorrect. This applies an incorrect ratio to ABAB.

Explanation

From 35=9DE\frac{3}{5} = \frac{9}{DE}, DE=15DE = 15.

Hard example
Triangles ABCABC and DEFDEF are similar. Given AB=10AB = 10, BC=14BC = 14, and DE=15DE = 15, what is the length of EFEF?
A
18
21
C
24
D
28

A: Incorrect. This does not result from the proportion.

B: Correct. From 1015=14EF\frac{10}{15} = \frac{14}{EF}, EF=21EF = 21.

C: Incorrect. This applies an incorrect ratio to BCBC.

D: Incorrect. This doubles BCBC instead of using the scale factor.

Explanation

From 1015=14EF\frac{10}{15} = \frac{14}{EF}, EF=21EF = 21.

The common traps

PatternWhat it doesThe tell
Mismatched correspondencePaired non-corresponding sides in the proportion.List the vertices in corresponding order before writing the ratio.
Area scaled by the side ratioMultiplied an area by the side ratio instead of its square.The area ratio is the square of the side ratio.
Similar treated as congruentAssumed equal side lengths when only the shape matches.Similar means proportional sides; congruent means equal sides.

Try it: two real questions

Question 1easy

If two angles of one triangle are congruent to two angles of another triangle, which criterion establishes that the triangles are similar?

Question 2easy

Triangle ABCABC is similar to triangle DEFDEF. Given that AB=6AB = 6, DE=9DE = 9, and BC=10BC = 10, what is the length of EFEF?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Right-Triangle Trigonometry
Right Triangles and the Pythagorean Theorem
Lines, Angles & Triangles
Angle Relationships
Area & Volume
Scale Factor for Area and Volume

Related reading

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How SAT wrong answers are built
From the blog
Which SAT skills move your score most

Common questions

What makes two triangles similar on the SAT?

Equal corresponding angles, which forces the corresponding sides into a constant ratio. Similar triangles have the same shape but not necessarily the same size.

How do I find a missing side with similar triangles?

Match the vertices in corresponding order, write a proportion of corresponding sides, and solve. The key is pairing sides that truly correspond.

How does area relate to the side ratio in similar figures?

The ratio of areas is the square of the ratio of sides. Sides in a 2 to 1 ratio give areas in a 4 to 1 ratio.

What is the difference between similar and congruent?

Similar figures share the shape and proportional sides; congruent figures are identical, with equal angles and equal side lengths. Congruent is the case where the ratio is 1.

Practice similarity and congruence the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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