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\(\cong \sim\)

Congruence &
Similarity

Master the reasoning behind congruent and similar figures: how to identify them, prove relationships, and solve for unknown measurements.

ISEE Upper Level

What Is Congruence?

Two figures are congruent when they have exactly the same shape and same size. Every pair of corresponding sides has equal length, and every pair of corresponding angles has equal measure.

A B C
\(5\)
\(7\)
\(6\)
\(\cong\)
 D E F
\(5\)
\(7\)
\(6\)

What Is Similarity?

Two figures are similar when they have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are in a constant ratio called the scale factor.

P Q R
\(8\)
\(10\)
\(6\)
\(\sim\)
 X Y Z
\(4\)
\(5\)
\(3\)
scale factor = \(\frac{1}{2}\)

Why Congruence & Similarity Matter

ISEE Tested
Quantitative Reasoning and Mathematics Achievement sections both test these ideas.
Foundation Skill
Proofs, coordinate geometry, and trigonometry all build on congruence and similarity.
Real-World Use
Architecture, engineering, and map reading rely on scale factors and proportional reasoning.

Triangle Congruence Criteria

You do not need to check all six measurements. Any of these shortcut criteria is enough to prove two triangles congruent:

SSS
All three pairs of corresponding sides are equal.
SAS
Two pairs of sides and the included angle are equal.
ASA
Two pairs of angles and the included side are equal.
AAS
Two pairs of angles and a non-included side are equal.

Triangle Similarity Criteria

Proving similarity requires less information than congruence because you only need to show that corresponding parts are proportional, not equal:

AA
Two pairs of corresponding angles are equal. (The third pair is automatic.)
SSS ~
All three pairs of corresponding sides are in the same ratio.
SAS ~
Two pairs of sides are proportional and the included angle is equal.

Congruent vs. Similar

Congruent ( \( \cong \) ) Similar ( \( \sim \) )
Shape Same Same
Size Same Can differ
Angles All equal All equal
Sides Equal Proportional (ratio \( k \))
Scale Factor \( k = 1 \) \( k \neq 1 \) (usually)
Key Insight Congruence is a special case of similarity Similarity is the broader concept

Step-by-Step Strategy

1
Identify the figures. Mark all given side lengths and angle measures on the diagram.
2
Match corresponding parts. List vertex-to-vertex correspondences carefully: order in the notation tells you the matching.
3
Decide: congruent or similar? If all sides are equal, they are congruent. If sides are proportional, they are similar.
4
Set up the equation. For congruence: set corresponding parts equal. For similarity: write a proportion \( \frac{a}{d} = \frac{b}{e} = \frac{c}{f} \).
5
Solve and verify. Cross-multiply to find unknowns, then check that all ratios match.

Example 1: Congruent Triangles

\(52^\circ\)
\(68^\circ\)
 A B C
\(9\)
\(7\)
\(8\)
\(\cong\)
\(52^\circ\)
\(68^\circ\)
 D E F
\(9\)
\(7\)
\(8\)

Example 2: Finding an Unknown Side

L M N
\(12\)
\(15\)
\(9\)
\(\sim\)
 X Y Z
\(8\)
\(?\)
\(6\)

Example 3: AA Similarity in Action

A B C D E
\(DE \parallel AB\)
\(6\)
\(4\)
\(15\)

Properties of Similar Figures

When two figures are similar with scale factor \( k \), several important relationships follow:

1
Corresponding sides are in ratio \( k \): if one side is \( a \), the matching side is \( ka \).
2
Perimeters are also in ratio \( k \): \( \frac{P_2}{P_1} = k \).
3
Areas are in ratio \( k^2 \): if the scale factor is \( 3 \), the area ratio is \( 9 \).
4
Corresponding angles remain equal: they do not scale.

Scale Factor, Perimeter, and Area

\(3\)
\(P = 12\)
\(A = 9\)
\(k = 2\)
\(6\)
\(P = 24\) (x2)
\(A = 36\) (x4)
 Ratios
Sides: \(k\)
Perimeter: \(k\)
Area: \(k^2\)

CPCTC: The Power Move

Corresponding Parts of Congruent Triangles are Congruent. Once you prove two triangles congruent, you can conclude that any pair of corresponding parts (sides or angles) are equal.

1
Prove congruence using SSS, SAS, ASA, or AAS.
2
State CPCTC to conclude that additional pairs of sides or angles are equal.
3
Use the result to find unknown measurements or complete a larger argument.

Common Mistakes

✕ Wrong Correspondence
Matching vertex A to the wrong vertex in the other triangle. Always follow the order in the notation.
✕ SSA "Proof"
Using two sides and a non-included angle to claim congruence. SSA does not guarantee a unique triangle.
✕ Area = Side Ratio
Assuming areas scale the same as sides. If sides scale by \( k \), areas scale by \( k^2 \), not \( k \).

Summary at a Glance

Criterion Congruence Similarity
SSS ✓ ✓ (proportional)
SAS ✓ ✓ (proportional + included angle)
ASA ✓ Implied by AA
AAS ✓ Implied by AA
AA Not sufficient (need size) ✓
SSA ✕ Not valid ✕ Not valid

Question 1 of 5

If \( \triangle JKL \cong \triangle MNO \) and \( JK = 11 \), \( KL = 14 \), \( JL = 9 \), what is the length of NO?

Question 2 of 5

Two similar triangles have corresponding sides in the ratio \( 3 : 7 \). What is the ratio of their areas?

Question 3 of 5

In \( \triangle PQR \), a line segment \( DE \) is drawn parallel to \( QR \), with \( D \) on \( PQ \) and \( E \) on \( PR \). If \( PD = 5 \), \( DQ = 10 \), and \( QR = 24 \), what is the length of DE?

Question 4 of 5

Which of the following is NOT a valid criterion for proving two triangles congruent?

Question 5 of 5

\( \triangle ABC \sim \triangle DEF \). The perimeter of \( \triangle ABC \) is \( 30 \) and the perimeter of \( \triangle DEF \) is \( 45 \). If \( AB = 8 \), what is the length of DE?