SAT — mathMedium3 min read

Congruent Triangles

Apply SSS, SAS, ASA, AAS congruence criteria for the Digital SAT. Use congruent triangles to find unknown sides and angles.

Tutor AI Team2026-02-25T01:35:34.096Z

Congruent triangles are identical in shape and size — all corresponding sides and angles are equal. The Digital SAT tests whether you can identify congruence using the standard criteria and use it to find unknown measurements.

Core Concepts

Congruence Criteria

Criterion	What Must Match
SSS	All three sides
SAS	Two sides and the included angle
ASA	Two angles and the included side
AAS	Two angles and a non-included side
HL	Hypotenuse and leg (right triangles only)

Note: SSA (two sides and a non-included angle) does NOT guarantee congruence (the ambiguous case).

Using Congruence

If $\triangle ABC \cong \triangle DEF$ , then:

$AB = DE$ , $BC = EF$ , $AC = DF$
$\angle A = \angle D$ , $\angle B = \angle E$ , $\angle C = \angle F$

CPCTC

"Corresponding Parts of Congruent Triangles are Congruent." Once you prove triangles congruent, all corresponding parts are equal.

Strategy Tips

Tip 1: Mark What You Know

Label equal sides with tick marks and equal angles with arcs.

Tip 2: Look for Shared Sides/Angles

Triangles that share a side (or angle) automatically have that part equal.

Tip 3: The Order Matters

$\triangle ABC \cong \triangle DEF$ means $A↔D$ , $B↔E$ , $C↔F$ .

Worked Example: Example 1

Problem

Two triangles share a side. Each has a right angle, and the hypotenuses are equal. Are they congruent?

Yes — by HL (hypotenuse-leg), since they share a leg and have equal hypotenuses.

Solution

Worked Example: SAT-Style

Problem

$\triangle ABC \cong \triangle XYZ$ . $AB = 7$ , $BC = 9$ , $\angle B = 50°$ . Find $XY$ and $\angle Y$ .

$XY = AB = 7$ . $\angle Y = \angle B = 50°$ .

Solution

Worked Example: Example 3

Problem

In quadrilateral $ABCD$ , diagonal $AC$ divides it into two triangles. If $AB = CD$ and $BC = DA$ , are the triangles congruent?

$\triangle ABC$ and $\triangle CDA$ share side $AC$ . $AB = CD$ , $BC = DA$ , $AC = AC$ . By SSS → congruent.

Solution

Practice Problems

Problem 1

What criterion proves congruence: two sides of 5 and 8 with an included angle of 60°?

Problem 2

Triangles with angles 40°, 60°, 80° and sides 5, 7, 9 vs. angles 40°, 60°, 80° and sides 5, 7, 9. Congruent?

Problem 3

$\triangle PQR \cong \triangle STU$ . $PQ = 12$ , $QR = 15$ , $PR = 9$ . Find $ST + TU + SU$ .

Want to check your answers and get step-by-step solutions?

Common Mistakes

Using SSA as a congruence criterion. SSA doesn't work (ambiguous case).
Mismatching corresponding vertices. Order matters in the congruence statement.
Forgetting shared sides/angles. A diagonal or shared side counts as an equal part.

Key Takeaways

✓
SSS, SAS, ASA, AAS, HL prove congruence.
✓
SSA does NOT prove congruence.
✓
Once congruent, all corresponding parts are equal (CPCTC).
✓
Look for shared sides and vertical angles as matching parts.

Tags:

#sat#math#geometry#triangles#congruence

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Core Concepts

Congruence Criteria

Using Congruence

CPCTC

Strategy Tips

Tip 1: Mark What You Know

Tip 2: Look for Shared Sides/Angles

Tip 3: The Order Matters

Worked Example: Example 1

Worked Example: SAT-Style

Worked Example: Example 3

Practice Problems

Problem 1

Problem 2

Problem 3

Common Mistakes

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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