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Geometry Unit 3: Congruent triangles & proof

Congruent triangles have all matching sides and angles equal. You don't need all six facts: a few shortcuts prove congruence.

Covers: SSS, SAS, ASA, AAS, HL, CPCTC, isosceles triangle theorem, two-column proofs

Study guide

Congruent triangles have exactly the same size and shape: all corresponding sides and angles are equal. You don't need to check all six parts. Five shortcuts prove congruence: SSS (three pairs of sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse and a leg of right triangles). SSA and AAA don't work: SSA can produce two different triangles, and AAA only proves similarity.

Once triangles are proven congruent, CPCTC (corresponding parts of congruent triangles are congruent) lets you conclude that any other matching sides or angles are equal. Many proofs first prove triangles congruent, then use CPCTC.

The isosceles triangle theorem says the base angles opposite the two equal sides are equal, and its converse says that if two angles are equal, the sides opposite them are equal. An equilateral triangle is also equiangular, with 60° angles.

A two-column proof lists statements on the left and reasons on the right. Reasons are givens, definitions, postulates and theorems, such as the reflexive property (a shared side equals itself), vertical angles, or angles formed by parallel lines. Mark the diagram as you go, and work toward the congruence shortcut you need.

The sum of any two sides of a triangle must be greater than the third side, so a missing side must be greater than the difference of the other two and less than their sum. Also, the longest side is opposite the largest angle. The three medians of a triangle meet at the centroid, which divides each median in a 2:1 ratio, with the longer part toward the vertex. Perpendicular bisectors meet at the circumcenter (equidistant from the vertices), angle bisectors meet at the incenter (equidistant from the sides), and altitudes meet at the orthocenter.

Key ideas

• Shortcuts: SSS, SAS, ASA, AAS, and HL for right triangles.
• SSA and AAA do not prove congruence.
• CPCTC: once triangles are congruent, all corresponding parts are congruent.
• Isosceles triangles have equal base angles.
• In a proof, every statement needs a reason: given, definition, postulate or theorem.

Formulas and rules

Common mistakes

• Using SSA as a congruence shortcut.
• Mismatching vertex order in a congruence statement.

Words to know

Five triangle congruence shortcuts

SSS, SAS, ASA, AAS, HL

SSA (not a congruence rule)

Two sides and a non-included angle can make two different triangles.

CPCTC

Corresponding parts of congruent triangles are congruent.

Isosceles triangle theorem

Base angles opposite equal sides are equal.

AAA

Proves similarity, not congruence.

Congruent figures

Same size and shape.

SSS

Three pairs of equal sides prove congruence.

SAS

Two sides and the included angle prove congruence.

ASA / AAS

Two angles and a side prove congruence.

HL

Hypotenuse and leg prove right triangles congruent.

Two-column proof

Statements in one column, reasons in the other.

Reflexive property

A segment or angle is congruent to itself (a shared side).

Sample practice questions

25 more questions like these, from level 1 to 5, are in the practice set.