Geometry

Triangle Congruence

Big Picture

One important relationship between triangles that we can prove is congruence, meaning that two triangles are exactly the same shape and size. congruent triangles have certain properties that are the same, namely their angle measurements and side lengths. Geometric theorems and postulates dictate the different characteristics of congruent triangles.

Key Terms

Congruent Triangles: Two triangles are congruent if the three corresponding angles and sides are congruent.

Corresponding Parts: The  parts  of  two  figures  that  have  congruent  measurements  (e.g.  corresponding  angles, corresponding sides, etc.).

Included Angle: When an angle is between two given sides of a triangle (or polygon).

Included Side: When a side is between two given angles of a triangle (or polygon).

Properties of a Triangle

△△ABC and DEF are congruent because:

\(\overline{AB} \cong \overline{DE}\)                \(\angle A \cong D\)

\(\overline{BC} \cong \overline{EF}\)     and     \(\angle B \cong E\)

\(\overline{AC} \cong \overline{DF}\)               \(\angle C \cong F\)

Properties of a Triangle

For congruent triangles, the order the vertices is listed in becomes important. Corresponding parts must be written in the same order in congruence statements.

Once we know two triangles are congruent, we also know that Corresponding  Parts of Congruent  Triangles are Congruent, often abbreviated CPCTC for short.

  • So \(\triangle ABC \cong \triangle DEF\) is correct, but \(\triangle ABC \cong \triangle EFD\) is wrong.

To identify corresponding parts, redraw the figures so that corresponding parts are easier to identify.

Properties of Congruent Triangles

The properties of congruence will apply to congruent triangles:

  • Reflexive Property: \(\triangle ABC \cong \triangle ABC\)
  • Symmetric Property: if \(\triangle ABC \cong \triangle DEF\), then \(\triangle DEF \cong \triangle ABC\)
  • Transitive Property: if \(\triangle ABC \cong \triangle DEF\), then \(\triangle DEF \cong \triangle JKL\), then \(\triangle ABC \cong \triangle JKL\)

Postulates and Theorems

Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also be congruent.

  • if \(\angle A \cong \angle X\) and \(\angle C \cong \angle Y\), then \(\angle B \cong \angle Z\)
Postulates and Theorems

Side-Side-Side (SSS) Congruence Postulate: If three sides in one triangle are congruent to those of another triangle, then the triangles are congruent.

  • Think of the SSS Postulate as a shortcut - now just showing three sets of sides are congruent (instead of three sets of sides and three sets of angles) is enough to say two triangles are congruent.
Congruence Postulate

Geometry

Triangle Congruence Cont.

Postulates and Theorems (cont.)

Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle in one triangle are congruent to those of another triangle, then the two triangles are congruent.

  • The placement of angle in SAS is important! It indicates that the angle must be between the two sides.
Side-Angle-Side (SAS) Congruence Postulate

The next two work because of the Third Angle Theorem:

Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent to those of another triangle, then the two triangles are congruent.

Angle-Side-Angle (ASA) Congruence Postulate

Angle-Angle-Angle and Side-Side-Angle (or Angle-Side-Side) do not show triangles are congruent!

Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to those of another triangle, then the triangles are congruent. Also Side-Angle-Angle (SAA) Congruence Theorem.

Angle-Angle-Side (AAS) Congruence

Special Triangles

Right Triangles

There are also some theorems that apply to right triangles only:

Right Triangles

Hypotenuse-Leg (HL) Congruence Theorem: If the hypotenuse and leg in one right triangle are congruent to those of another right triangle, then the two triangles are congruent.

Hypotenuse-Leg (HL) Congruence

Leg-Leg (LL) Theorem: If the legs of two right triangles are congruent, then the triangles are congruent.

Angle-Leg (AL) Theorem: If an angle and a leg of a right triangle are congruent to those of another right triangle, then the two triangles are congruent.

Hypotenuse-Angle (HA) Theorem: If an angle and the hypotenuse of a right triangle are congruent to those of another right triangle, then the two triangles are congruent.

Isosceles Triangles

Base Angles Theorem

Base Angles Theorem: The base angles of an isosceles triangle are congruent.

Base Angles Theorem Converse: If two angles in a triangle are congruent, then the op-posite sides are also congruent.