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Congruent Triangles
Triangles that have exactly the same size and shape are called congruent
triangles. The symbol for congruent is ≅. Two triangles are congruent when
the three sides and the three angles of one triangle have the same
measurements as three sides and three angles of another triangle. The
triangles in Figure 1 are congruent triangles.
Figure 1
Congruent triangles.
Corresponding parts
The parts of the two triangles that have the same measurements
(congruent) are referred to as corresponding parts. This means that
Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
Congruent triangles are named by listing their vertices in corresponding
orders. In Figure
1 , Δ BAT ≅ Δ ICE.
Example 1: If Δ PQR ≅ Δ STU which parts must have equal measurements?
These parts are equal because corresponding parts of congruent triangles
are congruent.
Tests for congruence
To show that two triangles are congruent, it is not necessary to show that all
six pairs of corresponding parts are equal. The following postulates and
theorems are the most common methods for proving that triangles are
congruent (or equal).
(SSS Postulate): If each side of one triangle is congruent to the
corresponding side of another triangle, then the triangles are congruent
(Figure
2 ).
Figure 2
The corresponding sides (SSS) of the two triangles are all congruent.
(SAS Postulate): If two sides and the angle between them in one triangle are
congruent to the corresponding parts in another triangle, then the triangles
are congruent (Figure 3 ).
Figure 3
Two sides and the included angle (SAS) of one triangle are congruent to the corresponding parts
of the other triangle.
(ASA Postulate): If two angles and the side between them in one triangle are
congruent to the corresponding parts in another triangle, then the triangles
are congruent (Figure 4 ).
Figure 4
Two angles and their common side (ASA) in one triangle are congruent to the corresponding parts
of the other triangle.
(AAS Theorem): If two angles and a side not between them in one triangle
are congruent to the corresponding parts in another triangle, then the
triangles are congruent (Figure 5 ).
Figure 5
Two angles and the side opposite one of these angles (AAS) in one triangle are congruent to the
corresponding parts of the other triangle.
(HL Postulate): If the hypotenuse and leg of one right triangle are congruent
to the corresponding parts of another right triangle, then the triangles are
congruent (Figure 6 ).
Figure 6
The hypotenuse and one leg (HL) of the first right triangle are congruent to the corresponding
parts of the second right triangle.
(HA Theorem): If the hypotenuse and an acute angle of one right triangle
are congruent to the corresponding parts of another right triangle, then the
triangles are congruent (Figure 7 ).
Figure 7
The hypotenuse and an acute angle (HA) of the first right triangle are congruent to the
corresponding parts of the second right triangle.
(LL Theorem): If the legs of one right triangle are congruent to the
corresponding parts of another right triangle, then the triangles are
congruent (Figure 8 ).
Figure 8
The legs (LL) of the first right triangle are congruent to the corresponding parts of the second right
triangle.
(LA Theorem): If one leg and an acute angle of one right triangle are
congruent to the corresponding parts of another right triangle, then the
triangles are congruent (Figure 9 ).
Figure 9
One leg and an acute angle (LA) of the first right triangle are congruent to the corresponding parts
of the second right triangle.
Example 2: Based on the markings in Figure 10 , complete the congruence
statement Δ ABC ≅Δ .
Figure 10
Congruent triangles.
Δ YXZ, because A corresponds to Y, B corresponds to X, and C corresponds,
to Z.
Example 3: By what method would each of the triangles in Figures
11 (a) through 11 (i) be proven congruent?
Figure 11
Methods of proving pairs of triangles congruent.

(a) SAS.

(b) None. There is no AAA method.

(c) HL.

(d) AAS.

(e) SSS. The third pair of congruent sides is the side that is shared by
the two triangles.

(f) SAS or LL.

(g) LL or SAS.

(h) HA or AAS.

(i) None. There is no SSA method.
Example 4: Name the additional equal corresponding part(s) needed to
prove the triangles in Figures 12 (a) through 12 (f) congruent by the
indicated postulate or theorem.
Figure 12
Additional information needed to prove pairs of triangles congruent.
(a) BC = EF or AB = DE ( but notAC = DF because these two sides lie

between the equal angles).

(b) GI = JL.

(c) MO = POandNO = RO.

(d) TU = WXandSU = VX.

(e) m ∠ T = m ∠ E andm ∠ TOW = m ∠ EON.
(f) IX = EN or SX = TN (but not IS = ET because they are
hypotenuses).

Example 1
ABC

XYZ
Two sides and the included angle are congruent
o AC = ZX (side)
ACB =
XZY (angle)
CB = ZY (side)
Therefore, by the Side Angle Side postulate, the triangles are congruent.
o
o

Included Angle
The included angle means the angle between two sides. In other words it is
the angle 'included between' two sides.
Identify Side Angle Side Relationships
In which pair of triangles pictured below could you use the Side Angle Side postulate (SAS) to prove the
triangles are congruent?
Answer
Pair four is the only true example of this method for proving triangles congruent. It is the only pair in which
the angle is an included angle.