Basic Mathematics for College Students

We can show that the triangles shown below are congruent by the SAS property:

756 Chapter 9 An Introduction to Geometry

ASA Property

If two angles and the side between them in one triangle are congruent,

respectively, to two angles and the side between them in a second triangle,

the triangles are congruent.

Since and , the segments are congruent.

Since and , the angles are congruent.

Since and , the segments are congruent.

Therefore,TVU  FGE.

UV  EG m(UV) 3 m(EG) 3

V  G m(V) 90 ° m(G) 90 °

TV  FG m(TV) 2 m(FG) 2

TU

E

G F

V

233

2

90 °

90 °

We can show that the triangles shown below are congruent by the ASA property:

RC

PQAB

9 82 °^82 °^9

60 ° 60 °

Since and , the angles are congruent.

Since and , the segments are congruent.

Since and , the angles are congruent.

Therefore,PQR  BAC.

R  C m(R) 82 ° m(C) 82 °

PR  BC m(PR) 9 m(BC) 9

P  B m(P) 60 ° m(B) 60 °

Caution! There is no SSA property. To illustrate this, consider the triangles

shown below. Two sides and an angle of are congruent to two sides and

an angle of. But the congruent angle is not between the congruent

sides.

We refer to this situation as SSA. Obviously, the triangles are not

congruent because they are not the same shape and size.

DEF

ABC

The tick marks indicate congruent

parts. That is, the sides with one tick

mark are the same length, the sides

with two tick marks are the same

length, and the angles with one tick

A D F mark have the same measure.

E

C

B

EXAMPLE (^2) Explain why the triangles in the figure on the following page
are congruent.
StrategyWe will show that two sides and the angle between them in one triangle
are congruent, respectively, to two sides and the angle between them in a second
triangle.