Geometry with Trigonometry

Sec. 1.5 Revision of geometrical concepts 17

A B

C P

D

Figure 1.17. Mid-line of an angle-support.

C A B

P

D

5. Any angle∠BACsuch that 0<|∠BAC|◦<90 is calledacute, such that
   |∠BAC|◦=90 is calledright, and such that 90<|∠BAC|◦<180 is calledobtuse.
   If∠BACis a right-angle, then the linesABandACare said to beperpendicularto
   each other, writtenAB⊥AC.

A

B

C 90

Figure 1.18. Perpendicular lines.

xA

B

y

C

z

A′

x

B′ y

z C′

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Figure 1.19. Congruent triangles.

1.5.4 Our treatment of congruence

If[A,B,C],[A′,B′,C′]are triangles such that

|B,C|=|B′,C′|,|C,A|=|C′,A′|,|A,B|=|A′,B′|,

|∠BAC|◦=|∠B′A′C′|◦,|∠CBA|◦=|∠C′B′A′|◦,|∠ACB|◦=|∠A′C′B′|◦,

then we say by way of definition that the triangle[A,B,C]iscongruentto the triangle
[A′,B′,C′]. Behind this concept is the physical idea that[A,B,C]can be placed on
[A′,B′,C′], fitting it exactly.
By observation if[A,B,C],[A′,B′,C′]are such that

|C,A|=|C′,A′|,|A,B|=|A′,B′|,|∠BAC|◦=|∠B′A′C′|◦,

then[A,B,C]is congruent to[A′,B′,C′]. This is known as the SAS (side, angle, side)
condition for congruence of triangles.
Similarly by observation if[A,B,C],[A′,B′,C′]are such that

|B,C|=|B′,C′|,|∠CBA|◦=|∠C′B′A′|◦,|∠BCA|◦=|∠B′C′A′|◦,