Appendix A

Geometry.

Some of the proofs in this book use basic geometrical facts to work out the relationships between various angles. Here are some of the facts that are used.

Angles in a triangle.

All the angles in a triangle must add up to \( 180^o \) .

fig1
Figure 1 The angles inside a triangle add up to \( 180^o \). In all these triangles, the sum \( a+b+c \) is \( 180^o \).
fig2
Figure 2 A right-triangle has one right angle ( \( 90^o \)in it, marked with a square. In that case, the other two angles have to add up to \( 90^o \). In these triangles \( a+b=90^o \) because the third (right) angle is \( 90^o \).

Similar Triangles.

Similar triangles are triangles that have the same shape, but different sizes. Two triangles are guaranteed to have the same shape if they have all angles the same. Since all angles must add up to \( 180^o \), it only needs two of the angles to be the same for all of them to be the same.

fig3
Figure 3 The larger triangle and the smaller triangle are similar because they have the same three angles.

When two triangles are similar, their corresponding sides are all the same ratio. In Figure 3, we have

\[ \frac{A^\prime}{A}=\frac{B^\prime}{B}=\frac{C^\prime}{C}\tag{1} \]

Here, "corresponding sides" means sides that join up the same angles.

Conversely, if we have two triangles all of whose sides have the same ratio (like Equation (1)), then they are similar and have the same angles.

Supplementary angles.

Supplementary angles are angles that add up to \( 180^o \) . You don't need to know the name "supplementary" but you do need to know the situations where these sorts of angles occur, which are generally where two lines touch or cross.

fig3
Figure 4 On the left, the angles \( a \) and \( b \) add up to \( 180^o \) . On the right, \( a+b=b+c=c+d=d+a=180^o \). These angles add up to \( 180^o \) simply because all the angles on one side of a line (that is, half a circle) have to add up to \( 180^o \).

On the right hand side, you can also prove that \( a=c \) and \( b=d \). Since \( a+b=b+c \) , subtracting \( b \) from both sides gives \( a=c \).

Alternating angles

Alternating angles occur when two parallel lines are crossed by a third line.

fig4
Figure 5 Two parallel lines are crossed by a third line. We have \( a=e \), \( b=f \), \( c=g \), and \( d=h \). Because \( a=c \) and \( b=d \), we end up with two sets of angles that are all equal: