Lesson

Previously, we the definition of betweenness of points. This definition is actually an extension of the segment addition postulate.

Segment addition postulate

Given two points $A$`A` and $C$`C`, a third point $B$`B` lies on $\overline{AC}$`A``C` if and only if the distances between the points satisfy the equation $AB+BC=AC$`A``B`+`B``C`=`A``C`.

We can apply the segment addition postulate, the **definition of congruent segments**, as well as the properties of equality and congruence to prove segment relationships in a diagram.

Suppose that points $A$`A`, $B$`B`, and $C$`C` are collinear, with point $B$`B` between points $A$`A` and $C$`C`. Solve for $x$`x` if $AC=21$`A``C`=21, $AB=15-x$`A``B`=15−`x` and $BC=4+2x$`B``C`=4+2`x`. Justify each step.

Given that the points $P$`P`, $Q$`Q`, $R$`R`, and $S$`S` are collinear, prove that $PQ=PS-QS$`P``Q`=`P``S`−`Q``S`.

In the image below, points $R$`R`, $S$`S`, $T$`T`, and $U$`U` are collinear. Given that $\overline{RT}$`R``T` is congruent to $\overline{SU}$`S``U`, prove that $\overline{RS}$`R``S` is congruent to $\overline{TU}$`T``U`.

Prove and apply theorems about lines and angles.