6.05 Proving segment relationships
Previously, we the definition of betweenness of points. This definition is actually an extension of the segment addition postulate.
Given two points $A$A and $C$C, a third point $B$B lies on $\overline{AC}$AC if and only if the distances between the points satisfy the equation $AB+BC=AC$AB+BC=AC.
$A$A, $B$B, and $C$C are collinear with $B$B between $A$A and $C$C
$AB+BC=AC$AB+BC=AC
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.
Practice questions
Question 1
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 $AC=21$AC=21, $AB=15-x$AB=15−x and $BC=4+2x$BC=4+2x. Justify each step.
Question 2
Given that the points $P$P, $Q$Q, $R$R, and $S$S are collinear, prove that $PQ=PS-QS$PQ=PS−QS.
Question 3
In the image below, points $R$R, $S$S, $T$T, and $U$U are collinear. Given that $\overline{RT}$RT is congruent to $\overline{SU}$SU, prove that $\overline{RS}$RS is congruent to $\overline{TU}$TU.
Outcomes
II.G.CO.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.