2.05 Proving segment relationships

Lesson

Practice

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 $\overline{AC}$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

NC.M2.G-CO.9

Prove theorems about lines and angles and use them to prove relationships in geometric figures including: • vertical angles are congruent. • When a transversal crosses parallel lines, alternate interior angles are congruent. • When a transversal crosses parallel lines, corresponding angles are congruent. • Points are on a perpendicular bisector of a line segment if and only if they are equidistant from the endpoints of the segment. • Use congruent triangles to justify why the bisector of an angle is equidistant from the sides of the angle.