Measure Segments and Use the Segment Addition Postulate | Geometry | LevelKite

You already know how to name points and lines. A line goes on forever, so it has no length. In this lesson, you will measure the part of a line that has two ends, decide when two such parts match, and add lengths along a line.

One rule holds for the whole lesson: a figure is not drawn to scale. Only marks placed on a figure and facts stated in the text are givens, the facts you may start from. You never read a length or any other fact from how a drawing looks.

Measuring rests on a postulate. A postulate is a statement you accept without proof. The Ruler Postulate says two things about distance along a line:

The points on a line can be matched one to one with the real numbers, so each point gets exactly one number and each number goes with exactly one point. A point's number is its coordinate.
The distance between two points is the absolute value of the difference of their coordinates. If points $A$ and $B$ have coordinates $a$ and $b$ , then the distance between them, written $AB$ , is $AB = |a - b|$ .

The absolute value of a number is its size without its sign, so $|-4| = 4$ and $|4| = 4$ . That is why you can subtract in either order and never get a negative distance.

Coordinates let you say exactly what between means. Point $B$ is between points $A$ and $C$ when all three lie on one line and their coordinates $a$ , $b$ and $c$ satisfy $a < b < c$ or $a > b > c$ .

Now you can define the figure you will measure. A line segment $\overline{AB}$ is made of two different points $A$ and $B$ on a line, called its endpoints, and all the points between them. Its length, or measure, is the distance $AB$ . With the bar, $\overline{AB}$ names the figure. Without the bar, $AB$ names a number.

To measure a segment, lay a ruler along it and read the number at each endpoint. Those readings are coordinates, so the Ruler Postulate gives you the length: subtract the readings, take the absolute value, and write the unit. In this lesson, every reading is stated in the text, because a figure that is not to scale cannot give you a length.

The number at the far end is the length only when the other endpoint sits at $0$ , as in the first row of the table. The table shows one segment with a centimeter scale placed three different ways. A scale may even give an endpoint a negative coordinate, as in the last row. The coordinates change, but the distance between the endpoints does not.

One segment, with a centimeter scale placed three different ways

Coordinate of A	Coordinate of B	Length AB in centimeters
$0$	$4$	$\|0 - 4\| = 4$
$3$	$7$	$\|3 - 7\| = 4$
$-1$	$3$	$\|-1 - 3\| = 4$

Worked example

Measure a segment from ruler readings

A centimeter ruler is placed along $\overline{AB}$ . It reads $2.5$ at $A$ and $8$ at $B$ . Find $AB$ .

Name the coordinates. By the Ruler Postulate, each point on a line has a coordinate, and each ruler reading gives the coordinate of that endpoint.
$a = 2.5, \quad b = 8$
Substitute the coordinates into the distance rule. The Ruler Postulate says the distance is the absolute value of the difference of the coordinates.
$AB = |a - b| = |2.5 - 8|$
Subtract inside the bars. Then take the absolute value, which removes the negative sign.
$AB = |-5.5| = 5.5$
Check by subtracting in the other order. You get the same positive result, and the length of a segment must be positive. The readings are in centimeters, so the length is too.
$|8 - 2.5| = |5.5| = 5.5$

Answer

Segment $\overline{AB}$ is $5.5$ cm long, so $AB = 5.5$ cm.

Once you can measure segments, you can compare them. Congruent segments are segments that have the same length. The symbol $\cong$ means "is congruent to." Only length matters, not where a segment sits or which way it points.

A definition works in both directions, so this one gives you two facts:

If $AB = CD$ , then $\overline{AB} \cong \overline{CD}$ .
If $\overline{AB} \cong \overline{CD}$ , then $AB = CD$ .

Write $\cong$ between segments, which are figures, and $=$ between lengths, which are numbers.

In a figure, tick marks state congruence: segments with the same number of ticks are given as congruent. In the figure below, $\overline{AB}$ and $\overline{CD}$ each have one tick, so $\overline{AB} \cong \overline{CD}$ and $AB = CD$ . Segments $\overline{EF}$ and $\overline{GH}$ each have two ticks, so $\overline{EF} \cong \overline{GH}$ and $EF = GH$ .

Notice that $\overline{CD}$ is drawn longer than $\overline{AB}$ . That does not matter, because the figure is not drawn to scale and the ticks are the given. The figure also states no relationship between a one-tick segment and a two-tick segment. If a figure has no ticks and the text states no lengths, you cannot call two segments congruent, however alike they look.

One tick each gives

\overline{AB} \cong \overline{CD}

, and two ticks each give

\overline{EF} \cong \overline{GH}

. The drawn lengths are not to scale.

Worked example

Identify congruent segments from their lengths

An inch ruler is placed along each of three segments. It reads $\tfrac{1}{4}$ at $P$ and $2\tfrac{3}{4}$ at $Q$ , then $3$ at $R$ and $5\tfrac{1}{2}$ at $S$ , then $1$ at $T$ and $3$ at $U$ . Which of the segments $\overline{PQ}$ , $\overline{RS}$ and $\overline{TU}$ are congruent?

Find PQ with the Ruler Postulate: take the absolute value of the difference of the readings, in inches.
$PQ = \left|2\tfrac{3}{4} - \tfrac{1}{4}\right| = \left|2\tfrac{2}{4}\right| = 2\tfrac{1}{2}$
Find RS with the Ruler Postulate.
$RS = \left|5\tfrac{1}{2} - 3\right| = \left|2\tfrac{1}{2}\right| = 2\tfrac{1}{2}$
Find TU with the Ruler Postulate. All three lengths are positive, as lengths must be.
$TU = |3 - 1| = |2| = 2$
Compare the lengths. PQ equals RS, so segments PQ and RS are congruent by the definition of congruent segments. This uses the direction from equal lengths to congruence.
$PQ = RS \quad\Rightarrow\quad \overline{PQ} \cong \overline{RS}$
TU has a different length. The other direction of the definition says congruent segments have equal lengths, so segment TU cannot be congruent to segment PQ or segment RS.
$2 \neq 2\tfrac{1}{2}$

Answer

Segments $\overline{PQ}$ and $\overline{RS}$ are congruent because each is $2\tfrac{1}{2}$ inches long. Segment $\overline{TU}$ is $2$ inches long, so it is congruent to neither.

Lengths along a line also add. Picture walking along a line from point $A$ to point $C$ , passing through point $B$ on the way. The walk from $A$ to $B$ and the walk from $B$ to $C$ together make the whole trip, with no gap and no overlap.

Numbers show the same thing. Suppose a ruler reads $1$ at $A$ , $3$ at $B$ and $8$ at $C$ , so $B$ is between $A$ and $C$ . By the Ruler Postulate, $AB = 2$ , $BC = 5$ and $AC = 7$ , and $2 + 5 = 7$ . One case does not prove a general rule, so geometry accepts this rule as a postulate.

Key concept

Segment Addition Postulate

The Segment Addition Postulate says: if point $B$ is between points $A$ and $C$ , then the lengths of the two parts add up to the length of the whole segment. You accept it without proof.

Use it only when you are given that $B$ is between $A$ and $C$ , which also means all three points lie on one line. A drawing that only looks that way does not count. The postulate runs from between to the equation, and that is the direction you use in this lesson.

AB + BC = AC

This figure marks the lengths of

\overline{AB}

and

\overline{BC}

. The text states their values:

AB = 4.5

cm and

BC = 7

cm. The figure is not to scale.

Worked example

Find the whole from its parts

In the figure above, you are given that point $B$ is between points $A$ and $C$ , that $AB = 4.5$ cm, and that $BC = 7$ cm. Find $AC$ .

Check the condition. You are given that B is between A and C, so the Segment Addition Postulate applies. Write its equation.
$AB + BC = AC$
Replace AB with 4.5 and BC with 7, since those are the given lengths.
$4.5 + 7 = AC$
Add the lengths. Both are in centimeters, so the sum is too.
$11.5 = AC$
Check the result against the geometry. AC is positive, and it is longer than each part, as it must be because each part has positive length.
$11.5 > 4.5 \text{ and } 11.5 > 7$

Answer

The whole segment $\overline{AC}$ is $11.5$ cm long, so $AC = 11.5$ cm.

The ticks give

\overline{AB} \cong \overline{CD}

, and the marked lengths are

AB = 3.5

cm and

AD = 12

cm. The figure is not to scale.

Worked example

Combine congruence with the Segment Addition Postulate

In the figure above, you are given that points $A$ , $B$ , $C$ and $D$ lie on one line in that order, that $AB = 3.5$ cm, and that $AD = 12$ cm. The tick marks give $\overline{AB} \cong \overline{CD}$ . Find $BC$ .

The ticks give segments AB and CD as congruent. By the definition of congruent segments, their lengths are equal.
$CD = AB = 3.5$
The stated order puts C between A and D, so use the Segment Addition Postulate on the whole segment AD. Substitute CD = 3.5 and AD = 12.
$AC + CD = AD \quad\Rightarrow\quad AC + 3.5 = 12$
Subtract 3.5 from both sides, by the Subtraction Property of Equality.
$AC + 3.5 - 3.5 = 12 - 3.5 \quad\Rightarrow\quad AC = 8.5$
The stated order also puts B between A and C, so use the Segment Addition Postulate on segment AC. Substitute AB = 3.5 and AC = 8.5.
$AB + BC = AC \quad\Rightarrow\quad 3.5 + BC = 8.5$
Subtract 3.5 from both sides, again by the Subtraction Property of Equality.
$3.5 + BC - 3.5 = 8.5 - 3.5 \quad\Rightarrow\quad BC = 5$
Check the result. BC is positive, and the three parts add up to the given whole, AD = 12. All these lengths are in centimeters.
$AB + BC + CD = 3.5 + 5 + 3.5 = 12$

Answer

The middle segment $\overline{BC}$ is $5$ cm long, so $BC = 5$ cm.

Here is the big idea: a segment's length is a number, and that number does the work. The Ruler Postulate gives you the length. Equal lengths mean congruent segments, by definition. When a point is between two others, the Segment Addition Postulate lets you add the parts to get the whole. Take your facts from the marks and the text, never from how a figure looks.

Next, you will find segment lengths using algebra.