Introduction

When two line segments have the same length, they are congruent segments. To indicate that the line segments \overline{AB} and \overline{CD} are congruent we write \overline{AB} \:\:{\color{red}\boldsymbol{\cong}}\:\: \overline{CD}, and we use special marks like in the diagram below.



We can also write AB \:{\color{blue}=}\: CD, which means that the measure of \overline{AB} is equal to the measure of \overline{CD}.

Watch out! The equation \overline{AB} = \overline{CD} is only true when both \overline{AB} and \overline{CD} correspond to the same exact segment. When the segments are distinct, we should use the symbol \cong to represent that the two segments are separate but that they have the same length.

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Example: Identifying Congruent Line Segments on a Grid

Which of the line segments shown below are congruent?



EXPLANATION

For two segments to be congruent, they must have the same length.

So, let's determine the length of each segment in the diagram. We see that:

  • AB = 3 units

  • CD=EF = 4 units

  • GH = 2 units

Therefore, the only pair of congruent line segments is \overline{CD}\cong\overline{EF}.

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Practice: Identifying Congruent Line Segments on a Grid

For the line segments shown above, which of the following statements are true?

  1. $\overline{AE} \cong \overline{AB}$
  2. $\overline{AB} \cong \overline{BC}$
  3. $\overline{BD} \cong \overline{DF}$
a
 I only
b
 III only
c
 II only
d
 II and III only
e
 I and II only
Practice: Identifying Congruent Line Segments on a Grid

Which of the line segments shown above are congruent?

a
 $\overline{CD}$ and $\overline{GH}$
b
 $\overline{CD}$ and $\overline{EF}$
c
 $\overline{AB}$ and $\overline{EF}$
d
 $\overline{AB}$ and $\overline{GH}$
e
 $\overline{AB}$ and $\overline{CD}$
Example: Identifying Congruent Segments on a Number Line



Given the number line above, which of the following statements are true?

  1. AC=BD
  2. \overline{AB} \cong \overline{CD}
  3. \overline{AB} is congruent to \overline{BD}
EXPLANATION

Let's go through the statements one by one.

  • Statement I is true. Indeed, we have that AC= BD=5 units.

  • Statement II is true. Since AB=CD=3 units, we have that \overline{AB} \cong \overline{CD} .

  • Statement III is false since AB=3 , while BD = 5. So, \overline{AB}\ncong \overline{BD} .

Therefore, only statements I and II are true.

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Practice: Identifying Congruent Segments on a Number Line

Given the number line above, which of the following statements are true?

  1. $PR=RS$
  2. $\overline{QR} \cong \overline{QS}$
  3. $\overline{PR}$ is congruent to $\overline{RS}$
a
 I, II and III
b
 I and III only
c
 II and III only
d
 I and II only
e
 II only
Practice: Identifying Congruent Segments on a Number Line

Given the number line above, which of the following statements are true?

  1. $AC=CD$
  2. $\overline{AB} \cong \overline{CD}$
  3. $\overline{AB}$ is congruent to $\overline{BD}$
a
 II only
b
 I only
c
 I and II only
d
 I and III only
e
 II and III only
Example: Identifying Line Segments on a Given Plane Figure

For the diagram shown below, which of the following statements are true?



  1. The segments \overline{BC} and \overline{BD} are congruent
  2. \overline{BC} \cong \overline{BE}
  3. BE = 4\,\textrm{cm}
EXPLANATION

Let's look at the statements one by one.

  • Statement I is true. In the picture, both \overline{BC} and \overline{BD} have the same double hash marks, so \overline{BC} \cong \overline{BD}.

  • Statement II is false. From the picture, we can see that \overline{BE} and \overline{BC} have different hash marks, so \overline{BC} \not\cong \overline{BE}.

  • Statement III is false. According to the picture, we have that \overline{BE} and \overline{BD} have different hash marks, so \overline{BE} \not\cong \overline{BD}. Hence, {BE} \neq {BD}, and since {BD} = 4\,\textrm{cm} , we have BE \neq 4\,\textrm{cm}.

Therefore, only statement I is true.

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Practice: Identifying Line Segments on a Given Plane Figure

For the diagram shown above, which of the following statements are true?

  1. The segments $\overline{AC}$ and $\overline{DE}$ are congruent
  2. $\overline{BC} \cong \overline{CD}$
  3. $BC = 6\,\textrm{cm}$
a
 I and III only
b
 II only
c
 I only
d
 I and II only
e
 III only
Practice: Identifying Line Segments on a Given Plane Figure

For the diagram shown above, which of the following statements are true?

  1. The segments $\overline{AB}$ and $\overline{CD}$ are congruent
  2. $\overline{AC} \cong \overline{AB}$
  3. $AC = 3\,\textrm{cm}$
a
 II only
b
 I and II only
c
 I and III only
d
 III only
e
 II and III only
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