Common Core: 8th Grade Math : Use Informal Arguments to Establish Facts about the Angle Sum and Exterior Angle of Triangles: CCSS.Math.Content.8.G.A.5

Study concepts, example questions & explanations for Common Core: 8th Grade Math

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Example Questions

Example Question #1 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ a}\) is equal to \(\displaystyle 135^\circ\), then which of the other angles is equal to \(\displaystyle 135^\circ?\)

2

Possible Answers:

\(\displaystyle \angle{\ d}\)

\(\displaystyle \angle{\ b}\)

\(\displaystyle \angle{\ c}\)

\(\displaystyle \angle{\ f}\)

Correct answer:

\(\displaystyle \angle{\ c}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ a}\). Based on our answer choices, angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles; thus, both angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are congruent and equal \(\displaystyle 135^\circ\)

Example Question #382 : Grade 8

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ a}\) is equal to \(\displaystyle 135^\circ\), then which of the other angles is equal to \(\displaystyle 135^\circ?\)

2

 

Possible Answers:

\(\displaystyle \angle{\ b}\)

\(\displaystyle \angle{\ h}\)

\(\displaystyle \angle{\ e}\)

\(\displaystyle \angle{\ f}\)

Correct answer:

\(\displaystyle \angle{\ e}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ a}\). Based on our answer choices, angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles; thus, both angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are congruent and equal \(\displaystyle 135^\circ\)

Example Question #2 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ b}\) is equal to \(\displaystyle 45^\circ\), then which of the other angles is equal to \(\displaystyle 45^\circ\)

2

 

 

Possible Answers:

\(\displaystyle \angle{\ g}\)

\(\displaystyle \angle{\ f}\)

\(\displaystyle \angle{\ c}\)

\(\displaystyle \angle{\ e}\)

Correct answer:

\(\displaystyle \angle{\ f}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ b}\). Based on our answer choices, angle \(\displaystyle \angle{\ b}\) and \(\displaystyle \angle{\ f}\) are corresponding angles; thus, both angle \(\displaystyle \angle{\ b}\) and \(\displaystyle \angle{\ f}\) are congruent and equal \(\displaystyle 45^\circ\)

Example Question #2 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ a}\) is equal to \(\displaystyle 135^\circ\), then which of the other angles is equal to \(\displaystyle 135^\circ?\)

2

 

Possible Answers:

\(\displaystyle \angle{\ h}\)

\(\displaystyle \angle{\ f}\)

\(\displaystyle \angle{\ g}\)

\(\displaystyle \angle{\ b}\)

Correct answer:

\(\displaystyle \angle{\ g}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ a}\). Based on our answer choices, angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are alternate exterior angles; thus, both angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are congruent and equal \(\displaystyle 135^\circ\)

Example Question #3 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ b}\) is equal to \(\displaystyle 45^\circ\), then which of the other angles is equal to \(\displaystyle 45^\circ\)

2

 

 

Possible Answers:

\(\displaystyle \angle{\ c}\)

\(\displaystyle \angle{\ h}\)

\(\displaystyle \angle{\ a}\)

\(\displaystyle \angle{\ g}\)

Correct answer:

\(\displaystyle \angle{\ h}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ b}\). Based on our answer choices, angle \(\displaystyle \angle{\ b}\) and \(\displaystyle \angle{\ h}\) are alternate exterior angles; thus, both angle \(\displaystyle \angle{\ b}\) and \(\displaystyle \angle{\ h}\) are congruent and equal \(\displaystyle 45^\circ\)

Example Question #3 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ c}\) is equal to \(\displaystyle 135^\circ\), then which of the other angles is equal to \(\displaystyle 135^\circ?\)

2

 

Possible Answers:

\(\displaystyle \angle{\ b}\)

\(\displaystyle \angle{\ h}\)

\(\displaystyle \angle{\ e}\)

\(\displaystyle \angle{\ f}\)

Correct answer:

\(\displaystyle \angle{\ e}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ c}\). Based on our answer choices, angle \(\displaystyle \angle{\ c}\) and \(\displaystyle \angle{\ e}\) are alternate interior angles; thus, both angle \(\displaystyle \angle{\ c}\) and \(\displaystyle \angle{\ e}\) are congruent and equal \(\displaystyle 135^\circ\)

Example Question #4 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). If angle \(\displaystyle \angle{\ d}\) is equal to \(\displaystyle 45^\circ\), then which of the other angles is equal to \(\displaystyle 45^\circ\)

2

 

 

Possible Answers:

\(\displaystyle \angle{\ g}\)

\(\displaystyle \angle{\ a}\)

\(\displaystyle \angle{\ f}\)

\(\displaystyle \angle{\ e}\)

Correct answer:

\(\displaystyle \angle{\ f}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is congruent to angle \(\displaystyle \angle{\ d}\). Based on our answer choices, angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\) are alternate interior angles; thus, both angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\) are congruent and equal \(\displaystyle 45^\circ\)

Example Question #3 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). Which angle is NOT equal to angle \(\displaystyle \angle {\ g}?\)

2

 

 

Possible Answers:

\(\displaystyle \angle{\ d}\)

\(\displaystyle \angle{\ c}\)

\(\displaystyle \angle{\ a}\)

\(\displaystyle \angle{\ e}\)

Correct answer:

\(\displaystyle \angle{\ d}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is not equal to to angle \(\displaystyle \angle{\ g}\). Let's look at our answer choices:

Angle \(\displaystyle \angle{\ g}\) and \(\displaystyle \angle{\ e}\) are vertical angles, which means they are congruent. 

Angle \(\displaystyle \angle{\ g}\) and  \(\displaystyle \angle{\ a}\) are alternate exterior angles, which means they are congruent. 

Angle \(\displaystyle \angle{\ g}\) and \(\displaystyle \angle{\ c}\) are corresponding angles, which means they are congruent. 

However, angle \(\displaystyle \angle{\ g}\) and \(\displaystyle \angle{\ d}\) do not share a common angle relationship; thus they are not congruent. 

Example Question #4 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). Which angle is NOT equal to angle \(\displaystyle \angle {\ f}?\)

2

 

Possible Answers:

\(\displaystyle \angle{\ c}\)

\(\displaystyle \angle{\ d}\)

\(\displaystyle \angle{\ b}\)

\(\displaystyle \angle{\ h}\)

Correct answer:

\(\displaystyle \angle{\ c}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is not equal to to angle \(\displaystyle \angle {\ f}\). Let's look at our answer choices:

Angle \(\displaystyle \angle {\ f}\) and \(\displaystyle \angle{\ h}\) are vertical angles, which means they are congruent. 

Angle \(\displaystyle \angle {\ f}\) and  \(\displaystyle \angle{\ d}\) are alternate interior angles, which means they are congruent. 

Angle \(\displaystyle \angle {\ f}\) and \(\displaystyle \angle{\ b}\) are corresponding angles, which means they are congruent. 

However, angle \(\displaystyle \angle {\ f}\) and \(\displaystyle \angle{\ c}\) do not share a common angle relationship; thus they are not congruent. 

Example Question #6 : Use Informal Arguments To Establish Facts About The Angle Sum And Exterior Angle Of Triangles: Ccss.Math.Content.8.G.A.5

The image provided contains a set of parallel lines, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\), and a transversal line, \(\displaystyle \overline{EF}\). Which angle is NOT equal to angle \(\displaystyle \angle {\ d}?\)

2

 

Possible Answers:

\(\displaystyle \angle{\ c}\)

\(\displaystyle \angle{\ f}\)

\(\displaystyle \angle{\ h}\)

\(\displaystyle \angle{\ b}\)

Correct answer:

\(\displaystyle \angle{\ c}\)

Explanation:

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other. 

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\) are parallel lines and line \(\displaystyle \overline{EF}\) is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

  • Vertical angles are congruent
  • Corresponding angles are congruent
  • Alternate interior angles are congruent
  • Alternate exterior angles are congruent

Let's look at angle \(\displaystyle \angle{\ a}\) in the image provided below to demonstrate our relationships. 

1

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ c}\) are vertical angles.

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ e}\) are corresponding angles. 

Angle \(\displaystyle \angle{\ a}\) and \(\displaystyle \angle{\ g}\) are exterior angles. 

Angle \(\displaystyle \angle{\ a}\) is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs \(\displaystyle \angle{\ e}\) and \(\displaystyle \angle{\ c}\) as well as angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ f}\)

For this problem, we want to find the angle that is not equal to to angle \(\displaystyle \angle{\ d}\). Let's look at our answer choices:

Angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ b}\) are vertical angles, which means they are congruent. 

Angle \(\displaystyle \angle{\ d}\) and  \(\displaystyle \angle{\ f}\) are alternate interior angles, which means they are congruent. 

Angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ h}\) are corresponding angles, which means they are congruent. 

However, angle \(\displaystyle \angle{\ d}\) and \(\displaystyle \angle{\ c}\) do not share a common angle relationship; thus they are not congruent. 

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