Common Core: High School - Geometry : Prove Geometric Theorems: Lines and Angles

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Example Question #1 : Prove Geometric Theorems: Lines And Angles

True or False: The Vertical Angle Theorem states that if two angles are vertical angles, then they have equal measure.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is a true statement.  If two lines are intersecting, then their vertical angles are congruent.  Vertical angles are the opposite angles formed when two lines intersect.  The figure below shows an example of this.

 

Example Question #2 : Prove Geometric Theorems: Lines And Angles

Which of the following is the Same Side Interior Angles Theorem?

Possible Answers:

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary

If a transversal intersects two parallel lines, then the exterior angles on the same side of the transversal are supplementary

If a transversal intersects two perpendicular lines, then the interior angles on the same side of the transversal are supplementary

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are complementary

Correct answer:

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary

Explanation:

Consider the figure below.  If a transversal line intersects two parallel lines, then the two interior angles on the same side of the transversal (the two consecutive, interior angles of the transversal) are supplementary.  This means that they add up to be 180.

 

Example Question #1 : Prove Geometric Theorems: Lines And Angles

True or False: The Corresponding Angles Theorem states that if two parallel lines are cut by a transversal then the pair of corresponding angles are supplementary.

Possible Answers:

False

True

Correct answer:

False

Explanation:

The Corresponding Angles Theorem states that if two parallel lines are cut by a transversal line then the pair of corresponding angles are congruent.  Corresponding angles are angles formed when a transversal line cuts two lines and they lie in the same position at each intersection.  The figure below illustrates corresponding lines.

 

Example Question #1 : Prove Geometric Theorems: Lines And Angles

Which of the following correctly state the Alternate Interior Angles Theorem?

Possible Answers:

If two parallel lines are cut by a transversal line, the corresponding interior angles are congruent

If two parallel lines are cut by a transversal line, the alternate interior angles are complementary

If two parallel lines are cut by a transversal line, the alternate interior angles are congruent

If two parallel lines are cut by a transversal line, the alternate interior angles are supplementary

Correct answer:

If two parallel lines are cut by a transversal line, the alternate interior angles are congruent

Explanation:

The correct answer is "If two parallel lines are cut by a transversal line, the alternate interior angles are congruent."

Alternate interior angles are pairs of angles on the inner sides of parallel lines but on the opposite sides of a transversal line which is intersecting parallel lines.  These angles are always congruent.  Alternate interior angles are illustrated in the figure below.

Example Question #1 : Prove Geometric Theorems: Lines And Angles

True or False: Alternate exterior angles are congruent.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This is true according to the Alternate Exterior Angle Theorem which can be proven in a similar way to the Alternate Interior Angle Theorem.  Alternate exterior angles are illustrated in the figure below.

 

Example Question #232 : Congruence

What does it mean for two angles to be complementary angles?

Possible Answers:

Complementary angles are any two angles that sum to be 90

Complementary angles are any two angles in a triangle that sum to be 90

Complementary angles are any two angles that sum to be 180

Complementary angles are any two angles in a triangle that sum to be 180

Correct answer:

Complementary angles are any two angles that sum to be 90

Explanation:

The definition of complementary angles is: any two angles that sum to 90.  We most often see these angles as the two angles in a right triangle that are not the right angle.  These two angles do not have to only be in right triangles, however.  Complementary triangles are any pair of angles that add up to be 90.

Example Question #236 : Congruence

Lines  and  are parallel.  Using this information, find the values for angles , and  .

Screen shot 2020 08 07 at 2.51.45 pm

Possible Answers:

Correct answer:

Explanation:

We must use the fact that lines  and are parallel lines to solve for the missing angles.  We will break it down to solve for each angle one at a time.

 

Angle :

We know that angle ’s supplementary angle.  Supplementary angles are two angles that add up to 180 degrees.  These two are supplementary angles because they form a straight line and straight lines are always 180 degrees.  So to solve for angle we simply subtract it’s supplementary angle from 180.

 

 

Angle :

We now know that angle  is 130 degrees.  We can either use that fact that angles  and  are opposite vertical angles to find the value of angle  or we can use that fact that angle ’s supplementary angle is the given angle of 50 degrees.  If we use the latter, we would use the same procedure as last time to solve for angle .  If we use the fact that angles  and  are opposite vertical angles, we know that they are congruent.  Since angle  then angle .

 

Angle :

To find angle  we can use the fact that angles  and  are corresponding angles and therefore are congruent or we can use the fact that angles  and  are alternate interior angles and therefore are congruent.  Either method that we use will show that .

 

Angle :

To find angle  we can use that fact that the given angle of 50 degrees and angle  are alternate exterior angles and therefore are congruent, or we can use the fact that angle  is angle 's supplementary angle.  We know that the given angle and angle  are alternate exterior angles so .

 

 

Example Question #1 : Prove Geometric Theorems: Lines And Angles

True or False: Lines AB and CD are parallel.

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Possible Answers:

True

False

Correct answer:

True

Explanation:

We know that lines AB and CD are parallel due to the information we get from the angles formed by the transversal line.  There are:

  • two pairs of congruent vertically opposite angles
  • two pairs of congruent alternate interior angles
  • two pairs of congruent alternate exterior angles
  • two pairs of congruent corresponding angles

Just using any one of these facts is enough proof that lines AB and CD are parallel.

Example Question #2 : Prove Geometric Theorems: Lines And Angles

Solve for angle 1.

Screen shot 2020 08 12 at 9.43.41 am

Possible Answers:

Correct answer:

Explanation:

Even though it may not be obvious at first, the given angle is actually a supplementary angle to angle .  This is because the given angle is corresponding (and therefore congruent) angles to the angle adjacent to angle .  Since they are supplementary we can set up the following equation.

 

Example Question #8 : Prove Geometric Theorems: Lines And Angles

Which of the following describes  and ?

Screen shot 2020 08 19 at 4.46.58 pm

Possible Answers:

These are vertically opposite angles, there is not enough information to determine any further relation

These are corresponding angles, there is not enough information to determine any further relation

These are vertically opposite angles, therefore they are equal

These are corresponding angles, therefore they are equal

Correct answer:

These are vertically opposite angles, therefore they are equal

Explanation:

To answer this question, we must understand the definition of vertically opposite angles.  Vertically opposite angles are angles that are formed opposite of each other when two lines intersect.  Vertically opposite angles are always congruent to each other.

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