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Geometry: Angles and Parallel Lines




 
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More Lessons for High School Geometry
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A series of free, online High School Geometry Video Lessons and solutions.
Videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

  • alternate interior angles
  • alternate exterior angles
  • same side interior and same side exterior angles
  • converse of the parallel lines theorem

Alternate Interior Angles

Alternate interior angles are formed by a transversal intersecting two lines. They are located between the two lines, but on opposite sides of the transversal, creating two pairs (four total angles) of alternate interior angles. If the two lines are parallel then the alternate interior angles are congruent, meaning they have equal measure.

How to define alternate interior angles and their special properties?
Learn what alternate interior angles are and their relationship with parallel lines. How to use alternate interior angles to find the measures of angles?
Example:
∠BDE = 30°, ∠ADB = 61°.
Find the measure of ∠CDA, ∠DAB and ∠ABD.



Alternate Exterior Angles

Alternate exterior angles are formed by a transversal intersecting two lines. They are located "outside" the two lines but on opposite sides of the transversal, creating two pairs (four total angles) of alternate exterior angles. If the two lines are parallel then the alternate exterior angles are congruent, meaning they have equal measure.

How to define alternate exterior angles and their special properties?
Learn what alternate exterior angles are and their relationship with parallel lines. How to Find an Angle Using Alternate Exterior Angles?


 

Same Side Interior and Same Side Exterior Angles

When two parallel lines are intersected by a transversal, same side interior (between the parallel lines) and same side exterior (outside the parallel lines) angles are formed. Since alternate interior and alternate exterior angles are congruent and since linear pairs of angles are supplementary, same side angles are supplementary.

How to describe same side interior and same side exterior angles and their special properties? Description and Examples of Transversals and Angles (Alternate Interior, Alternate Exterior, Same Side Interior, Same Side Exterior, Corresponding)
If a transversal crosses parallel lines:
• corresponding angles are congruent.
• alternate interior angles are congruent.
• alternate exterior angles are congruent.
• same side interior angles are supplementary.
• same side exterior angles are supplementary.

Converse of Parallel Lines Theorem

If two lines are intersected by a transversal, then alternate interior angles, alternate exterior angles, and corresponding angles are congruent. The converse of the theorem is true as well. If two corresponding angles are congruent, then the two lines cut by the transversal must be parallel. Similarly, if two alternate interior or alternate exterior angles are congruent, the lines are parallel.

How to use the converse of the parallel lines theorem?
This lesson investigates and use the converse of alternate interior angles theorem, the converse of alternate exterior angles theorem, the converse of corresponding angles postulate, the converse of same side interior angles theorem and the converse of same side exterior angles theorem.

Converse of the Corresponding Angles Postulate
If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel.

Converse of the Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent then the two lines are parallel.

Converse of the Same-Side Interior Angles Theorem
If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.

Example:
A rectangular wooden frame has a diagonal metal brace. The angles indicated were measured to be equal. Which sides are parallel? Explain. How to prove lines are parallel?
This lesson investigates and use the converse of alternate interior angles theorem, the converse of alternate exterior angles theorem, the converse of corresponding angles postulate, the converse of same side interior angles theorem and the converse of same side exterior angles


 

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