Geometry - Chapter 10 Parallel Lines and Angles Key Concepts: It is a good idea to try to remember these concepts from Lesson 10: ! Exterior Angles ! Interior Angles ! Transversal ! Alternate Angles ! Parallel Lines ! Skew Lines In this lesson, we are going to learn about angle and line relationships. The next few lessons are totally devoted to parallel lines. We are going to discover all sorts of interesting things that can be done with parallel lines!
Section 10-1 In this section, we are learning the difference between the exterior angles of a triangle and its remote interior angles. The exterior angle is found by
extending one of the lines of the triangle. The angle created is the exterior angle (in purple above). The remote interior angles are found on the inside of this triangle. They are the two angles that are NOT supplementary to this angle. The most important thing that you should try to remember from this section is the Exterior Angle Inequality Theorem. This tells us that the exterior angle is greater than either of its two remote interior angles. Theorem 10-1 Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles. Let’s try a few problems! 1
Problem: Which angles in this diagram are definitely larger than ∠CBA ?
C
1
3
B A
2
Solution: Using the Exterior Angle Inequality Theorem, we know that the measure of an exterior angle is larger than that of either of its interior angles. Since ∠CBA is a remote interior angle of both ∠1 and ∠3, it is smaller than both of those angles.
Problem: For what triangle is ∠GHF an exterior angle?
G H
E
F
Solution: ∠GHF is an exterior angle for ∆EHF , because ∠EHF and ∠GHF are a linear pair, and EG is an extension of EH .
Section 10-2
In section 10-2, we learn about parallel lines and skew lines. Because parallel lines never cross, they have very interesting properties. Especially
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when these lines are cut by something called a transversal. The theorems described in this section show us some of the qualities of parallel lines.
Theorem 10-2 Alternate Interior Angles Theorem If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Theorem 10-3 Converse of the Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Problem: If ∠1 ≅ ∠2 which two segments are parallel?
B
C 2
A
1
D
Solution: Using the result of Theorem 10-2, we can see that AC is a transversal. Since angles 1 and 2 are alternate interior angles, and ∠1 ≅ ∠2 , we can assume that BC is parallel to AD .
Problem: If these two lines are parallel, and the segment is a transversal, what do we know about the relationship between ∠1 and ∠2 ?
1 2
3
Solution: Well, the first thing we notice is that ∠1 and ∠2 are alternate interior angles. Now we can see that, by Theorem 10-3, ∠1 and ∠2 are congruent.
Section 10-3 In this section, we are building on the knowledge we have acquired so far about angles. It should be really interesting, especially since we are already so good at figuring out the measures of angles! Theorem 10-4 Supplementary Interior Angles Theorem If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, then the lines are parallel. Theorem 10-5
Converse of the Supplementary Interior Angles Theorem If two parallel lines are cut by a transversal, then interior angles on the same side of the transversal are supplementary.
Theorem 10-6 Common Perpendicularity Theorem In a plane, if two lines are both perpendicular to a third line, then the lines are parallel. Theorem 10-7 Converse of the Common Perpendicularity Theorem In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also.
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Problem: If these lines are parallel, what is m∠1 + m∠2 ? 2 1
Solution: The sum of the measures of angles 1 and 2 is 180. This is because Theorem 10-4 states that these two angles must be supplementary, and the sum of two supplementary angles is 180.
Problem: Assume the relevant lines are parallel. If m∠1 = x + 5 m∠2 = 3x + 3 Find the measures of angles 1 and 2.
2 1
1
Solution: Due to Theorem 10-4, we know that m∠1 + m∠2 = 180. From this we can substitute in: x + 5 + 3x + 3 =180 4x + 8 = 180 4x = 172 x = 43 If we plug in 43 for x in the equations, we find that: m∠1 = 43 + 5 = 48 m∠2 = 129 + 3 = 132
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Problem: What can we conclude about the measure of ∠2 if: m∠1 = 41, m∠2 = 65 , and ∠1 and ∠ABC are a linear pair.
A
D
1
2
3
B
C
Solution: We know that the measure of angles 1, 2 and 3 must add up to 180. So, ∠2 = 180 − 41 − 65 = 74 .
We are now finished with Lesson 10. This is the part of geometry that is really fun. Solving these types of problems is just like playing games. In the next few lessons we are going to look at more of these types of puzzles and how they relate to triangles and polygons. Good luck on your submission!
Wonderful Website --- You simply MUST try it! The Mathematical Atlas
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Glossary Alternate Angles Alternate angles are nonadjacent angles on opposite sides of a transversal. Angles 1 and 4 are alternate interior angles. Angles 2 and 3 are alternate exterior angles. 2 1
4 3
Exterior Angle – An angle that forms a linear pair with an interior angle of a polygon. ∠1 is an exterior angle.
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1
2
3
Interior Angle – An angle on the inside of a polygon. In the figure above ∠ 2, ∠3, and ∠4 are interior angles. Linear Pair - A linear pair is a pair of adjacent angles such that two of the rays are opposite rays. ∠1 and ∠2 are a linear pair. 1
2
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Transversal A transversal is a line that intersects (or cuts) each of two other lines in different points. The blue line a transversal.
Parallel Lines Parallel lines are lines in the same plane that do not intersect. These lines are parallel.
Skew Lines Skew lines are lines that do not intersect, are not parallel, and are not coplanar. The lines on these planes do not intersect, but they are not parallel because they are in different planes (coplanar) so they are skew.
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