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Chapter 6 – Parallel Lines

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Chapter 6 – Parallel Lines

By Grace Grimaldi and Matt O’Donnell

- symmetric
- two points are symmetric with respect to a lineiff the line is the perpendicular bisector of the line segment connecting the two points

- In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment

- Construction
- To construct a line perpendicular to a given line through a point

- parallel
- two lines are parallel iff they line in the same plane and do not intersect

- transversal
- a line that intersects two or more lines in different points

l1

l2

t

Ways to Prove Lines Parallel

- equal corresponding angles mean that lines are parallel
- equal alternate interior angles mean that lines are parallel
- supplementary same-side interior angles mean that lines are parallel

- In a plane, two lines perpendicular to a third line are parallel to each other

- Construction
- To construct a line parallel to a given line through a given point

- The Parallel Postulate
- Through a point not on a line, there is exactly one line parallel to the given line

. P

parallel

not parallel

- In a plane, two lines parallel to a third line are parallel to each other.

parallel

parallel

parallel

parallel

parallel

parallel

- parallel lines ⟷ equal corresponding angles

- parallel lines ⟷ equal alternate interior angles

- In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other
- This is easy to determine because if one of the perpendicular lines forms a 90 degree angle with the transversal than the corresponding angle on the parallel line must also be 90 degrees making it perpendicular to the transversal

- parallel lines ⟷ supplementary same-side interior angles

- The Sum Angle Theorem
- the sum of the angles of a triangle is 180°
- the third angle of any triangle can be found by adding up the sum of the two known angles and subtracting that from 180

- If two angles of one triangle are equal to two angles of another triangle, the third angles are equal.

- The acute angles of a right triangle are complementary

- Each angle of an equilateral triangle is 60°
- 180/3 = 60

- Exterior angle - the angle between any side of a polygon and an extended adjacent side
- An exterior angle of a triangle is equal to the sum of the remote interior angles.

- AAS Congruence
- If two angles and the side opposite one of them are equal to the corresponding parts of another triangle, the triangles are congruent.
- If two of the angles are equal it follows that the third angles will be equal by corollary one of the angle sum theorem.

- Hypotenuse-Leg (HL) Congruence
- If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right triangle than the triangles are congruent.

- HL Congruence (continued)
- If the two triangles are joined along the two equal legs, an isosceles triangle is formed and the far bottom corners are shown to be equal, which shows the triangles to be congruent.

- This chapter focused on parallel lines and the relationships with angles that parallel lines form. The majority of the Chapter 6 Test were problems that dealt with finding the measures of angles in a figure given one, two or no measures of other angles in the figure.
- Important to Know
- Angle Sum Theorem
- AAS Congruence
- HL Congruence
- Exterior Angle Equality
- Parallel lines and the angles they form