Geometry Journal 3

1. Geometry Journal 3 Andres Cofiño

2. Parallel lines and parallel planesSkew lines • Parallel lines are lines that are coplanar (same plane) and never intersect with each other. • Parallel planes are planes that never intersect with each other. • Skew lines are lines that are not coplanar, not parallel and neither intersect with each other.

3. B A E F H C G AC and BH are parallel lines AFB and CGH are parallel planes ED and BF are skew lines D

4. Transversal • A transversal is a certain line that intersects two coplanar lines at two different points. When the transversal crosses two parallel lines, the angles formed are either congruent or supplementary. • Transversal  Transversal  • Transversal 

5. Types of Angles formed by a transversal • Corresponding angles- they lie on the same sides of the transversal and same sides of the lines being cut by the transversal. • Alternate Exterior angles- they lie on the opposite sides of the transversal and outside the lines being cut by transversal. • Alternate Interior angles- they are nonadjacent angles that lie on opposite sides of the transversal and between the lines being cut by the transversal. • Consecutive Interior angles- also known as “same-side interior angles are the ones that lie on the same side of the transversal and between the lines being cut by the transversal.

6. 1 and 3 are corresponding angles 1 and 8 are alternate exterior angles 2 and 7 are alternate interior angles 2 and 3 are consecutive interior angles/same-side interior angles

7. Corresponding Angles Postulate and Converse • The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal, the corresponding angles left are congruent. Its convers is also true and states that if two parallel lines are cut so they end up with congruent corresponding angles, then the two lines that are cut by the transversal are parallel.

8. 1 2 4 3 e a w l b m If l is parallel to m, then 1 and 2 are congruent. 6 5 If 4 and 3 are congruent, then w and e are parallel. If a is parallel to b, then 5 and 6 are congruent.

9. Alternate Interior Angles Theorem and Converse • The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal, the alternate interior angles that are left, are congruent. Its convers is also true and states that if two parallel lines are cut by a transversal so they end up with congruent alternate interior angles, then the two lines that are cut by the transversal are parallel.

10. If l is parallel to m, then 2 is congruent to 7 and 6 is congruent to 3. l 1 2 m l m If 2 and 8; 3 and 5 are congruent, then k is parallel to l.  3 4 If l is parallel to m, then 1 is congruent to 4 and 2 is congruent to 3.

11. Alternate Exterior Angles Theorem and Converse • The Alternate Exterior Angles Theorem states that, when two parallel lines are cut by a transversal, the alternate exterior angles left are congruent. The converse is also true and states that if two lines are cut by a transversal and leave us with congruent alternate exterior angles, then the lines cut by the transversal are parallel to each other.

12. If l is parallel to m, then 1 is congruent to 8 and 5 is congruent to 4. l 1 2 m l m If 1 is congruent to 7 and 4 is congruent to 6, then k is parallel to l.  3 4 If l is parallel to m, then 1 is congruent to 4 and 2 is congruent to 3.

13. Consecutive/Same-side interior Angles Theorem • If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. Its converse is also true and states that if the pairs of the consecutive interior angles formed are supplementary, then the two lines that are cut by a transversal are parallel.

14. If l is parallel to m, then 2 is supplementary to 3 and 6 is supplementary to 7. l 1 2 m l m If 2 is supplementary to 5 and 3 is supplementary to 8, then k is parallel to l. 3 4 If l is parallel to m, then 2 is supplementary to 1 and 4 is supplementary to 3.

15. Perpendicular Transversal Theorem • In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. • In a plane, if 2 lines are perpendicular to the same line, then the lines are parallel. If J is perpendicular to K and K is parallel to H, then J is perpendicular to H. H K J

16. If k is perpendicular to a and h is perpendicular to a, then k is parallel to h. If n is perpendicular to m and m is parallel to l, then n is perpendicular to l. h l m k n a

17. Transitive Property for Perpendicular and Parallel Lines • This property also applies to parallel and perpendicular lines because for example two lines are parallel to each other and another parallel to the second is added, then the third will also be parallel with the first line because of transitive property. Also to perpendicular lines, if there are two perpendicular lines and another is added; the first and third lines will be perpendicular as well. 4 3 3 2 2 1 1 3 is parallel to 2 and 2 is parallel to 1 so 3 is parallel to 1. 1 is parallel to 2 and 2 is parallel to 3 so 3 is parallel to 1.

18. z x c z isparallelto x

19. Slope Formula: m= y2 – y1 x2 – x1 You use the slope formula to find the steepness of a line. Relations with lines. Parallel lines have the same slope Two lines are parallel iff they have same slope 2 lines are perpendicular if slopes are negative reciprocals of each other Ex. (2,3) (8,5) = 5-3=2 8-2=6 = 2/6 (9,7) (1,3) = 3-7= -4 1-9=-8 =-4/8 = ½ (5,6) (3,4) = 4-6= -2 3-5=-2 = -2/-2=2/2= 1

20. Slope/intercept and Point/slope Forms The slope/intercept form is written like this : y=mx+b This form is best used when you already know the y-intercept and slope of the equation. The point/slope intercept form is written like this: Y-Y₁=m(X-X₁) This form is best used when you want to find out the slope of a line from point to point. Slope/intercept form ex. Y=3/2x-1 Go one down, 3 up and 2 right. Y=5/4x+3 Go 3 up, 5 up and 4 right Y=8/9x-2 Go 2 down, 8 up and 9 right Point/Slope form ex. Y+1= 3(x-4) Slope should be 3 and line goes through point ¼. Y-3=2/3(x+5) Slope should be 2/3 and line goes through point 3/5 Y-2=1/2(x-7) Slope should be ½ and line goes through point 2/7