3-1, 3-2, 3-3, 3-4, 3-5, 3-6 This section is all about congruency

There are 4 postulates and theorems

All of them are dependent on if 2 parallel lines are cut by a transversal -Corresponding Angles Postulate:

If 2 parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. You can usually see that the corresponding angles are congruent by just looking at the angles, but make sure the lines that are cut by the transversal are parallel. Angles Formed by Parallel Lines and Transversals 3-2

-Alternate Interior Angles Theorem:

If 2 parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

-Alternate Exterior Angles Theorem:

If 2 parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent.

-Same-Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the 2 pairs of same-side interior angles are supplementary. 3-1 Continued -Transversal: A line that intersects two coplanar lines at two different points. For example, in Figure B, line FE is a transversal. Figure B -Corresponding Angles: Angles that lie on the same sided of the transversal and on the same side of the two parallel lines. In Figure B, angle 6 and angle 2 are corresponding angles. -Alternate Interior Angles: Nonadjacent angles that lie on opposite sides of the transversal and are inside the two parallel lines. For example, in Figure B, angle 3 and angle 6 are alternate interior angles. -Alternate Exterior Angles: Angles that lie on opposite sides of the transversal and lie outside the two parallel lines. In Figure B, angles 1 and 8 are alternate exterior angles. -Same-Side Interior Angle: (a.k.a. consecutive interior angles) Angles that lie on the same side of the transveral and are inside the two parallel lines. For example, in Figure B, angles 4 and 6 are same-side interior angles. Transversal These angles are in between

the two parallel lines (interior) 3-1 Example Problems Identify one pair of the following: 1. Skew Lines Lines EF and GC Identify one pair of the following: 2. Alternate Interior Angles Angles 4 and 6 3. Corresponding Angles Angles 1 and 5 All angles outside the

interior are considered exterior angles 3-2 Example Problems Chapter 3 Parallel and Perpendicular Lines By Michael Yoon, Umesh Narayan, Armaun Rouhi, and Francis Nguyen 3-1 Lines and Angles -Parallel Lines- Coplanar lines that never intersect. Coplanar lines are lines that are in the same plane. For example, in Figure A, line CG is parallel to line DH and both of them lie on the same plane. -Perpendicular Lines- Lines that intersect at 90° angles. In Figure A, lines CB and AB are perpendicular. -Skew Lines- Lines that are not coplanar, do not intersect one another, and are not parallel. In Figure A, lines BF and EH are skew lines. -Parallel Planes- Planes that do not intersect. In Figure A, plane DAB & HEF are parallel planes. Figure A 3-3 3-4 Find each angle measure: -Perpendicular Bisector: A line perpendicular to a segment at the segment’s midpoint. 1. m. ang. FCE -The shortest segment from a point to a line is perpendicular to the line. The length of the perpendicular segment from the point to the line. Line AP is the Perpendicular bisector of line CB (say that CP is congruent to PB), and it is also the shortest segment. 3-6 2. m. ang. DCB m. ang. GBE=70°

m. ang. DCB=70° Given

Alternate Interior Angles Theorem 3-6 Equations for Lines 3. m. ang. EDG -Explanation of Thm. 3-4-1:

Basically, this theorem states that if two intersecting lines form a pair of congruent angles, then those lines have intersected each other at a 90 degree angle, making them perpendicular to one another. -Explanation of the Perpendicular Transversal Thm. :

In this theorem, it says that if a transversal intersects one of two parallel lines at a 90 degree angle, then it is intersecting the second parallel line at a 90 degree angle as well. -Explanation of Thm. 3-4-3:

In simpler terms, this theorem is the converse of the Perpendicular Transversal Theorem. So, this theorem states that if two lines are perpendicular to the same line/transversal, then the two lines must be parallel to eachother. Again, this theorem is the converse of the Perpendicular Transversal Theorem. Theorems Figure A Figure B A B m. ang. FBC=75°

m. ang. EDG=75° Given

Alternate Exterior Angles Theorem Parallel lines have same slopes 4. m. ang. GDB Explanation: If two lines on the same plane are cut by a tranversal and the corresponding angles are congruent then the lines are parallel. Explanation: If two lines on the same plane are cut by a transversal and the alternate interior angles are congruent than the two lines are parallel. Intersecting lines have different slopes Explanation: If two lines on the same plane are cut by a transversal and the alternate exterior angles are congruent then the two lines are parallel. Coinciding lines have the same slopes and y-intercept Explanation: If two lines of the same plane are cut by a transversal and the same-side interior angles are supplementary then the two lines are parallel. AIDS 3-3 Example Problems 3-4 Example Problems What Information do you need to conclude that K is parallel to L? Explanation- Slope-intercept form is taking the slope and the y-intercept of a line,and making it into an equation. Substitute both the slope and the y-intercept for their correct variables. 3-5 Slopes of Lines Explanation- Point slope form is taking a point and the slope of a line, and making it into an equation. Substitute both the point and slope for their correct variables. Find x and y in the diagram: 10x=90 x=9 8(9)+4y=90 72+4y=90z 4y=18 y=4.5 Formulas x=5, y=4 x=9, y=4.5 x=6, y=9 Remember for same-side interiors angles they are SUPPLEMENTARY not congruent Real Life Examples Different Types of Lines This section is the reverse (converse) of 3-2. 3-2 tells if the angles are congruent while 3-3 tells if the lines are parallel. Determine whether YX is perpendicular to VU. YX is perpendicular to VU because the product of the slopes equals -1. Same Side Interior Angles

Simplify Explanation: For two lines to be parallel in a coordinate plane they have to have the same slope. Explanation: For two lines to be perpendicular in a coordinate plane the product of the two slopes has to be equal to -1. 2x - 135 + x = 180

3x -135 = 180

+135 +135

3x = 135

x = 105

m. ang. GDB = 105° m. ang. GBE=70°

m. ang. FCE=70° Given

Corresponding Angles Postulate Example Problem Angle 7 is supplementary to Angle 6

Angle 1 is congruent to Angle 6

Angle 3 is congruent to Angle 5

Angle 4 is congruent to Angle 2 This section is all about perpendicular lines within transversals.

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