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1 Geo Ch 3 Angles formed by Lines Parallel lines are two coplanar lines that do not intersect. Skew lines are that are not coplanar and do not intersect. Transversal is a line that two or more lines at different points. Corresponding angles are two angles on the same side of the transversal, one interior, one exterior that are nonadjacent The following pairs of angles are corresponding angles: 2 and 6 4 and 8 1 and 5 3 and 7 Alternate interior angles are two angles that lie between the two lines on opposite sides of the transversal that are nonadjacent. The following pairs of angles are alternate interior angles: 3 and 6 4 and 5
2 Same side interior angles (consecutive interior) are two angles that lie between the two lines on the same side of the transversal that are nonadjacent. The following pairs of angles are same side interior angles: 3 and 5 4 and 6 Alternate exterior angles are two angles that lie outside the two lines on opposite sides of the transversal that are nonadjacent. The following pairs of angles are alternate exterior angles: 1 and 7 2 and 8 Perpendicular Lines If two lines intersect to form a linear pair of congruent angles, the lines are perpendicular. If two sides of two adjacent acute angles are perpendicular, The angles are complementary. If two lines are perpendicular, then they intersect to form Four right angles.
3 Example Find the value of x. x 60 Since the exterior sides of the two adjacent angles are perpendicular, the angles are complementary. x + 60 = 90 x = 30 Angles Pairs formed by Parallel lines Something interesting occurs if the two lines being cut by a transversal happen to be parallel. It turns out that every time I measure the corresponding angles, they turn out to be equal. You might use a protractor to measure the corresponding angles below. Since that seems to be true all the time and we can t prove it, we ll write it as an axiom a statement we believe without proof.
4 Postulate If two parallel lines are cut by a transversal, the corresponding angles are congruent. m l 2 1 t 1 2 Now let s take this information and put it together and see what we can come up with. Proofs: Alternate Interior Angles Let s see, we ve already learned vertical angles are congruent and corresponding angles are congruent if they are formed by parallel lines. Using this information we can go on to prove alternate interior angles are also congruent if they are formed by parallel lines. What we need to remember is drawing the picture will be extremely helpful to us in the body of the proof. Let s start. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent. By drawing the picture of parallel lines being cut by a transversal, we ll label the alternate interior angles. t 2 l 1 m The question is, how do we go about proving 1 2? Now this is important. We need to list on the picture things we know about parallel lines. Well, we just learned that corresponding angles are congruent when they are formed by parallel lines. Let s use that information and label an angle in our picture so we have a pair of corresponding angles.
5 2 t 3 l 1 m Since the lines are parallel, 1 and 3 are congruent. 2 and 3 are vertical angles. They are congruent. That means 1 3 because they are corresponding angles and 2 3 are congruent because they are vertical angles, that means 1 must be congruent to 3. That would suggest that Now we have to write that in two columns, the statements on the left side, the reasons to back up those statements on the right side. Let s use the picture and what we labeled in the picture and start with what has been given to us, line l is parallel to m. Statements 1. l ll m Given 1 and 2 are alt int s Reasons 2. 1 and 3 Def of corr. s are corr. s Two ll lines, cut by t, corr. s Vert s Transitive Prop
6 Is there a trick to this? Not at all. Draw your picture, label what s given to you, then fill in more information based on your knowledge. Start your proof with what is given, the last step will always be your conclusion. Now, we have proved vertical angles are congruent, we accepted corresponding angles formed by parallel lines are congruent, and we just proved alternate interior angles are congruent. Could you prove alternate exterior angles are congruent? Try it. Write the theorem, draw the picture, label the alternate exterior angles, add more information to your picture based on the geometry you know, identify what has been given to you and what you have to prove. Let s write that as a theorem. If two parallel lines are cut by a transversal, the alternate exterior angles are congruent. If we played some more in the world of angles being formed by parallel lines, we might find an interesting relationship between the same side interior angles. Let s take a look If you filled in all the angles formed by those parallel lines being cut by a transversal, what relationship do you see when looking at the same side interior angles? Let s write that as a theorem. If two parallel lines are cut by a transversal, the same side interior angles are supplementary. Summarizing, we have: If two parallel lines are cut by a transversal:, then the corresponding s are congruent. then the alt. int. s are congruent. then the alt. ext. s are congruent. then the same side int. s are supplementary
7 Example Find the value of x if the two lines are parallel. 2x 130 Since the lies are parallel, the alternate exterior angles are congruent. 2x = 130 x = 65 Example Given the lines are parallel, find the value of x. 5x 30 x + 60 Since the lines are parallel, the same side interior angles are supplementary. 5x 30 + x + 60 = 180 6x + 30 = 180 6x = 150 x = 25
8 Showing lines are Parallel Postulate If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. This is the converse of the postulate that read; if two parallel lines are cut by a transversal, the corresponding angles are congruent. Now what I will accept as true is if the corresponding angles are congruent, the lines must be parallel. The converse of a conditional is not always true, so this development is fortunate. As it turns out, the other three theorems we just studied about having parallel lines converses are also true. If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel. If two lines are cut by a transversal so that the alternate exterior angles are congruent, then the lines are parallel. If two lines are cut by a transversal so that the same side interior angles are supplementary, then the lines are parallel. 4 Ways to Show Lines are Parallel Now I have four ways to show lines are parallel, corresponding s congruent, alternate interior s congruent, alternate exterior s congruent, same side interior s supplementary. Proving the converse of the Alternate Interior Angles theorem. If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel. Given : l Prove: l m 3 m
9 Using the postulate, if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. I need to have corresponding angles in the proof (picture), so I insert l 3 m 1 3 was given. 1 and 2 are vertical angles. 1 2 By substitution, 2 3. By postulate, if the corresponding angles are congruent, the lines are parallel. Statements Reasons Given 2. 1 and 2 are vertical angles Def. vertical angles Vert angles are congruent 4. 2 and 3 are corr. angles Def. corr. Angles Substitution 6. l m Corr. angles congruent
10 If two lines are parallel to the same line, then they are parallel to each other. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Slope Slope is the ratio of the vertical change (rise) to the horizontal change (run). slope = rise run slope = y 2 y 1 x 2 x 1 Example Find the slope of the line that passes through (2, 1) and (6, 5). m = m = 4 4 m = 1
11 Parallel & Perpendicular Lines Postulate Two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Postulate Two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Vertical and horizontal lines are perpendicular. Writing an Equation of a Line Slope has been defined as slope = y 2 y 1 x 2 x 1. Using that relationship, we have y y 1 x x 1 = m Multiplying both sides the common denominator y y 1 = m(x x 1 ) Point-Slope form of a Line Example Find an equation of a line that passes through (2, 5) with slope 3. Using the Point-Slope form of a Line; y y 1 = m(x x 1 ) and substituting the given values, we have y 5 = 3(x 2) y 5 = 3x 6 y = 3x 1 Example Find an equation of a line that passes through (5, 1) and is perpendicular to y = 1 3 x + 2 Using the Point-Slope form of a Line; y y 1 = m(x x 1 ) and substituting the given values, we have y + 1 = 3(x 5) y + 1 = 3x + 15 y = 3x + 14
12 When solving an equation for y, the coefficient of the x-term is the slope of the line. In the equation; y = 3x 1, the slope is 3, the y intercept is (0, 1). Any lines parallel to that line will have slope 3. In the equation y = 1 x + 2, the coefficient of the x-term is 1/3, the slope is 1/3, the y 3 intercept is (0, 2). Any line parallel to that line will have slope 1/3. Any line perpendicular to that line will have slope 3/1 or 3.
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