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Transcript

Chapter 3 Perpendicular and Parallel Lines Chapter Objectives Identify parallel lines Define angle relationships between parallel lines Develop a Flow Proof Use Alternate Interior, Alternate Exterior, Corresponding, & Consecutive Interior Angles Calculate slopes of Parallel and Perpendicular lines Lesson 3.1 Lines and Angles Lesson 3.1 Objectives Identify relationships between lines. Identify angle pairs formed by a transversal. Compare parallel and skew lines. Lines and Angle Pairs Alternate Exterior Angles – because they lie outside the two lines and on opposite sides Transversal 1 2 4 3 5 6 8 7 Corresponding Angles – because they lie in corresponding positions of each intersection. of the transversal. Alternate Interior Angles – because they lie inside the two lines and on opposite sides of the transversal. Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal. Example 1 Determine the relationship between the given angles 1) 3 and 9 1) 2) 13 and 5 2) 3) Alternate Interior Angles 5 and 15 4) 5) Corresponding Angles 4 and 10 3) 4) Alternate Interior Angles Alternate Exterior Angles 7 and 14 5) Consecutive Interior Angles Parallel versus Skew Two lines are parallel if they are coplanar and do not intersect. Lines that are not coplanar and do not intersect are called skew lines. These are lines that look like they intersect but do not lie on the same piece of paper. Skew lines go in different directions while parallel lines go in the same direction. Example 2 Complete the following statements using the words parallel, skew, perpendicular. 1) Line WZ and line XY are _________. 1) 2) Line WZ and line QW are ________. 2) 3) perpendicular Line TS and line ZY are __________. 6) 7) parallel Plane RQT and plane WQR are _________. 5) 6) skew Plane WQR and plane SYT are _________. 4) 5) perpendicular Line SY and line WX are _________. 3) 4) parallel skew Line WX and plane SYZ are __________. 7) parallel. Parallel and Perpendicular Postulates: Postulate 13-Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point that is parallel to the given line. Parallel and Perpendicular Postulates: Postulate 14-Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Homework 3.1 In Class 2-9 p132-135 Homework 10-31 Due Tomorrow Lesson 3.2 Proof and Perpendicular Lines Lesson 3.2 Objectives Develop a Flow Proof Prove results about perpendicular lines Use Algebra to find angle measure Flow Proof 5 7 6 A flow proof uses arrows to show the flow of a logical argument. Each reason is written below the statement it justifies. GIVEN: 5 and 6 are a linear pair 6 and 7 are a linear pair PROVE: 5 7 5 and 6 are a linear pair 6 and 7 are a linear pair Given 1. 5 and 6 are a linear pair 6 and 7 are a linear pair 1. Given 2. 5 and 6 are supplementary 6 and 7 are supplementary 2. Linear Pair Postulate 3. 5 7 3. Congruent Linear Pair Supplements Postulate Theorem 5 and 6 are supplementary Given 6 and 7 are supplementary Linear Pair Postulate 5 7 Congruent Supplements Theorem Theorem 3.1: Congruent Angles of a Linear Pair If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. g h So g h Theorem 3.2: Adjacent Angles Complementary If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Theorem 3.3: Four Right Angles If two lines are perpendicular, then they intersect to form four right angles. Homework 3.2 None! Move on to Lesson 3.3 Lesson 3.3 Parallel Lines and Transversals Lesson 3.3 Objectives Prove lines are parallel using tranversals. Identify properties of parallel lines. Postulate 15: Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. You must know the lines are parallel in order to assume the angles are congruent. 1 2 4 3 5 6 8 7 Theorem 3.4: Alternate Interior Angles If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Again, you must know that the lines are parallel. If you know the two lines are parallel, then identify where the alternate interior angles are. Once you identify them, they should look congruent and they are. 1 2 4 3 5 6 8 7 Theorem 3.5: Consecutive Interior Angles If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Again be sure that the lines are parallel. They don’t look to be congruent, so they MUST be supplementary. 1 2 4 3 180o = + + = 180o 5 6 8 7 Theorem 3.6: Alternate Exterior Angles If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Again be sure that the lines are parallel. 1 2 4 3 5 6 8 7 Theorem 3.7: Perpendicular Transversal If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Again you must know the lines are parallel. That also means that you now have 8 right angles! Example 3 Find the missing angles for the following: 120o 60o 120o 120o 140o 140o 105o 110o 70o 105o 110o Homework 3.3 In Class 3-6 p146-149 Homework 8-26, 34-44 even Due Tomorrow Quiz Wednesday Lessons 3.1-3.3 Emphasis on 3.1 & 3.3 Lesson 3.4 Proving Lines are Parallel Lesson 3.4 Objectives Prove that lines are parallel Recall the use of converse statements Postulate 16: Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. You must know the corresponding angles are congruent. It does not have to be all of them, just one pair to make the lines parallel. 1 2 4 3 5 6 8 7 Theorem 3.8: Alternate Interior Angles Converse If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Again, you must know that alternate interior angles are congruent. 1 2 4 3 5 6 8 7 Theorem 3.9: Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. Be sure that the consecutive interior angles are supplementary. 1 2 4 3 180o = + + = 180o 5 6 8 7 Theorem 3.10: Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then lines are parallel. Again be sure that the alternate exterior angles are congruent. 1 2 4 3 5 6 8 7 Example 4 Is it possible to prove the lines are parallel? If so, explain how. Yes they are parallel! Yes they are parallel! Because Alternate Interior Angles are congruent. Because Corresponding Angles are congruent. Corresponding Angles Converse Alternate Interior Angles Converse Yes they are parallel! Because Alternate Exterior Angles are congruent. Alternate Exterior Angles Converse No they are not parallel! No relationship between those two angles. Example 5 Find the value of x that makes the m n. x = 2x – 95 (AIA) 100 = 4x – 28 (CA) (3x + 15) + 75 = 180(CIA) -x = –95 (SPOE) 128 = 4x (APOE) 3x + 90 = 180 (CLT) x = 95 (DPOE) x = 32 (DPOE) 3x = 90 (SPOE) x = 30 (DPOE) Directions do not ask for reasons, I am showing you them because I am a teacher!! Homework 3.4 In Class 1, 3-9 p153-156 Homework 10-35, 37, 38 Due Tomorrow Lesson 3.5 Using Properties of Parallel Lines Lesson 3.5 Objectives Prove more than two lines are parallel to each other. Identify all possible parallel lines in a figure. Theorem 3.11: 3 Parallel Lines Theorem If two lines are parallel to the same line, then they are parallel to each other. This looks like the transitive property for parallel lines. If p // q and q // r, then p // r. p q r Theorem 3.12: Parallel Perpendicular Lines Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If m p and n p, then m // n. m n p Finding Parallel Lines Find any lines that are parallel and explain why. b a x y z 125o 55o c Results x // y Corresponding Angles Converse Postulate 16 y // z Consecutive Interior Angels Converse Theorem 3.9 x // z 3 Parallel Lines Theorem Theorem 3.11 b // c Alternate Exterior Angles Converse Theorem 3.10 Homework 3.5 In Class 16, 19 p160-163 Homework 8-24, 33-36, 43-51 Due Tomorrow Lesson 3.6 Parallel Lines in the Coordinate Plane Lesson 3.6 Objectives Review the slope of a line Identify parallel lines based on their slopes Write equations of parallel lines in a coordinate plane Slope Recall that slope of a nonvertical line is a ratio of the vertical change divided by the horizontal change. It is a measure of how steep a line is. The larger the slope, the steeper the line is. Slope can be negative or positive whether or not the lines slants up or down. Remember that slope is often referred to as Which really means y2 – y1 x2 – x1 rise run =m Which we find it in an equation by looking for m. y = mx + b Example of Slope You are given two points: A(1 , 2 ) B(3 , 8 ) 1 Now label each point as 1 and 2. 2 Then substitute as the formula for slope tells you. y2 – y1 x2 – x1 = 8 3 – – 2 1 = 6 2 = 3 Postulate 17: Slopes of Parallel Lines Postulate In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. m = -1 m = -1 m = undefined Writing an Equation in Slope Intercept Form You will be given Slope Or at least two points so you can calculate slope y-intercept slope y = mx + b Your final answer should always appear in this form. y-intercept This is the point at which the line touches the y-axis. Writing an Equation Given Slope and 1 Point For this, you will be given the slope Or have to determine it from and equation Or determine it from a set of two points You will also be given 1 point through which the line passes To solve, use your slope-intercept form to find b y = mx+b Plug in Slope for m. The x-value from your point for x. The y-value from your point for y. Solve for b using algebra When finished, be sure to rewrite in slope intercept form using your new m and b. Leave x and y as x and y in your final equation. Example Example 5 p167 Write an equation of the line through the point (2,3) that has a slope of 5. y = mx + b y = 5x + b 3 = 5(2) + b 3 = 10 + b b = -7 So, your final answer is: y = 5x - 7 Homework 3.6 WS Due Tomorrow Lesson 3.7 Perpendicular Lines in the Coordinate Plane Lesson 3.7 Objectives Use slope to identify perpendicular lines Write equations of perpendicular lines Postulate 18: Slopes of Perpendicular Lines Postulate In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is –1. Vertical and horizontal lines are perpendicular. Identifying Perpendicular Lines There are two ways to identify perpendicular lines The product of the slopes equal –1 (3/2)(-2/3) = -1 Or to get from one slope to the other, you find the negative reciprocal Remember that reciprocal flips the number. Well now you flip it and make it negative! 3/2 -2/3 Determining Perpendicular Lines Remember there are two ways to identify perpendicular lines Multiply the slopes together If the answer is –1, they are perpendicular Verify that the slopes are negative reciprocals of each other Take one of the slopes, flip it over, and make it negative. If the answer matches the other slope, they are perpendicular. Example (#25 p176) y = 3x y = - 1 /3 x – 2 (3)(-1/3) = -1 Perpendicular Or 3 1/3 -1/3 Check! Tougher Example Need to change into slope-intercept form. y = mx + b Example 4 P173 Decide whether the lines are perpendicular Line s: 5x + 4y = 3 Line r: 4x + 5y = 2 5y = -4x + 2 y = - 4 /5 x + 2 /5 Subtract x-term from both sides 4y = -5x + 3 Get y to be alone by dividing off the coefficient y = - 5 /4 x + 3 /4 Now multiply their slopes (-4/5)(-5/4) = 20/ 20 =1 NOT Perpendicular Homework 3.7 WS Due Tomorrow Test Thursday January 29th