5 and Parallel and Perpendicular Lines

Transcription

1 Ch 3: Parallel and Perpendicular Lines 3 1 Properties of Parallel Lines 3 Proving Lines Parallel 3 3 Parallel and Perpendicular Lines 3 Parallel Lines and the Triangle Angles Sum Theorem 3 5 The Polgon Angle Sum Theorem 3 6 Lines in the Coordinate Plane 3 7 Slopes of Parallel and Perpendicular Lines 3.8 Constructing Parallel and Perpendicular Lines 3 1 Properties of Parallel Lines: Focused Learning Target: I will be able to Standards: Geom 7.0 Identifing angles formed b two lines and a transversal Proving and using properties of parallel lines Vocabular: Corresponding Angles Transversal Two Column Proof Alternate Interior Angles Alternate Eterior Angles Same Side Interior Angles Same side Eterior Angles 5 and 8 are alternate eterior angles 5 and 7 are same side eterior angels

2 Identifing angles: I ll do one: We ll do one together: You tr: Using the diagram. Identif which angle forms a pair of alternate interior angels with 1. Identif which angle forms a pair of alternate eterior with. Identif which angle forms a pair of same side eterior angle with 5. Identif which angel forms a pair of corresponding angle with. Writing Two column proofs:

3 I ll do one: Statement Reason We ll do one together: Given: a b Prove: 1 and 3 are supplementar Statement Reason You tr one: Given: a b Prove: 1 and are supplementar Statement Reason

4 Using algebra to find angle measures I ll do one: We ll do one: You tr: Find the value of and in the diagram. Justif each answer. Find the value of a, b and c. Justif each answer. Given: l m Find the values and. Justif each answer. Then find the measures of the angles. 3 Proving Lines Parallel Focused Learning Target: I will be able to Standards: Geom 7.0 Using a transversal in proving lines parallel

5 I ll do one: We ll do one together: You tr: Which lines or segments are parallel? Justif our answer with a theorem or postulate. Which lines or segments are parallel? Justif our answer with a theorem or postulate. Which lines or segments are parallel? Justif our answer with a theorem or postulate. Using Algebra I ll do one: We ll do one together: You tr: Find the value of for which l m Find the value of for which a b Find the value of for which l m

6 3 3 Parallel and Perpendicular Lines: Focused Learning Target: I will be able to Relate parallel and perpendicular lines Standards: Geom 7.0 I ll do one: Prove Theorem r t, s t 1. Given We ll do one together: Using Theorem 3 11

7 1. a b, b c, and c d 1. Given We ll do another one: Prove Theorem 3 11: 1. a b 1. Given.. Definition of right angles 3. b c 3. Given.. Corresponding Angles Postulate 5. a c 5. Now that we have proven Theorems 3 10 and 3 11, we can use them in an proofs from now on! You Tr: 1. t r, t s 1.. r s. You Tr: 1. a b, b c 1. a c c d 3. a d.

8 Let s tr one more together: 1. q p 1. Given p r q r 6. 3 Parallel Lines and the Triangle Sum Theorem: Focused Learning Target: I will be able to Standards: Classif triangles based on their side and angle measures Geom 1.0 Find the measures of angles in a triangle Geom 13.0 Use eterior angles of triangles to find the measures of missing interior and eterior angles of a triangle Vocabular: Equiangular Triangle Equilateral Triangle Acute Triangle Isosceles Triangle Right Triangle Scalene Triangle Obtuse Triangle Eterior Angle of a Triangle Remote Interior Angles of a Triangle Triangles classified b their angles:

9 Triangles classified b their sides: Each triangle can now be classified based on the characteristics of its angles and sides, like a first and last name. I ll tr one: We ll tr one: You Tr one: Some combinations of triangle classifications are not possible, can ou think of an? No matter what tpe of triangle ou have, the all have the same interior angle sum, which is described in the following theorem. I ll tr one: We ll tr one together: You tr: Find the value of. Then find the Find the values of,, and z. Find the values of,, and z. measures of the angles. If ever triangle has 3 interior angles, the must also have 3 eterior angles. An eterior angle of a triangle is an angle formed b etending a side. As a result, the two non adjacent interior angles of an eterior angle are called the remote interior angles.

10 An eterior angle of a triangle has an interesting relationship with its remote interior angles. I ll do one: We ll tr one: You Tr one: Find the missing angle measure. Find the missing angle measure. Find the missing angle measure. Note: An equilateral polgon has all sides congruent. An equiangular polgon has all angles congruent. A regular polgon is both equilateral and equiangular. 3 6 Lines in the Coordinate Plane: Focused Learning Target: I will be able to Standards: Geom 17.0 Graph lines given their equations Write equations of lines Vocabular: Slope intercept form Standard form Graphing in slope intercept form: I ll do one: Graph 3 X intercept 1. Identif the slope (m) and the intercept (b). m = b =. locate the intercept and plot the point Point slope form 3. Use the slope to find the net two points.. Draw the line through the points We ll do one: You tr:

11 Graph 3 Graph: 3 m = b= m b m b You Tr: Graph: Graph: You Tr: Graphing Lines Using Intercepts: Standard form of a linear equation: a linear equation written in the form A B C, where A and B are not both zero and A, B and C are all real numbers. E: 3 5 is written in standard form. 5 intercept: the intercept is the point at which the line crosses the ais. I ll do one: We ll do one: You tr:

12 intercept = intercept = intercept = To find the & intercepts algebraicall: 1. Substitute 0 in place of, then solve for. The result is the intercept.. Substitute 0 in place of, then solve for. The result is the intercept. I ll do one: We ll do one: You tr: Find the & intercepts: Find the & intercepts: Find the & intercepts: Graph an equation b finding the & intercepts: I ll do one: Graph: We ll do one together: Graph: You Tr: To transform an equation of a line into slope intercept form, solve for : I ll do one: We ll do one together: You tr: Write the equation in slopeintercept form: Write the equation in slopeintercept form: 5 7 Write the equation in slopeintercept form: 5 0

13 Writing the equation of a line given two points on the line: find the slope of the line, then use the slope and one of the points and plug them into point slope form of the line. This will give ou the intercept. I ll do one: We ll do one together: You tr: Write the equation of the line that passes through (, 3) and (5, 5) Write the equation of the line that passes through (3, ) and (, 5) Write the equation of the line that passes through (5, 7) and (18, 13) Equations of horizontal and vertical lines: horizontal and vertical lines are special cases. Draw the line. When drawn, one of the coordinates will repeat. The repeating coordinate is the equation. I ll do one We ll do one together: You tr: Write the equation of the horizontal Write the equation of the vertical Write the equation of the horizontal line that passes through (5, 7). line that passes through (3, 5). line that passes through (0, 7). 3 7 Slopes of Parallel and Perpendicular Lines: Focused Learning Target: I will be able to Relate slope and parallel lines Relate slope and perpendicular lines Write converses of conditional statements Slopes of Parallel Lines: If two nonvertical lines are parallel, their slopes are equal. If the slopes of two distinct nonvertical lines are equal, then the lines are parallel An two vertical lines are parallel You can test whether nonvertical lines are parallel b comparing slopes.

14 Eample 1: Checking for Parallel Lines I ll do one: We ll do one together: You tr: Are lines l 1 and l parallel? Eplain. Are lines l 1 and l parallel? Eplain. Are lines l 1 and l parallel? Eplain. Slope intercept form allows ou to compare slopes easil in order to decide whether lines are parallel. Slope intercept form is =m+b where m is the slope. Eample : Writing equations of parallel lines: I ll do one: We ll do one together: ou tr: Write an equation for the line Write an equation for the line parallel to = that contains ( 3, 5). parallel to = that contains (6, 1). Write an equation for the line parallel to = +3 that contains (1, ). Slopes of perpendicular lines: If two nonvertical lines are perpendicular, the product of their slopes is 1. If the slopes of two lines have a product of 1, the lines are perpendicular. An horizontal line and vertical line are perpendicular. To find the slope of a line containing the points ( 1, 1 ) and (, ), use the formula: m = 1 1

15 Eample 3: Checking for Perpendicular Lines: I ll do one: We ll do one together: You tr: Lines l 1 and l are neither vertical nor horizontal. Are the perpendicular? Eplain. Lines l 1 and l are neither vertical nor horizontal. Are the perpendicular? Eplain. Lines l 1 and l are neither vertical nor horizontal. Are the perpendicular? Eplain.

16 You can write an equation for a line perpendicular t a given line. Eample : Writing Equations for Perpendicular Lines I ll do one: We ll do one together: You tr: Write an equation for the line Write an equation for the line Write an equation for the line perpendicular to XY that contains perpendicular to XY that contains perpendicular to XY that contains point Z. point Z. point Z. XY : 3 + = 6, Z(3, ) 3 XY : = +, Z(1, 8) XY : + = 0, Z(, 1) 3 8 Constructing Parallel and Perpendicular Lines: Focused Learning Target: I will be able to Standards: Geom 16.0 Construct parallel lines Construct perpendicular lines You can use what ou know about parallel lines, transversals, and corresponding angles to construct parallel lines. To construct the perpendicular to a given line through a given point not on the line: 1. Open our compass to a size greater than the distance from Q to l. With the compass point on point Q, draw an arc that intersects l at two points. Label the points E and F. Place the compass on point E and make an arc 3. Keep the same compass setting. With the compass tip on F, draw an arc that intersects the arc from Step. Label the point of intersection G.. Draw line QG I ll do one: We ll do one together: You tr: Construct a line perpendicular to line l through point Q. Construct a line perpendicular to line l through point Q. Construct a line perpendicular to line l through point Q.

17 To construct the perpendicular to a given line at a given point on the line: 1. Put the compass point on point T. Draw arcs intersecting l in two points. Label the points A and B.. Open the compass wider. With the compass tip on A, draw an arc above point T. 3. Without changing the compass sitting, place the compass point on point B. Draw an arc that intersects the arc from Step. Label the point of intersection C.. Draw line CT I ll do one: We ll do one together: You Tr: Construct a line perpendicular to Construct a line perpendicular to Construct a line perpendicular to line l at point T. line l at point T. line l at point T. To construct a line parallel to a given line and through a given point not on the line: 1. Label two points H and J on line l. Draw line HK. Construct 1 with verte at K so that m 1 = m KHJ and the two angles are corresponding angles. Label the line ou just constructed m I ll do one: We ll do one together: You tr: Construct a line parallel to line l and Construct a line parallel to line l and Construct a line parallel to line l and through point K. through point K. through point K.

18 Now ou can put our constructions together to construct polgons I ll do one: Construct a quadrilateral with one pair of parallel sides of lengths a and the other pair length b. We ll do one together: Construct a right triangle with leg lengths of b and c. You Tr: Construct a square with side lengths of b.