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Parallel

Parallel. Chapter 3. Name ________________________. Geometry Block. Perpendicular lines. 3.1 identify Pairs of Lines and Angles. Vocabulary:. Parallel lines – two lines that do not intersect and are coplanar. The symbol // means “is parallel to.”.

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Parallel

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  1. Parallel Chapter 3 Name ________________________ Geometry Block Perpendicularlines

  2. 3.1 identify Pairs of Lines and Angles Vocabulary: Parallel lines – two lines that do not intersect and are coplanar. The symbol // means “is parallel to.” Perpendicular lines – two lines that intersect to form a right angle. Skew lines – two lines that do not intersect and are not coplanar. Parallel planes – two planes that do not intersect. A transversal is a line that intersects two or more coplanar lines at different points.

  3. 3.1 identify Pairs of Lines and Angles Lines m and n are parallel lines . Symbol: m ǁ n k Lines mand k are skew lines. n m T U Lines k and n are intersecting lines, And there is a plane (not shown) containing them. Planes Tand U are parallel planes. Symbol: T ǁ U

  4. Example 1 Identify relationships in space Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure appear to fit the description? Line(s) parallel to and containing point A. Lines(s) skew to and containing point A. Line(s) perpendicular to and containing point A. Plane(s) parallel to plane EFG and containing point A.

  5. Parallel and Perpendicular Lines P • l P • l

  6. Photography The given line markings show how the roads are related to one another. Example 2 Identify parallel and perpendicular lines

  7. Angles and Transversals Corresponding angles - have corresponding positions on the lines and the transversal. Alternate Interior angles - lie between the two lines and opposite sides of the transversal. Alternate Exteriorangles - lie outside the two lines and opposite sides of the transversal. Consecutive Interior angles - lie between the two lines and on the same side of the transversals.

  8. Identify all pairs of angles of the given type. Example 3 Identify angle relationships Corresponding Alternate Interior Alternate Exterior Consecutive Interior SOLUTION

  9. 3.2 Use Parallel Lines and Transversals

  10. The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning. Example 1 Identify congruent angles SOLUTION

  11. ALGEBRA Find the value of x. Example 2 Use properties of parallel lines SOLUTION

  12. Example 3 Prove the Alternate Interior Angles Theorem Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. SOLUTION

  13. Science When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, . What is ? How do you know? Example 4 Solve real-world problem

  14. 3.4 Prove Lines are Parallel How can you Prove that Lines are Parallel?

  15. Corresponding Angles Converse: If two lines are cut by a transversal so the corresponding angles are congruent (≅), then the lines are parallel. Examples: Find the value of x that will make line u and v parallel. a)b)

  16. Prove that if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel. gh PROVE : Example 1 SOLUTION GIVEN :

  17. Alternate Interior Angles Converse: If two lines are cut by a transversal so the alternate interior angles are congruent (≅), then the lines are parallel. 2 6 Examples: Find the value of x that will make line u and v parallel. a)b)

  18. In the figure, and Prove: p ⃦ q Example 2 Prove the Alternate Interior Angles Converse SOLUTION

  19. Alternate Exterior Angles Converse: If two lines are cut by a transversal so the alternate exterior angles are congruent (≅), then the lines are parallel. Examples: Find the value of x that will make line u and v parallel. a)b)

  20. Example 3 Prove the Alternate Exterior Angles Converse r s In the figure, and . Prove: p ⃦ q p 1 q 8 16

  21. Consecutive Interior Angles Converse: If two lines are cut by a transversal so the consecutive interior angles are supplementary (1800),then the lines are parallel. Examples: Find the value of x that will make line u and v parallel. a) b)

  22. Example 4 Prove the Consecutive Interior Angles Converse In the figure, and . Prove: p ⃦ q r s 10 p 4 12 6 q

  23. Example: U.S. Flag The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe. Transitive Property of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other. p q r If and then .

  24. 3.4 Find and Use Slopes of Lines Key Concepts: Slope the ratio of vertical change (rise) to horizontal change (run) between any two points on the line ‘m’ (lower case m) is the symbol used to represent the slope Negative slope  falls from the left to the right Positive slope  rises from left to right Zero slope (m = 0) horizontal (line) Undefined slope vertical (line)

  25. Find the slope of line aand line d. Example 1 Find slopes of lines in coordinate plane Solution

  26. Picture this! Two perpendicular lines form a right angle. Two parallellines never intersect. The slopes of these two lines are the same. The slopes of these two lines are exact opposite reciprocals of each other (the product of their slopes is -1.)

  27. Find the slope of each line. Which lines are parallel? Example 2 Identify parallel lines

  28. Example 3 Graphing Line hpasses through (3, 0) and (7, 6). Graph the line perpendicular to hthat passes through the point (2, 5). Solution

  29. 3.5 Write and Graph Equations of Lines Slope-intercept form of a linear equation is = slope = y-intercept Rewrite the following equations in slope-intercept form. 1) 2)

  30. Example 1 Write an equation of a line from a graph Write an equation of the line in slope-intercept form. a)b)

  31. How to write an equation of a parallel line • Write an equation of the line passing through the point (4, 4) that • is parallel to the given line with the equation . • Steps: • Determine the slope of the given line. m = ______ • Determine the slope of the parallel line. m = ______ • Plug in the given information (4, 4) into the slope-intercept form. • x = ___ and y = ____ y = mx + b______________ • Solve for the y-intercept of the desired equation. • Write the equation in slope-intercept form: _________________

  32. Example 2 Write an equation of a parallel line 1) Given line equation: ; passing through point (2, 1). 2) Given line equation: ; passing through point (0, -3).

  33. How to write an equation of a perpendicular line • Write the slope-intercept form of the equation of the line that is perpendicular to the line and contains the point (-2, 4). • Steps: • Determine the slope of the given line. m = ________ • Determine the slope of the perpendicular line. m = _______ • Plug in the information into the slope-intercept form. • x = ____ and y = ____ y = mx + b____________ • Solve for the y-intercept of the desired equation. • Write the equation in slope-intercept form: ______________

  34. Example 3 Write an equation of a perpendicular line 1) Given line equation: ; passing through point (1, 5). 2) Given line equation: ; passing through point (-2, 5).

  35. 3.6 Distance from a Line The distance from a point to a line is the length of the perpendicular segment from the point to the line. This perpendicular segment is the shortest distance between the point and the line. E m C > A > F D p k B Distance between two parallel lines. Distance from a point to a line.

  36. Example 1 Find the distance between two parallel lines.

  37. Example 2

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