Use slopes to identify parallel and perpendicular lines. 3.5 Slopes of Lines Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines.

Roles in each Group Recorder- everyone is writing but someone records what goes in the middle of placemat (must write neat and big enough to see)-makes placemat Presenter- speak about what your group’s results Responsible for materials-cleaning at the end and putting them back where they belong 4. Group leader- makes sure everyone is on task and understands correct answer

Materials Must have 4 different color (bigger) markers The materials person from each group grab the following materials for your group Must have 4 different color (bigger) markers Post it poster paper

Recorder Creates Placemat!

What are some things we know about slope? Each person in the group will write what they can recall about slope in their section of the placemat (2-3 minutes) When told: each person will read what they wrote-Recorder will write common ideas in the middle of the placemat (2-3 minutes) When told: each group’s presenter will share one idea with the class (different from other groups until everything is shared) (3-5 minutes) When told: materials person will put their groups placemat on the side of the room (making room for next activity) (quickly/quietly!!!) When told: Group Leader will grab 3.5 Notes for each member of their group-making sure everyone is recording when we go through examples

Slope Formula The slope of a line is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Rise is the difference in the y-values of two points on a line Run is the difference in the x-values of two points on a line Slope of a line is the ratio of the rise to run

Use the slope formula to determine the slope of JK through J(3, 1) and K(2, –1). Substitute (3, 1) for (x1, y1) and (2, –1) for (x2, y2) in the slope formula and then simplify.

Zero-Horizontal line Undefined-Vertical line

Parallel or Perpendicular? Parallel lines theorem- same slope Perpendicular line theorem- opposite reciprocal (slopes have a product of -1) Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear.

Use the slope formula to determine the slope of JK through J(3, 1) and K(2, –1). Substitute (3, 1) for (x1, y1) and (2, –1) for (x2, y2) in the slope formula and then simplify. A line parallel to JK needs a slope of? A line perpendicular to JK needs a slope of? Parallel m= 2 Perpendicular m= -1/2

UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Group Problems Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3) (notes example: 7) GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4) (notes example: 8) CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3) (notes example: 9) WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3) (notes example: 10) KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1) (notes example: 11)

The products of the slopes is –1, so the lines are perpendicular. Example 7: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3) The products of the slopes is –1, so the lines are perpendicular.

GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4) Example 8: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4) The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

The lines have the same slope, so they are parallel. Example 9: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3) The lines have the same slope, so they are parallel.

Vertical and horizontal lines are perpendicular. Example 10: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3) Vertical and horizontal lines are perpendicular.

KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1) Example 11: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1) The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.