essential question: how can you use slope to solve ... following theorem states an important...

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Explore Proving the Slope Criteria for Parallel Lines

The following theorem states an important connection between slope and parallel lines.

Theorem: Slope Criteria for Parallel Lines

Two nonvertical lines are parallel if and only if they have the same slope.

Follow these steps to prove the slope criteria for parallel lines.

A First prove that if two lines are parallel, then they have the same slope.

Suppose lines m and n are parallel lines that are neither vertical nor horizontal.

Let A and B be two points on line m, as shown. You can draw a horizontal line through A and a vertical line through B to create the “slope triangle,” △ABC.

You can extend _ AC to intersect line n at point D and then extend it to point F

so that AC = DF. Finally, you can draw a vertical line through F intersecting line n at point E.

Mark the figure to show parallel lines, right angles, and congruent segments.

B When parallel lines are cut by a transversal, corresponding angles are congruent, so

∠BAC ≅ .

△BAC ≅ by the Triangle Congruence Theorem.

By CPCTC, _ BC ≅ and BC = .

The slope of line m = _ AC , and the slope of line n = _ DF .

The slopes of the lines are equal because

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Module 10 495 Lesson 1

10.1 Slope and Parallel LinesEssential Question: How can you use slope to solve problems involving parallel lines?

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C

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CNow prove that if two lines have the same slope, then they are parallel.

Suppose lines m and n are two lines with the same nonzero slope. You can set up a figure in the same way as before.

Let A and B be two points on line m, as shown. You can draw a horizontal line through A and a vertical line through B to create the “slope triangle,” △ABC .

You can extend _ AC to intersect line n at point D and then extend it to point F

so that AC = DF. Finally, you can draw a vertical line through F intersecting line n at point E.

Mark the figure to show right angles and congruent segments.

D Since line m and line n have the same slope, _ AC = _ DF .

But DF = AC, so by substitution, _ AC = _ AC .

Multiplying both sides by AC shows that BC = .

Now you can conclude that △BAC ≅ by the Triangle Congruence Theorem.

By CPCTC, ∠BAC ≅ .

Line m and line n are two lines that are cut by a transversal so that a pair of corresponding angles are congruent.

You can conclude that .

Reflect

1. Explain why the slope criteria can be applied to horizontal lines.

2. Explain why the slope criteria cannot be applied to vertical lines even though all vertical lines are parallel.

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Explain 1 Using Slopes to Classify Quadrilaterals by Sides

You can use the slope criteria for parallel lines to analyze figures in the coordinate plane.

Example 1 Show that each figure is the given type of quadrilateral.

A Show that ABCD is a trapezoid.

Step 1 Find the coordinates of the vertices of quadrilateral ABCD.

A (−1, 1) , B (2, 3) , C (3, 1) , D (−3, −3)

Step 2 Use the slope formula to find the slope of _ AB and the

slope of _ DC .

slope of _ AB =

y 2 - y 1 _ x 2 - x 1 = 3 - 1 _ 2 - (-1)

= 2 _ 3

slope of _ DC =

y 2 - y 1 _ x 2 - x 1 =

1 - (-3) _

3 - (-3) = 4 _ 6 = 2 _ 3

Step 3 Compare the slopes.

Since the slopes are the same, _ AB is parallel to

_ DC .

Quadrilateral ABCD is a trapezoid because it is a quadrilateral with at least one pair of parallel sides.

B Show that PQRS is a parallelogram.

Step 1 Find the coordinates of the vertices of quadrilateral PQRS.

P (-3, 4) , Q (1, 2) , R ( , ) , S ( ,

) Step 2 Use the slope formula to find the slope of each side.

_ PQ :

y 2 - y 1 _ x 2 - x 1 = 2 - 4 _

1 - (-3) = -2 _ 4 = - 1 _ 2

_ QR :

y 2 - y 1 _ x 2 - x 1 =

- 2 _

- 1 = _ =

_ RS :

y 2 - y 1 _ x 2 - x 1 =

- _

-

= _ = - _ _ SP :

y 2 - y 1 _ x 2 - x 1 = 4 - _

-3 -

= _ =

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Step 3 Compare the slopes.

Since the slope of _ PQ is the same as the slope of ,

_ PQ is parallel to .

Since the slope of _ QR is the same as the slope of ,

_ QR is parallel to .

Quadrilateral PQRS is a parallelogram because .

\ Reflect

3. What If? Suppose you know that the lengths of _ PQ and

_ QR in the figure in Example 1B

are each √_

20 . What type of parallelogram is quadrilateral PQRS? Explain.

Your Turn

Show that each figure is the given type of quadrilateral.

4. Show that JKLM is a trapezoid.

5. Show that ABCD is a parallelogram.

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Explain 2 Using Slopes to Find Missing Vertices

Example 2 Find the coordinates of the missing vertex in each parallelogram.

A ▱ABCD with vertices A (1, −2) , B (−2, 3) , and D (5, −1)

Step 1 Graph the given points.

Step 2 Find the slope of _ AB by counting units from A to B.

The rise from -2 to 3 is 5. The run from 1 to -2 is -3.

Step 3 Start at D and count the same number of units.

A rise of 5 from −1 is 4. A run of −3 from 5 is 2.

Label (2, 4) as vertex C.

Step 4 Use the slope formula to verify that _ BC || _ AD .

slope of _ BC = 4 - 3 _

2 - (-2) = 1 _ 4

slope of _ AD = -1 - (-2)

_ 5 - 1 = 1 _ 4

The coordinates of vertex C are (2, 4).

B ▱PQRS with vertices P (−3, 0) , Q (−2, 4) , and R (2, 2)

Step 1 Graph the given points.

Step 2 Find the slope of _ PQ by counting units from Q to P.

The rise from 4 to 0 is . The run from −2 to −3 is .

Step 3 Start at R and count the same number of units.

A rise of from 2 is . A run of from 2 is .

Label ( , ) as vertex S.

Step 4 Use the slope formula to verify that _ QR || _ PS .

slope of _ QR =

- _

-

= - _ slope of _ PS =

- _

-

= - _

The coordinates of vertex S are ( , ) .

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Reflect

6. Discussion In Part A, you used the slope formula to verify that ― BC ∥ ― AD . Describe another way you can check that you found the correct coordinates of vertex C.

Your Turn

Find the coordinates of the missing vertex in each parallelogram.

7. ▱JKLM with vertices J (−3, −2) , K (0, 1) , and M (1, −3)

8. ▱DEFG with vertices E (−2, 2) , F (4, 1) , and G (3, −2)

Elaborate

9. Suppose you are given the coordinates of the vertices of a quadrilateral. Do you always need to find the slopes of all four sides of the quadrilateral in order to determine whether the quadrilateral is a trapezoid? Explain.

10. A student was asked to determine whether quadrilateral ABCD with vertices A (0, 0) , B (2, 0) , C (5, 7) , and D (0, 2) was a parallelogram. Without plotting points, the student looked at the coordinates of the vertices and quickly determined that quadrilateral ABCD could not be a parallelogram. How do you think the student solved the problem?

11. Essential Question Check-In What steps can you use to determine whether two given lines on a coordinate plane are parallel?

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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

1. Jodie draws parallel lines p and q. She sets up a figure as shown to prove that the lines must have the same slope. First she proves that △JKL ≅ △RST by the ASA Triangle Congruence Theorem. What should she do to complete the proof?

Show that each figure is the given type of quadrilateral.

2. Show that ABCD is a trapezoid.

3. Show that KLMN is a parallelogram.

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Find the coordinates of the missing vertex in each parallelogram. Use slopes to check your answer.

4. ▱ABCD with vertices A (3, −3) , B (−1, −2) , and D (5, −1)

5. ▱STUV with vertices S (-3, -1) , T (-1, 1) and V (0, 0)

6. Show that quadrilateral ABCD is not a trapezoid. 7. Show that quadrilateral FGHJ is a trapezoid, but is not a parallelogram.

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Determine whether each statement is always, sometimes, or never true. Explain your reasoning.

8. If quadrilateral ABCD is a trapezoid and the slope of _ AB is 3, then the slope

of _ CD is 3.

9. A parallelogram has vertices at (0, 0) , (2, 0) , (0, 2) , and at a point on the line y = x.

10. If the slope of _ PQ is 1 __ 3 and the slope of

_ RS is - 1 __ 3 , the quadrilateral PQRS is a

parallelogram.

11. If line m is parallel to line n and the slope of line m is greater than 1, then the slope of line n is greater than 1.

12. If trapezoid JKLM has vertices J (-4, 1) , K (−3, 3) , and L (−1, 4) , then the coordinates of vertex M are (2, 4) .



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Explain whether the quadrilateral determined by the intersections of the given lines is a trapezoid, a parallelogram, both, or neither.

13. 18.

Line EquationLine ℓ y = 2x + 3

Line m 2y = −x + 6

Line n y = x - 3

Line p x + y = - 3

14.

Line EquationLine ℓ y = x + 3

Line m y - x = 0

Line n x + 2y = 6

Line p y = −0.5x - 3

15. 18.

Line EquationLine ℓ 2y = x + 4

Line m y + 5 = 2x

Line n -2x + y = 2

Line p x + 2y = -6

16.

Line EquationLine ℓ 3x + y = 4

Line m y + 3 = 0

Line n y = 3x + 5

Line p y = 3

Algebra Find the value of each variable in the parallelogram.

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20. Use the slope-intercept form of a linear equation to prove that if two lines are parallel, then they have the same slope. (Hint: Use an indirect proof. Assume the lines have different slopes, m 1 and m 2 . Write the equations of the lines and show that there must be a point of intersection.)

21. Critique Reasoning Mayumi was asked to determine whether quadrilateral RSTU is a trapezoid given the vertices R (−2, 3) , S (1, 4) , T (1, -4) , and U (−2, 1) . She noticed that the slopes of

_ RU and

_ ST are undefined, so she concluded that the

quadrilateral could not be a trapezoid. Do you agree? Explain.

22. Kaitlyn is planning the diagonal spaces for the parking lot at a mall. Each space is a parallelogram. Kaitlyn has already planned the spaces shown in the figure and wants to continue the pattern to draw the next space to the right. What are the endpoints of the next line segment she should draw? Explain your reasoning.

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23. Multi-Step Two carpenters are using a coordinate plane to design a tabletop in the shape of a trapezoid. They have already drawn the two sides of the tabletop shown in the figure. They want side ― AD to lie on the line x = −2. What is the equation of the line on which side

_ CD will lie? Explain your reasoning.

24. Quadrilateral PQRS has vertices P(−3, 2), Q(−1, 4), and R(5, 0). For each of the given coordinates of vertex S, determine whether the quadrilateral is a parallelogram, a trapezoid that is not a parallelogram, or neither. Select the correct answer for each lettered part.a. S (0, 0) Parallelogram Trapezoid but not Neither

parallelogramb. S (3, −2) Parallelogram Trapezoid but not Neither

parallelogramc. S (2, −1) Parallelogram Trapezoid but not Neither

parallelogramd. S (6, -4) Parallelogram Trapezoid but not Neither

parallelograme. S (5, −3) Parallelogram Trapezoid but not Neither

parallelogram

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H.O.T. Focus on Higher Order Thinking

25. Explain the Error Tariq was given the points P (0, 3) , Q (3, −3) , R (0, -4) , and S (−2, −1) and was asked to decide whether quadrilateral PQRS is a trapezoid. Explain his error.

slope of _ SP =

3 - (-1) _

0 - (-2) = 4 _ 2 = 2

slope of _ QP =

3 - (-3) _ 3 - 0 = 6 _ 3 = 2

Since at least two sides are parallel, the quadrilateral is a trapezoid.

26. Analyze Relationships Four members of a marching band are arranged to form the vertices of a parallelogram. The coordinates of three band members are M (−3, 1) , G (1, 3) , and Q (2, −1) . Find all possible coordinates for the fourth band member.

27. Make a Conjecture Plot any four points on the coordinate plane and connect them to form a quadrilateral. Find the midpoint of each side of the quadrilateral and connect consecutive midpoints to form a new quadrilateral. What type of quadrilateral is formed? Repeat the process by starting with a different set of four points. Do you get the same result? State a conjecture about your findings.



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esLesson Performance Task

Suppose archeologists uncover an ancient city with the foundations of 16 houses. The locations of the houses are as follows:

(2, 2) (-5, 6) (3, -6) (-1, 0) (5, -8) (3, 5) (-3, 3) (0, 5)

(-8, 1) (4, -1) (1, -3) (-4, -3) (8, -7) (-5, -4) (-2, 8) (6, -4)

a. How could you show that the streets are parallel? Explain.

b. Are the streets parallel?

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