Five-Minute Check (over Lesson 2–8) Mathematical Practices Then/Now Postulate 2.4: Converse of Corresponding Angles Postulate Postulate 2.5: Parallel Postulate Theorems: Proving Lines Parallel Example 1: Identify Parallel Lines Example 2: Standardized Test Example: Use Angle Relationships Example 3: Real-World Example: Prove Lines Parallel Lesson Menu

What is the slope of the line MN for M(–3, 4) and N(5, –8)? B. C. D. 5-Minute Check 1

What is the graph of the line that has slope 0 and contains the point (–3, –4)? A. B. C. D. 5-Minute Check 2

What is the equation of the line with slope 3 containing the point (–2, 7) in point-slope form? A. y = 3x + 7 B. y = 3x – 2 C. y – 7 = 3x + 2 D. y – 7 = 3(x + 2) 5-Minute Check 3

containing the point (4, –6) in slope-intercept form? B. C. D. 5-Minute Check 4

A. B. C. D. 5-Minute Check 5

What is the slope of a line perpendicular to MN for M(–3, 4) and N(5, –8)? B. C. D. 5-Minute Check 6

Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. Content Standards G.CO.9 Prove theorems about lines and angles. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). MP

You used slopes to identify parallel and perpendicular lines. Recognize angle pairs that occur with parallel lines. Prove that two lines are parallel. Then/Now

Concept

Concept

Concept

1 and 3 are corresponding angles of lines a and b. Identify Parallel Lines A. Given 1  3, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. 1 and 3 are corresponding angles of lines a and b. Answer: Example 1

1 and 3 are corresponding angles of lines a and b. Identify Parallel Lines A. Given 1  3, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. 1 and 3 are corresponding angles of lines a and b. Answer: Since 1  3, a║b by the Converse of the Corresponding Angles Postulate. Example 1

1 and 4 are alternate interior angles of lines a and c. Identify Parallel Lines B. Given m1 = 103 and m4 = 100, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. 1 and 4 are alternate interior angles of lines a and c. Answer: Example 1

1 and 4 are alternate interior angles of lines a and c. Identify Parallel Lines B. Given m1 = 103 and m4 = 100, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer. 1 and 4 are alternate interior angles of lines a and c. Answer: Since 1 is not congruent to 4, line a is not parallel to line c by the Converse of the Alternate Interior Angles Theorem. Example 1

D. It is not possible to prove any of the lines parallel. A. Given 1  5, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ ║ n B. Yes; m ║ n C. Yes; ℓ ║ m D. It is not possible to prove any of the lines parallel. Example 1

D. It is not possible to prove any of the lines parallel. A. Given 1  5, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ ║ n B. Yes; m ║ n C. Yes; ℓ ║ m D. It is not possible to prove any of the lines parallel. Example 1

D. It is not possible to prove any of the lines parallel. B. Given m4 = 105 and m5 = 70, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ ║ n B. Yes; m ║ n C. Yes; ℓ ║ m D. It is not possible to prove any of the lines parallel. Example 1

D. It is not possible to prove any of the lines parallel. B. Given m4 = 105 and m5 = 70, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ ║ n B. Yes; m ║ n C. Yes; ℓ ║ m D. It is not possible to prove any of the lines parallel. Example 1

Find m ZYN so that || . Identify the postulate or theorem you used. Use Angle Relationships Find m ZYN so that || . Identify the postulate or theorem you used. Read the Test Item From the figure, you know that mWXP = 11x – 25 and mZYN = 7x + 35. You are asked to find mZYN. Example 2

m WXP = m ZYN Alternate exterior angles Use Angle Relationships Solve the Test Item WXP and ZYN are alternate exterior angles. For line PQ to be parallel to line MN, the alternate exterior angles must be congruent. So mWXP = mZYN. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to find mZYN. m WXP = m ZYN Alternate exterior angles 11x – 25 = 7x + 35 Substitution 4x – 25 = 35 Subtract 7x from each side. 4x = 60 Add 25 to each side. x = 15 Divide each side by 4. Example 2

Now use the value of x to find mZYN. Use Angle Relationships Now use the value of x to find mZYN. mZYN = 7x + 35 Original equation = 7(15) + 35 x = 15 = 140 Simplify. Answer: Example 2

Now use the value of x to find mZYN. Use Angle Relationships Now use the value of x to find mZYN. mZYN = 7x + 35 Original equation = 7(15) + 35 x = 15 = 140 Simplify. Answer: mZYN = 140 Check Verify the angle measure by using the value of x to find mWXP. mWXP = 11x – 25 = 11(15) – 25 = 140 Since mWXP = mZYN, WXP  ZYN and || . Example 2

ALGEBRA Find x so that || . A. x = 60 B. x = 9 C. x = 12 D. x = 12 Example 2

ALGEBRA Find x so that || . A. x = 60 B. x = 9 C. x = 12 D. x = 12 Example 2

Prove Lines Parallel CONSTRUCTION In the window shown, the diamond grid pattern is constructed by hand. Is it possible to ensure that the wood pieces that run the same direction are parallel? If so, explain how. If not, explain why not. Answer: Example 3

Prove Lines Parallel CONSTRUCTION In the window shown, the diamond grid pattern is constructed by hand. Is it possible to ensure that the wood pieces that run the same direction are parallel? If so, explain how. If not, explain why not. Answer: Measure the corresponding angles formed by two consecutive grid lines and the intersecting grid line traveling in the opposite direction. If these angles are congruent, then the grid lines that run in the same direction are parallel by the Converse of the Corresponding Angles Postulate. Example 3

GAMES In the game Tic-Tac-Toe, four lines intersect to form a square with four right angles in the middle of the grid. Is it possible to prove any of the lines parallel or perpendicular? Choose the best answer. A. The two horizontal lines are parallel. B. The two vertical lines are parallel. The vertical lines are perpendicular to the horizontal lines. All of these statements are true. Example 3

GAMES In the game Tic-Tac-Toe, four lines intersect to form a square with four right angles in the middle of the grid. Is it possible to prove any of the lines parallel or perpendicular? Choose the best answer. A. The two horizontal lines are parallel. B. The two vertical lines are parallel. The vertical lines are perpendicular to the horizontal lines. All of these statements are true. Example 3