Proving Lines Parallel LESSON 3–5

Lesson Menu Five-Minute Check (over Lesson 3–4) TEKS Then/Now Postulate 3.4:Converse of Corresponding Angles Postulate Postulate 3.5:Parallel Postulate Theorems:Proving Lines Parallel Example 1:Identify Parallel Lines Example 2:Use Angle Relationships Example 3:Real-World Example: Prove Lines Parallel

Over Lesson 3–4 5-Minute Check 1 A. B. C. D. containing the point (5, –2) in point-slope form?

Over Lesson 3–4 5-Minute Check 2 A.y = 3x + 7 B.y = 3x – 2 C.y – 7 = 3x + 2 D.y – 7 = 3(x + 2) What is the equation of the line with slope 3 containing the point (–2, 7) in point-slope form?

Over Lesson 3–4 5-Minute Check 3 A.y = –3x B.y = –3x C.y – 2.5 = –3x D.y = –3(x + 2.5) What equation represents a line with slope –3 containing the point (0, 2.5) in slope-intercept form?

Over Lesson 3–4 5-Minute Check 4 A. B. C. D. containing the point (4, –6) in slope-intercept form?

Over Lesson 3–4 5-Minute Check 5 A.y = 3x + 2 B.y = 3x – 2 C.y – 6 = 3(x – 2) D.y – 6 = 3x + 2 What equation represents a line containing points (1, 5) and (3, 11)?

Over Lesson 3–4 5-Minute Check 6 A. B. C. D.

TEKS Targeted TEKS G.5(B) Construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge. G.6(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and these relationships to solve problems. Mathematical Processes G.1(E), G.1(F)

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

Concept

Example 1 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. Answer: Since  1   3, a║b by the Converse of the Corresponding Angles Postulate.  1 and  3 are corresponding angles of lines a and b.

Example 1 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. 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.  1 and  4 are alternate interior angles of lines a and c.

Example 1 A.Yes; ℓ ║ n B.Yes; m ║ n C.Yes; ℓ ║ m 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?

Example 1 A.Yes; ℓ ║ n B.Yes; m ║ n C.Yes; ℓ ║ m 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?

Example 2 Find m  ZYN so that ||. Show your work. Read the Item From the figure, you know that m  WXP = 11x – 25 and m  ZYN = 7x You are asked to find m  ZYN. Use Angle Relationships

Example 2 m WXP= m ZYN Alternate exterior angles 11x – 25= 7x + 35Substitution 4x – 25= 35Subtract 7x from each side. 4x=60Add 25 to each side. x= 15Divide each side by 4. Solve the 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. Use Angle Relationships

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

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

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. 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. Example 3 Prove Lines Parallel

Example 3 A.The two horizontal lines are parallel. B.The two vertical lines are parallel. C.The vertical lines are perpendicular to the horizontal lines. D.All of these statements are true. 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.

Proving Lines Parallel LESSON 3–5