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3.4 Warm Up 1. Find the values of x and y. Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 2. m = 2, x = 3, and y = 0 3. m = -1, x = 5, and y = -4 October 28, 2015 3.3 Proofs with Parallel Lines Geometry 3.4 Proofs with Perpendicular Lines Essential Question What conjectures can you make about perpendicular lines? October 28, 2015 3.4 Proofs with Perpendicular Lines Distance between a Point and 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. October 28, 2015 3.4 Proofs with Perpendicular Lines Example 1 d = (x2 - x1 )2 + (y2 - y1 )2 October 28, 2015 3.4 Proofs with Perpendicular Lines Perpendicular Bisector October 28, 2015 3.4 Proofs with Perpendicular Lines Theorem 3.10 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. m 1 2 October 28, 2015 n Linear Pair Perpendicular Theorem 3.4 Proofs with Perpendicular Lines Theorem 3.10: The Proof If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Given: 1 & 2 form a linear pair. 1 2. Plan: Prove: m n. m 1 2 October 28, 2015 n Show that 1 and 2 are supplementary, and then since the angles are congruent, each is 90°. This means the lines are perpendicular. Geometry 3.2 Proof and Perpendicular Lines 8 Theorem 3.10: 2-column proof m Given: 1 & 2 form a linear pair. 1 2. Prove: m n. 12 Statements Reasons 1. 1 & 2 are lin. pr. 2. 1 supp. 2 3. m1 + m2 =180 4. 1 2 5. m1 = m2 6. m1 + m1 = 180 7. 2m1 = 180 8. m1 = 90 9. 1 is a right angle 10. m n 1. Given 2. Lin. Pr. Post. 3. Def. Supp. 4. Given 5. Def. s 6. Substitution 7. Simplify 8. Division 9. Def Rt 10. Def. lines n QED October 28, 2015 Geometry 3.2 Proof and Perpendicular Lines 9 Theorem 3.10: Paragraph Proof m Given: 1 & 2 form a linear pair. 1 2. 12 n Prove: m n. Since 1 & 2 form a linear pair, their sum is 180 degrees. They are also congruent, which means they have the same measure. So, each angle must be 90 degrees. This means the angles are right angles, and hence the lines are perpendicular by definition. October 28, 2015 Geometry 3.2 Proof and Perpendicular Lines 10 Theorem 3.10: Flow Proof Given: 1 & 2 form a linear pair. 1 2. Prove: m n. 1 & 2 are lin. pr. Given 1 2 Given m1 + m2 =180 Lin. Pr. Post m1 + m1 = 180 Substitution 2m1 = 180 Simplify m1 = 90 Division mn Def. of October 29, 2015 Geometry 3.2 Proof and Perpendicular Lines 11 Theorem 3.11 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Perpendicular Transversal Theorem October 28, 2015 3.4 Proofs with Perpendicular Lines Proving the Perpendicular Transversal Theorem (3.11) If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Given: h || k, j h Prove: j k. October 28, 2015 Plan: j 1 2 34 h 5 6 7 8 k Show that by knowing h and k are parallel we know something about corresponding angles, then show 2 and 6 are congruent and measure 90°. This means the lines are perpendicular. 3.4 Proofs with Perpendicular Lines Theorem 3.11: 2-Column Proof j 1 34 Given: h || k, j h Prove: j k. 5 6 7 8 Statements Reasons 1. h || k, j h 2. m2 = 90 3. 2 6 4. m2 = m6 5. m6 = 90 6. j k QED October 28, 2015 𝟏. 𝑮𝒊𝒗𝒆𝒏 𝟐. 𝑫𝒆𝒇. 𝒐𝒇 𝟑. 𝑪𝑨 ≅ 𝟒. 𝑫𝒆𝒇 ≅ ∠ 𝟓. 𝑺𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒊𝒐𝒏 𝟔. 𝑫𝒆𝒇. 𝒐𝒇 3.4 Proofs with Perpendicular Lines 2 h k Theorem 3.12 In a plane, if two lines are perpendicular to the same line, then the lines are parallel. mt t m nt m || n n Lines Perpendicular to a Transversal Theorem October 28, 2015 3.4 Proofs with Perpendicular Lines Example 1 True or False? Line a is perpendicular to line b. Line b is perpendicular to line c. Therefore, line a is parallel to line c. a c b October 28, 2015 Th. 3.12 3.4 Proofs with Perpendicular Lines Example 2 True or False? Line a is perpendicular to line b. Line b is perpendicular to line c. Therefore, line a is perpendicular to line c. a c b October 28, 2015 3.4 Proofs with Perpendicular Lines False Example 3: Building a Fence The horizontal slats are perpendicular to all of the vertical slats. The vertical slats are parallel. Why are the horizontal slats also parallel? Theorem 3.12 If two lines are perpendicular to the same line, then the lines are parallel. October 28, 2015 3.4 Proofs with Perpendicular Lines Example 4 The photo shows the layout of a neighborhood. Determine which lines, if any, must be parallel in the diagram. Explain your reasoning. s t u p q October 28, 2015 3.4 Proofs with Perpendicular Lines To Summarize Linear Pair Perpendicular Theorem (3.10): If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Perpendicular Transversal Theorem (3.11): If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other. Lines Perpendicular to a Transversal Theorem (3.12): If two lines are perpendicular to the same line, then those lines are also parallel. October 29, 2015 3.4 Proofs with Perpendicular Lines Assignment October 28, 2015 3.4 Proofs with Perpendicular Lines