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3.4 Proof and Perpendicular Lines

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1 3.4 Proof and Perpendicular Lines
Geometry 3.4 Proof and Perpendicular Lines EQ: How do you construct a line perpendicular to a line passing through a point not on the line?

2 Geometry 3.4 Proof and Perpendicular Lines
Goals Learn about different types of proof. Prove statements about perpendicular lines. Find the distance from a point to a line. Construct perpendicular lines November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

3 Distance from a point to a line.
Defined as the length of the perpendicular segment between the point and the line. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

4 Distance from a point to a line.
Defined as the length of the perpendicular segment between the point and the line. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

5 Geometry 3.4 Proof and Perpendicular Lines
Use (-4, -3) and (1, 2) November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

6 The perpendicular bisector of a segment can be a segment.
A M B S RS is a  bisector of AB. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

7 The perpendicular bisector of a segment can be a line.
RS is a  bisector of AB. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

8 The perpendicular bisector of a segment can be a ray.
RS is a  bisector of AB. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

9 Constructing perpendicular Lines
Construct a line  to line m through point p. m November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

10 Constructing Perpendicular Bisector
A B November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

11 Theorem 3.10 Linear Pair Perp Theorem
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. Prove: m  n. 1 2 m n November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

12 Geometry 3.4 Proof and Perpendicular Lines
Theorem 3.10: How to prove 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. Prove: m  n. An explanation without two column proof. Show that 1 and 2 are supplementary, and then since the angles are congruent, each is 90°. This means the lines are perpendicular. 1 2 m n November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

13 Theorem 3.10: 2-column proof
Given: 1 & 2 form a linear pair. 1  2. Prove: m  n. Statements Reasons 1. 1 & 2 are lin. pr. 1. Given 2. 1 supp. 2 2. Lin. Pr. Post. 3. m1 + m2 =180 3. Def. Supp. 4. 1  2 4. Given 5. m1 = m2 5. Def.  s 6. m1 + m1 = 180 6. Substitution 7. 2m1 = 180 7. Simplify 8. m1 = 90 8. Division 9. 1 is a right angle 9. Def Rt  10. m  n 10. Def.  lines November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

14 Theorem 3.10: Paragraph Proof
2 m n Given: 1 & 2 form a linear pair. 1  2. 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. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

15 Theorem 3.11 Perp Transversal Theorem
In a plane, if a transversal is Perpendicular to one of two parallel lines, then it is perpendicular to the other line. If h || k and j  h then j is  k j h k November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

16 Geometry 3.4 Proof and Perpendicular Lines
Theorem 3.12 In a plane, if two lines are perpendicular to the same line, then the lines are parallel. m  t n  t t m || n m n November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

17 Geometry 3.4 Proof and Perpendicular Lines
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. November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

18 Geometry 3.4 Proof and Perpendicular Lines
Problem True or False? Line a is parallel to line b. Line b is parallel to line c. Therefore, line a is parallel to line c. a b c True November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

19 Geometry 3.4 Proof and Perpendicular Lines
Problem 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 False b November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

20 Geometry 3.4 Proof and Perpendicular Lines
Problem 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 Th. 3.12 b November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines

21 Geometry 3.4 Proof and Perpendicular Lines
To do proofs, you must: Know the definitions we have studied so far. Know the properties. Know all of the postulates. Know the theorems. There are no shortcuts! November 13, 2018 Geometry 3.4 Proof and Perpendicular Lines


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