2. 4.7Theorems On Perpendicular Lines Theorem 4.7: If two lines intersect to form a linear pair of congruent angles, then the lines are ______________. g h 1 2

3. 4.7Theorems On Perpendicular Lines Theorem 4.8: If two lines are perpendicular, then they intersect to form four ______________. b a 12 3 4

4. 4.7Theorems On Perpendicular Lines Theorem 4.9: If two sides of two adjacent acute angles are perpendicular, then the angles are ________________. 2 1 A B C

5. 4.7Theorems On Perpendicular Lines Theorem 4.10: Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is _____________ to the other. j h k

6. 4.7Theorems On Perpendicular Lines Theorem 4.11: Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are _________ to each other. mn p

7. 4.7Theorems On Perpendicular Lines J K L 13 N M g h 1 2 2 2 1

8. 4.7Theorems On Perpendicular Lines Example 1 Application of the Theorems Find the value of x. a b x o a. b. l p x o 55 o Solution a.Because a and b are _______________, all four angles formed are right angles by _______________. perpendicular Theorem 4.8 By definition of a right angle, x = ____. 90 b.Because l and p are perpendicular, all four angles formed are right angles by ____________. Theorem 4.8 By ___________, the 55 o angle and the x o angle are ______________. Thus x + 55 = 90, so x = ____. complementary Theorem 4.9 35

9. 4.7Theorems On Perpendicular Lines Checkpoint. Find the value of x. 1. By Theorem 4.8, all angles are right angles.

10. 4.7Theorems On Perpendicular Lines Checkpoint. Find the value of x. By Theorem 4.8, all angles are right angles. 2.

11. 4.7Theorems On Perpendicular Lines Example 2 Find the distance between a point and a line What is the distance from point B to line q? B q Solution You need to find the slope of line q. Using the points (3, 2) and (6, 5), the slope of line q is The distance from point B to line q is the length of the perpendicular segment from point B to line q. The slope of the perpendicular segment from point B to line q is the negative reciprocal of ___, or _____ = ____.

12. 4.7Theorems On Perpendicular Lines Example 2 Find the distance between a point and a line What is the distance from point B to line q? B q Solution The segment from (6, 5) to (3, 8) has a slope of ____. So, the segment is perpendicular to line q. Find the distance between (6, 5) and (3, 8). The distance from point B to line q is about ____ units.

13. 4.7Theorems On Perpendicular Lines

14. 5.2Use Perpendicular Bisectors Theorem 5.3: Converse of the Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it is on the ____________ _________ of the segment. A B C P D

15. 5.2Use Perpendicular Bisectors Example 1 Use the Perpendicular Bisector Theorem AC is the perpendicular bisector of BD AC is the perpendicular bisector of BD. Find AD. A B C 7x – 6 4x4x D Solution

16. 5.2Use Perpendicular Bisectors Example 2 Use perpendicular bisectors In the diagram, KN is the perpendicular bisector of JL In the diagram, KN is the perpendicular bisector of JL. a.What segment lengths in the diagram are equal? b.Is M on KN? J K L 13 N M Solution a.KN bisects JL, so ____ = ____. Because K is on the perpendicular bisector of JL, ____ = ____ by the Theorem 5.2. The diagram shows that ____ = ____ = 13. b.Because MJ = ML, M is ____________ from J and L. equidistant So, by the __________________________________________, M is on the perpendicular bisector of JL, which is KN. Converse of the Perpendicular Bisector Theorem

17. 5.2Use Perpendicular Bisectors Checkpoint. In the diagram, JK is the perpendicular bisector of GH. H G F K x + 1 2x2x 4.1 J 1.What segment lengths are equal? 2.Find GH.

18. 5.2Use Perpendicular Bisectors Theorem 5.4: Concurrency of Perpendicular Bisector of a Triangle The perpendicular bisector of a triangle intersects at a point that is equidistant from the vertices of the triangle. A B C If PD, PE, and PF are perpendicular bisectors, then PA = _____ = _____. D F E P

19. 5.2Use Perpendicular Bisectors Example 3 Use the concurrency of perpendicular bisectors The perpendicular bisectors of MNO meet at point S. Find SN. M N O 7 P Q R S 92 Solution Using ______________, you know that point S is _____________ from the vertices of the triangle. Theorem 5.4 equidistant So, _____ = _____ = _____. Theorem 5.4. _____ = _____ Substitute.

20. 5.2Use Perpendicular Bisectors Checkpoint. Complete the following exercise. 3.The perpendicular bisector of ABC meet at point G. Find GC. A C B 12 D E F G 6 15 By ______________, Theorem 5.4 _____ = _____ = _____.

21. 5.2Use Perpendicular Bisectors Pg. 284, 5.2 #1-10