Geometry Chapter 3 3-6: PROVE THEOREMS ABOUT PERPENDICULAR LINES

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1 Geometry Chapter 3 3-6: PROVE THEOREMS ABOUT PERPENDICULAR LINES

2 Warm-Up 1.) What is the distance between the points (2, 3) and (5, 7). 2.) If < 1 and < 2 are complements, and m < 1 = 49, then what is m < 2? 3.) If m < DBC = 90, then what is m < ABD? D A B C

3 Prove Theorems about Perpendicular Lines Objective: Students will be able to prove properties of perpendicular lines, using them to solve problems. Agenda Theorems Proofs Example

4 For the Theorems List Theorem 3.8: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 1 g 2 If < 1 < 2, Then g h h

5 For the Theorems List Theorem 3.9: If two lines are perpendicular, then they intersect to form four right angles. g 1 2 h If g h, Then < 1, < 2, < 3, and < 4 are right angles 3 4

6 Example 1 In the diagram, AB BC. What can you conclude about < 1 and < 2. A D 1 B 2 C

7 Example 1 In the diagram, AB BC. What can you conclude about < 1 and < 2. < 1 and < 2 are right angles by theorem 3.9 D A 1 B 2 C

8 For the Theorems List Theorem 3.10: If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. A 1 2 If BA BC, Then < 1 and < 2 are complementary B C

9 Example 2 In the diagram, < ABC < ABD. What can you conclude about < 3 and < 4. Explain how you know? A C B 3 4 D

10 Example 2 In the diagram, < ABC < ABD. What can you conclude about < 3 and < 4. Explain how you know? < 3 and < 4 are complementary by Thm 3.10 A C B 3 4 D

11 For the Theorems List Theorem 3.11 Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. j If h k and j h, then j k h k

12 For the Theorems List Theorem 3.12 Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m n If m p and n p, then m n p

13 Example 4: Drawing Conclusions from Diagrams Use the diagram to answer the following questions: 1.) Is b a? Explain your reasoning using the theorems. a b c d

14 Example 4: Drawing Conclusions from Diagrams Use the diagram to answer the following questions: a b 1.) Is b a? Explain your reasoning using the theorems. Ans: Lines a and b are both perpendicular to line d. Thus, b a by Thm 3.12 (Line Perpendicular to a Transversal Theorem) c d

15 Example 4: Drawing Conclusions from Diagrams Use the diagram to answer the following questions: a b 2.) Is b c? Explain your reasoning using the theorems. c d

16 Example 4: Drawing Conclusions from Diagrams Use the diagram to answer the following questions: a b 2.) Is b c? Explain your reasoning using the theorems. Ans: c d by thm Thus, b c by thm 3.11 (Perpendicular Transversal Theorem) c d

17 Example 3:Drawing Conclusions from Diagrams From the given diagram, determine which lines, if any, must be parallel. Explain your reasoning using the theorems as reference. m t n Sample Answer: p m t by thm 3.12 because they are both perpendicular to line q. q

18 Distance from a line The distance from a point to a line is the length of the perpendicular segment form the point to the line. This perpendicular segment is the shortest distance between the point and the line. Ex: The distance between point A and line k is AB A B k

19 Distance from a line The distance between two parallel lines is the length of any perpendicular segment joining the two lines. Ex: The distance between line p and line m is CD or EF C E m D F p

20 Example 5 What is the distance from line c to line d? This problem can be solved using the following steps:

21 Example 5 Step 1: Find the slope of either line. Through counting spaced, you should see that the slope of either line is m 1 = 2

22 Example 5 Step 2: Find the slope of a perpendicular line. Change the slope you just found to that of a line that would be perpendicular to it: m 2 = 1 2

23 Example 5 Step 3: Use the slope you just found to graph a perpendicular line that will go through the two lines.

24 Example 5 Step 4: Identify the points that the perpendicular line crossed on each of the other two lines (round when needed) Points: ( 1, 0) and (0.75, 0.5)

25 Example 5 Step 5: Use the distance formula to find the distance between the intersected points: d = (x 2 x 1 ) 2 +(y 2 y 1 ) 2 d = ( ) 2 +(0 ( 0.5)) 2 d = ( 1.75) 2 +(0.5) 2 RS = d =

right angle an angle whose measure is exactly 90ᴼ

right angle an angle whose measure is exactly 90ᴼ m B = 90ᴼ B two angles that share a common ray A D C B Vertical Angles A D C B E two angles that are opposite of each other and share a common vertex two