3-6: Prove Theorems about Perpendicular Lines Geometry Chapter 3 3-6: Prove Theorems about Perpendicular Lines

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? 𝑪 𝑫 𝑩 𝑨

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

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

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

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

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

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

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

Example 2 <𝟑 and <𝟒 are complementary by Thm 3.10 𝟒 𝟑 𝑩 𝑫 𝑨 𝑪 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 𝟒 𝟑 𝑩 𝑫 𝑨 𝑪

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

For the Theorems List 𝒎 𝒏 𝒑 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. 𝒏 𝒎 𝒑 𝑰𝒇 𝒎⊥𝒑 𝒂𝒏𝒅 𝒏⊥𝒑, 𝒕𝒉𝒆𝒏 𝒎∥𝒏

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 𝒄 𝒅 𝒃

Example 4: Drawing Conclusions from Diagrams Use the diagram to answer the following questions: 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) 𝒂 𝒄 𝒅 𝒃

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

Example 4: Drawing Conclusions from Diagrams Use the diagram to answer the following questions: 2.) Is 𝑏⊥𝑐? Explain your reasoning using the theorems. Ans: 𝑐∥𝑑 by thm 3.12. Thus, 𝑏⊥𝑐 by thm 3.11 (Perpendicular Transversal Theorem) 𝒂 𝒄 𝒅 𝒃

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. 𝒏 𝒎 𝒑 𝒒 𝒕 Sample Answer: 𝒎∥𝒕 by thm 3.12 because they are both perpendicular to line 𝒒.

Distance from a line 𝒌 𝑨 𝑩 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 𝒌 𝑨 𝑩

Distance from a line 𝒎 𝑪 𝒑 𝑫 𝑬 𝑭 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 𝒎 𝑪 𝒑 𝑫 𝑬 𝑭

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

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

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

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

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)

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= (−1−0.75) 2 + (0−(−0.5)) 2 𝑑= (−1.75) 2 + (0.5) 2 𝑅𝑆= 3.0625+0.25 𝑑= 3.3125 ≈𝟏.𝟖