Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Trigonometric functions wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Euler angles wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Cartesian coordinate system wikipedia , lookup
Contour line wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Rational trigonometry wikipedia , lookup
1 Lesson Plan #30 Date: Wednesday November 23rd, 2016 Class: Geometry Topic: How do we prove lines are perpendicular? Aim: How do we prove lines are perpendicular? Do your HW the same day that this lesson is completed. Start by taking a blank sheet of paper, and completing a proper heading at the top of the paper, as per the rules sheet given out the 1st day of class. Then complete the HW assignment on that sheet of paper. Have the HW ready for collection by the next school day. HW #30: Note: Substitution Postulate for inequalities If π, π and π are real numbers, such that π > π and π = π, then π > π. For example 7 > 5 and π₯ = 5, therefore 7 > π₯. Postulate If a, b, c and d are real numbers, such that π > π and π = π, then π + π > π + π For example 7 > 5 and 2 = 2, then 7 + 2 > 5 + 2 Theorem (presented without proof) - The measure of an exterior angle of a triangle is greater than the measure of either non adjacent interior angle. Example: Objectives: 1) Students will be able to prove two lines perpendicular. Do Now: 1. Construct a line perpendicular to AB that goes through point D. D 2 PROCEDURE: Write the Aim and Do Now Get students working! Take attendance Give Back HW Collect HW Go over the Do Now Assignment #1: Given: Ξπ΄π΅πΆ with < π΄π·πΆ β < π΅π·πΆ Prove: Μ Μ Μ Μ πΆπ· β₯ Μ Μ Μ Μ π΄π΅ Statements 1. < π΄π·πΆ+< π΅π·πΆ = π π‘ππππβπ‘ πππππ 2. π < π΄π·πΆ + π < π΅π·πΆ = 180 3. < π΄π·πΆ β < π΅π·πΆ 4. π < π΄π·πΆ = π < π΅π·πΆ 5. π < π΄π·πΆ + π < π΄π·πΆ = 180 Or 2π < π΄π·πΆ = 180 6. π < π΄π·πΆ = 90 7. π < π΅π·πΆ = 90 8. <ADC and <BDC are right angles 9. Μ Μ Μ Μ πΆπ· β₯ Μ Μ Μ Μ π΄π΅ Reasons 1.Definition of a linear pair 2. 3. Given 4 5. Substitution Postulate ( ) 6. 7. Substitution Postulate ( ) 9. Definition of perpendicular lines ( ) Theorem: If two intersecting lines form congruent adjacent angles, the lines are perpendicular. Assignment #1: For what values of x and y will the lines L1 and L2 be perpendicular? Assignment #2: Statements Given: DCF , ACB AD ο BD AF ο BF Prove: 1) οADF ο οBDF 2) οΌ ADC οοΌ BDC 3) οADC ο οBDC 4) DF bisects AB 5) DF ο AB Reasons 3 Theorem: If two points are each equidistant from the endpoints of a line segment, the points determine the perpendicular bisector of the line segment. Corollary: Any point on the perpendicular bisector is equidistant from the endpoints of the line segment. Summary of ways to prove that lines are perpendicular 1) When two lines intersect, they form right angles. 2) When two lines intersect, they form congruent adjacent angles 3) There are two points on a line segment each of which is equidistant from the endpoints of the other line segment. Given: Prove: Assignment: Do the proofs below on a separate sheet of paper. 4 Sample Test Question: 1) 5 If enough time: 1) 2) 3) 4) 5) Final Summary: What are ways to prove that lines are perpendicular?