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28 Aug 2015 9:50 - 11:20 Geometry Agenda 1st 10 minutes - corrections on Quiz - looking for all correct use of geometry notation Station Reports - Station 1 Vertical Angles Homework In Geometry some things can be assumed to be true and some things can be proven. Most things that we assume to be true are known as postulates. Most things that are proven true are known as theorems. We are going to use a postulate to prove something on this page. So let’s start with a postulate (something we assume to be true.) Postulate: Two adjacent angles (angles that share a side and vertex) that form a straight angle have a sum of 180°. a° b° c° 1) What can we say about the sum of angles a and b? a°+ b° = a° b° c° 1) What can we say about the sum of angles a and b? a°+ b° = 180° and b° + c° = a° b° c° 1) What can we say about the sum of angles a and b? a°+ b° = 180° and b° + c° = 180° a° b° c° 2) What can we say about the relationship between (a°+ b°) and (b° + c°) ? a° b° c° 2) What can we say about the relationship between (a°+ b°) and (b° + c°) ? (a°+ b°) = (b° + c°) a° b° c° 3) Looking at your answers to #1 and #2, explain below how you know that a° = c°. Since a°+ b° = b° + c° We can then subtract b° from each side of the equation so a°+ b° - b° = b° + c° - b° a° b° c° 3) Looking at your answers to #1 and #2, explain below how you know that a° = c°. Since a°+ b° = b° + c° a°+ b° - b° = b° + c° - b° Which simplifies to a° = c° or a = c Theorem: Vertical Angles are Congruent a° b° c° Vertical angles - two nonadjacent angles formed by intersecting lines Theorem: Vertical Angles are Congruent m∠ABC = m∠EBD C A B E ∠ABC D ∠EBD Parallel lines and Transversals For the picture below we are going to assume that line p and line q are parallel. Line t is a transversal. For all of the questions on this page, we are going to assume the fact below. This is one of the basic geometry postulates. (Things we assume to be true.) Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. In this case, this means e° = h° 1. If we know the e° = h° by the postulate above, what can we say about the relationship between f° and h° ? On the previous page we stated that vertical angles are congruent. In this example f°=e°, these are a pair of vertical angles. We also know that e°= h°, Therefore we can state that f° = h°. 2. Using the postulate above and our conclusion in #1, what can we say about the relationship between g° and h°? Explain how we know. e° + g°= 180°, these are supplementary angles Since we also know that e°= h°, we can substitute h° into the equation to get h° + g° = 180°. So h and g are also supplementary angles. 1. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. 2. If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. Equations and Transversals Each figure consists of two parallel lines and a transversal. Find the values of x and y. Show reasoning. Same side interior angles are 1. supplementary 128° 2x° Equations and Transversals Each figure consists of two parallel lines and a transversal. Find the values of x and y. Show reasoning. Same side interior angles are 1. 128° + 2x° = 180° supplementary 128° 2x° Equations and Transversals Each figure consists of two parallel lines and a transversal. Find the values of x and y. Show reasoning. Same side interior angles are 1. 128° + 2x° = 180° supplementary 128° 2x° = 180°- 128° 2x° x° = 52° 2 x = 26° Equations and Transversals Each figure consists of two parallel lines and a transversal. Find the values of x and y. Show reasoning. Complete 2, 3, 4, 5, and 6. Homework - Vertical Angles Chapter 2, page 130 - 131 Questions 8, 9, 10, 11, 12, 13, and 28. Use paper put into your binder after the Equations and Transversals page. Draw the diagram each time. Show your reasoning.