- Properties of Parallel Lines Common Core State Standards G-CO.C. Prove theorems about lines and angles. Theorems include:... when a transversal crosses parallel lines, alternate interior angles are congruent... MP, MP, MP, MP Objectives To prove theorems about parallel lines To use properties of parallel lines to find angle measures You see the vertical angles, right? Keep looking! There are other angle pairs. MATHEMATICAL PRACTICES Look at the map of streets in Clearwater, Florida. Nicholson Street and Cedar Street are parallel. Which pairs of angles appear to be congruent? Palm Bluff St. Cedar St. Nicholson St. N. Garden Ave. In the Solve It, you identified several pairs of angles that appear congruent. You already know the relationship between vertical angles. In this lesson, you will explore the relationships between the angles you learned about in Lesson - when they are formed by parallel lines and a transversal. Essential Understanding The special angle pairs formed by parallel lines and a transversal are congruent, supplementary, or both. Postulate - Same-Side Interior Angles Postulate Postulate If a transversal intersects two parallel lines, then same-side interior angles are supplementary. If... / } m m Then... m + m = m + m = Chapter Parallel and Perpendicular Lines
How do you find angles supplementary to a given angle? Look for angles whose measure is the difference of and the given angle measure. Problem Identifying Supplementary Angles The measure of j is. Which angles are supplementary to j? How do you know? By definition, a straight angle measures. If m a + m b =, then a and b are supplementary by definition of supplementary angles. - =, so any angle x, where m x =, is supplementary to. m = by the definition of a straight angle. m = by the Same-Side Interior Angles Postulate. m = m by the Vertical Angles Theorem, so m =. m = m by the Vertical Angles Theorem, so m =. Got It?. Reasoning If you know the measure of one of the angles, can you always find the measures of all angles when two parallel lines are cut by a transversal? Explain. You can use the Same-Side Interior Angles Postulate to prove other angle relationships. Theorem - Alternate Interior Angles Theorem Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. If... / } m m Then... Theorem - Corresponding Angles Theorem Theorem If a transversal intersects two parallel lines, then corresponding angles are congruent. If... / } m m Then... You will prove Theorem - in Exercise. Lesson - Properties of Parallel Lines
Proof Proof of Theorem -: Alternate Interior Angles Theorem Given: / } m Prove: m Statement Reasons ) / } m ) Given ) m + m = ) Supplementary Angles ) m + m = ) Same-Side Interior Angles Postulate ) m + m = m + m ) Transitive Property of Equality ) m = m ) Subtraction Property of Equality ) ) Definition of Congruence Proof Problem Proving an Angle Relationship Given: Prove: a } b and are supplementary. a a } b From the diagram you know and are corresponding and form a linear pair b and are supplementary, or m + m =. Statements Show that and that m + m =. Then substitute m for m to prove that and are supplementary. Reasons ) a } b ) Given ) ) If lines are }, then corresp. are. ) m = m ) Congruent have equal measures. ) and are supplementary. ) that form a linear pair are suppl. ) m + m = ) Def. of suppl. ) m + m = ) Substitution Property ) and are supplementary. ) Def. of suppl. Got It?. Using the same given information and diagram in Problem, prove that. Chapter Parallel and Perpendicular Lines
In the diagram for Problem, and are alternate exterior angles. In Got It, you proved the following theorem. Theorem - Alternate Exterior Angles Theorem Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent. If... / } m Then... m If you know the measure of one of the angles formed by two parallel lines and a transversal, you can use theorems and postulates to find the measures of the other angles. Problem Finding Measures of Angles What are the measures of j and j? Which theorem or postulate justifies each answer? How do j and j relate to the given angle? and the given angle are alternate interior angles. and the given angle are same-side interior angles. p q Since p } q, m = by the Alternate Interior Angles Theorem. m Since / } m, m + = by the Same-Side Interior Angles Postulate. So, m = - =. Got It?. Use the diagram in Problem. What is the measure of each angle? Justify each answer. a. b. c. d. e. f. Lesson - Properties of Parallel Lines
You can combine theorems and postulates with your knowledge of algebra to find angle measures. Problem Algebra What is the value of y? Finding an Angle Measure What do you know from the diagram? You have one pair of parallel lines. The angle and the angle formed by the and y angles are same-side interior angles. By the Angle Addition Postulate, y + is the measure of an interior angle. (y + ) + = Same-side interior of } lines are suppl. y + = y = y Simplify. Subtract from each side. Got It?. a. In the figure at the right, what are the values of x and y? b. What are the measures of the four angles in the figure? (x ) x y (y ) Lesson Check Do you know HOW? Use the diagram for Exercises.. Identify four pairs of congruent angles. (Exclude vertical angle pairs.). Identify two pairs of supplementary angles. (Exclude linear pairs.) f g Do you UNDERSTAND? MATHEMATICAL PRACTICES. Compare and Contrast How are the Alternate Interior Angles Theorem and the Alternate Exterior Angles Theorem alike? How are they different?. In Problem, you proved that and, in the diagram below, are supplementary. What is a good name for this pair of angles? Explain.. If m =, what is m?. If m = and m = x, what is the value of x? a b Chapter Parallel and Perpendicular Lines
Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice Identify all the numbered angles that are congruent to the given angle. See Problem. Justify your answers..... Developing Proof Supply the missing reasons in the two-column proof. Given: a } b, c } d Prove: a b c d See Problem. Statements Reasons ) a } b ) Given ) and are supplementary. ) a.? ) c } d ) Given ) and are supplementary. ) b.? ) ) c.?. Write a two-column proof for Exercise that does not use. Proof Find mj and mj. Justify each answer.. t. a b. q m A D B See Problem. C Algebra Find the value of x. Then find the measure of each labeled angle.. x. (x ). See Problem. (x ) (x ) x x Lesson - Properties of Parallel Lines
B Apply Algebra Find the values of the variables.... x y (p ) w x y y v y. Think About a Plan People in ancient Rome played a game called terni lapilli. The exact rules of this game are not known. Etchings on floors and walls in Rome suggest that the game required a grid of two intersecting pairs of parallel lines, similar to the modern game tick-tack-toe. The measure of one of the angles formed by the intersecting lines is. Find the measure of each of the other angles. Justify your answers. How can you use a diagram to help? You know the measure of one angle. How does the position of that angle relate to the position of each of the other angles? Which angles formed by two parallel lines and a transversal are congruent? Which angles are supplementary?. Error Analysis Which solution for the value of x in the figure at the right is incorrect? Explain. A. B. x x x x (x ) x x x x (x ). Outdoor Recreation Campers often use a bear bag at night to avoid attracting animals to their food supply. In the bear bag system at the right, a camper pulls one end of the rope to raise and lower the food bag. a. Suppose a camper pulls the rope taut between the two parallel trees, as shown. What is m? b. Are and the given angle alternate interior angles, same-side interior angles, or corresponding angles?. Writing Are same-side interior angles ever congruent? Explain. Chapter Parallel and Perpendicular Lines
. Write a two-column proof to prove the Proof Corresponding Angles Theorem (Theorem -). Given: / } m Prove: and are congruent. m. Write a two-column proof. Proof Given: a } b, Prove: a b C Challenge Use the diagram at the right for Exercises and.. Algebra Suppose the measures of and are in a ratio. Find their measures. (Diagram is not to scale.) ( x) (x ). Error Analysis The diagram contains contradictory information. What is it? Why is it contradictory? PERFORMANCE TASK Apply What You ve Learned MATHEMATICAL PRACTICES MP Look back at the blueprint on page of the plan for a city park. Choose from the following words, numbers, and expressions to complete the following sentences. alternate interior alternate exterior vertical corresponding congruent same-side interior supplementary The measure of is a.?, because alternate interior angles formed by parallel lines and a transversal are b.?. The measure of is c.?, because d.? angles formed by parallel lines and a transversal are congruent. The measure of is e.?, because f.? angles formed by parallel lines and a transversal are g.?. The measure of h.? is, because i.? angles formed by parallel lines and a transversal are congruent. Lesson - Properties of Parallel Lines