B C E F Given: A D, AB DE, AC DF Prove: B E Proof: Either or Assume.

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1 Geometry -Chapter 5 Parallel Lines and Related Figures 5.1 Indirect Proof: We ve looked at several different ways to write proofs. We will look at indirect proofs. An indirect proof is usually helpful when a direct proof would be difficult to use. Example: A D B C E F Given: A D, AB DE, AC DF Prove: B E Proof: Either or Assume. Indirect-Proof Procedures 1. List the for the conclusion. 2) Assume the of the desired conclusion is correct. 3) Write a chain until you reach an. This will be a of either a) or b) a theorem, definition or other known fact. 4) State the remaining as the desired conclusion.

2 P 2 Given: RS PQ S PR QR R Prove: RS does not bisect PRQ Q Either or Assume Then Homework #2 J Given: P is not the midpoint of HK HJ JK Prove: JP does not bisect HJK H P K Either or Assume Then -

3 3 Homework #5 Given: O OB is not an altitude o Prove: OB does not bisect AOC A C B Either or Assume Then Homework # 6 ODEF is a square In terms of a, find y axis a)coordinates of F and E b)area of Square F E c) Midpoint of FD d) Midpoint of OE o (0,0) D ( 2a, 0) x axis

4 4 5.2 Proving That Lines Are Parallel: The exterior angle of a triangle is formed whenever a side of the triangle is extended to form an angle supplementary to the adjacent interior angle. adjacent exterior interior angle angle remote interior angles Theorem #30 The measure of Theorem # 31 If two lines are cut by a transversal ( short form: alt inter s ll lines) Theorem #32 If two lines cut by a transversal such ( short form: alt ext s ll lines ) Theorem #33 If two lines are cut by a transversal such ( short form: corres. s ll lines ) Theorem # 34 If two lines are cut by a transversal such

5 5 Theorem # 35 If two lines are cut by a transversal such Theorem # 36 If two coplanar lines are Pg 219 1a) 1b) 1c) 2a) 2b) 2c) 3) list all the pairs of angles that will prove a ll b. 6) Q D Given: QD ll UA 1 Prove : U A Either or Assume Then

6 6 #11 Complete the inequality that shows the restrictions on x. < x < xº 110º 5.3 Congruent Angles Associated with Parallel Lines: In this section we shall see the converses of many of the theorems in Section 5.2 are also true. Parallel Postulate #8 Through a point not on a line there is exactly one parallel to the given line. We will assume that the Parallel Postulate is true. In this section we will learn that the converse is true- that is if we start with parallel lines, then we can conclude that alternate interior angles are congruent. In fact many pairs of congruent angles are determined by parallel lines cut by a transversal. Theorem # 37 If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. ( Short Form: ll lines alt int s )

7 7 Theorem # 38 If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary. a ll b, and let x be the measure of any one of the angles line a xº line b Find all the measures of the other seven angles, algebraically, based on the Theorem #38. Theorem # 39 If two parallel lines ( Short Form: ) Theorem # 40 IF two parallel lines are cut ( Short Form: ) Theorem # 41 If two parallel lines are cut by a transversal

8 8 Theorem # 42 If two parallel lines are cut by a transversal, each pair of Theorem # 43 In a plane, if a line is to one of the, it is to the other. Create a drawing that shows this theorem. Then state the givens that support it. Given Can prove Theorem # 44 If two lines are to a line, they are to each other. ( Transitive Property of lines.) Summary of if two parallel lines are cut by a transversal, then : each pair of angles are congruent each pair of angles are congruent each pair of angles are congruent each pair of angles on the same side of the transversal are each pair of angles on the same side of the transversal are

9 9 Given AB DC A B AB ll DC Prove: AD BC D C Statement Reason Given EF ll GH E G EF GH J Prove EJ JH F H Statements Reasons

10 10 Given: a ll b, 30º angle as shown- find the seven remaining angles 30º Homework # 4 R S Given RS ll NP Prove: Δ NPR is isosceles N P Statements Reasons Homework # 5 ( 2x + 5 ) º Given: a ll b 1 Find m 1 (3x 13)º Do Crook Problem Pg 229 #4

11 Four-Sided Polygons Polygons are figures. The following are example of polygons. Decide which are convex and which are not convex. A B C d Why is PLAN not a polygon? P N L A Look at bottom pg. 234 and top of pg. 235 to help you. Define convex polygons. Be complete and specific How do we name polygons? Definition # 42 A convex polygon is a polygon in which each interior angle has a measure less than 180º Explain why polygon C above is not convex. Diagonals of Polygons- Draw all the diagonals in each polygon.

12 12 Definition # 43 A diagonal of a polygon is any segment that connects two nonconsecutive ( nonadjacent) vertices of the polygon. A quadrilateral A parallelogram A rectangle A rhombus A kite A square A trapezoid An isosceles trapezoid

13 13 Turn to page Do problems # 1-3 below ) In the isosceles trapezoid shown ST ll RV Name: S T a) The base b) The diagonals c) The legs R V d) The lower base angles e) The upper base angles f) All pairs of congruent alternate interior angles 8) Write S-sometines, A-always, N-never for each statement below. a) A square is a rhombus. b) A rhombus is a square. c) A kite is a parallelogram. d) A rectangle is a polygon. e) A polygon has the same number of vertices as sides. f) A parallelogram has three diagonals. g) A trapezoid has three bases.

14 14 10) Cut out any size rectangle. Then listen for more directions. Explain how the formula for a rectangle can be used to find the formula for a parallelogram. 11) If the sum of the measures of the angles of a triangle is 180º, what is the sum of the measures of all the angles in a) a quadrilateral b) a pentagon 5.5 Properties of Quadrilaterals Get out the long paper we have begun and we are going to add the characteristics of parallelograms, rectangles, rhombus, kites, squares, and isosceles trapezoids.

15 15 Homework #1 D C Given ABCD ( is a rectangle) Conclusion: Δ ABC Δ CDA Statement A Reason B Homework #2 J H Given : EFHJ 1 2 K G Conclusion: KH EG E 2 F Statements Reasons 1)EFHJ is a rectangle 1) 2) J F 2) 3) JH EF 3) 4) 1 2 4) 5) Δ KJH Δ GFE 5) 6)KH EG 6)

16 16 Homework #3 S R Given: Rectangle MPRS MO PO M O P Prove: Δ ROS is isosceles Statement Reason Homework #4 D C Given: ABCD F AE CF Conclusion: DE BF A E B Statement Reason

17 Proving that a Quadrilateral is a Parallelogram 1) If both pairs of of a quadrilateral are, then the quadrilateral is a (reverse of the definition) 2) If of opposite sides of a are, then the is a parallelogram. (converse of a property) 3) If of opposite sides of a quadrilateral are both and, then the quadrilateral is a parallelogram. 4) If the of a quadrilateral each other, then the quadrilateral us a.( converse of a property) 5) If pairs of of a quadrilateral are, then the quadrilateral is a parallelogram, ( converse of a property) Homework #1 ( Look at the diagrams in your book pg. 251) a) b) c) d) e)

18 18 Homework #2 T V Given: XRV RST RSV TVS Conclusion: RSTV is a S R X Statements Reasons Homework #3 S R Given O O Conclusion: SMPR is a M P Statements Reasons

19 Proving That Figures Are Special Quadrilaterals Proving that a quadrilateral is a rectangle E H 1) F G 2) 3) Proving that a quadrilateral is a kite. K 1) E J 2) T Proving that a quadrilateral is a rhombus. J O 1) K M 2) 3)

20 Proving that a Quadrilateral is a Square. N S 1) 20 P R Proving that a Trapezoid is Isosceles. A D 1) B C 2) 3) Problem # 6 a) A quadrilateral with diagonals that are perpendicular bisectors of each other. b) A rectangle that is also a kite. c) A quadrilateral with opposite angles supplementary and consecutive angles supplementary. d) A quadrilateral with one pair of opposite sides congruent and the other pair of opposite sides parallel.

21 21 Problem #13 What is the most descriptive name for each quadrilateral below? a c e g b d f h 8 10 Problem #17 a) If a quadrilateral is symmetrical across both diagonals, it is a? b) If a quadrilateral is symmetrical across exactly one diagonal, it is a? c) Which quadrilateral has four axes of symmetry? Homework #1 Locate points Q=( 2,4), U=( 2,7), A=( 10,7), and D=(10,4) on a graph. Then give the most descriptive name for QUAD. Turn to your book and lets begin # 2 5 on your own paper.

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