Chapter 4 - Lines in a Plane. Procedures for Detour Proofs

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Berniece Leonard
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1 Chapter 4 - Lines in a Plane 4.1 Detours and Midpoints Detour proofs - To solve some problems, it is necessary to prove pair of triangles congruent. These we call detour proofs because we have to prove one set of triangles congruent, first, before we can get to the triangles we need to prove congruent. Procedures for Detour Proofs 1) Determine which triangles you must prove congruent 2) Attempt to prove these triangles congruent. 3) Identify the parts that you must prove to be congruent 4) Find a pair of triangles that: a) b) 5) Prove that the triangles found in step #4 6) Use and complete the proof planned in step 1. MIDPOINT FORMULA In some coordinate geometry problems you will need to locate the midpoint of a line segment. The formula applies an averaging process to develop a formula called the midpoint formula Theorem 22 If A = ( x 1,y 1 ) and B = ( x 2,y 2, then the midpoint M = (x m,y m ) of AB can be found by using the midpoint formula: m = ( x 1 + x 2 2, y + y 1 2 ) 2

2 2 Find the midpoint of the points A ( -1,3) and B ( 7,6) Homework #1 X Given: WX WZ, XY ZY W A Y Prove: Δ XAY Δ ZAY Z Statements Reasons Homework #2 Given: MN NS MP PS P Q N Prove: MQP SQP S Statement Reason M

3 Homework #3 A Given: A is equidistant from B and D (that is AB =AD) AC bisects BAD Prove: AC bisects BD B E D 3 Statements Reasons C 4.2 The Case of the Missing Diagram Some of the geometry problems you encounter will not be accompanied by a diagram. It is important that you be able to set up the problemthat is draw a diagram that accurately represents the problem, and decide on the given s and the prove. Example 1 If two altitudes of a triangle are congruent, then the triangle is isosceles. Given: 1) and are to and of ΔACD 2) Drawing: Prove: Δ is

4 4 Example 2 (First rewrite this statement in if, then format.) The medians of a triangle are congruent if the triangle is equilateral. Given: Δ is Drawing:,, and are medians Prove: Example 3 If an isosceles triangle has an altitude to the base then it bisects the vertex angle. Given: Drawing; Prove: Homework #4 (We will not write the proof) The bisector of the vertex angle of an isosceles triangle is perpendicular to the base. Given: Drawing Prove:

5 4.3 A Right-Angle Theorem: Proving that lines are perpendicular depends on proving that they form. Theorem #23 If two angles are BOTH supplementary and congruent, then they are right angles. (* We will assume that whenever two angles form a angle they are supplementary. No formal statement will be necessary.) Homework # 1 Given: P S is the midpt. of QR P Prove: PS QR Q R S Statement Reason 5 Homework#3 Given: AB BC CD AD (that is, ABCD is a rhombus) B E D Conclusion: AC BD (Hint: Use a detour) C Statement Reason A

6 4.4 The Equidistance Theorems Definition # 32 The distance between two objects is the length of the shortest path joining them. Postulate # 7 A line segment is the shortest path between two points. 6 If two points P and Q are the same distance X from a third point, X, they are said to be equidistant from X P Q PX QX means that X is equidistant from P and Q A A A C D B C D B C B D (Please highlight segment CD and put a circle around points A and B.) These diagrams have something in common. In each, both points A and B are equidistant from the endpoints and of segment. You can prove that line AB is the perpendicular bisector of segment CD. Definition # 33 The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment. Theorem # 24 If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment Theorem # 25 If a point is on the perpendicular bisector of a segment, then it( the point) is equidistant from the endpoints of that segment.

7 Given: Prove: AE bis. BD Statements A B 1 2 D 3 4 C Reasons 7 Problem #3 Given: O AB AC B Prove: AD bis. BC ( Hint: Show that A and O are each equi- D A distant from B and C.) C Statement Reason

8 Homework #4 R Given: s P and Q P Q Prove: PQ bis. RS S Statements Reasons 8 Homework #5 A Given: AD bis. BC Prove: ΔABE ΔACE E B D Statements Reasons C

9 4.5 Introduction to Parallel Lines Planes In order to explain parallel lines adequately, we must first acquaint you with the meaning of plane. Definition # 34 A plane is a surface such that if any two points on the surface are connected by a line all points of the line are also in the plane. A plane has only two dimensions- and Definition # 35 If,,, and so forth, lie in the same plane we call them. Points. lines and and so forth, that do not line in the same plane are called t exterior region In the figure, line t, is a a transversal of lines interior region a and b. b exterior region Definition # 36 A transversal is a line that intersects two coplanar lines in two distinct points. The region that lies between lines a and b is the interior of the figure. The region that lies above and below lines a and b is called the exterior of the figures. 9 Definition # 37 Alternate interior angles are a pair of angles formed by two lines and a transversal. The angles must both lie in the interior of the figure, must lie on alternate sides of the transversal, and must have different vertices.

10 10 Definition # 38 Alternate exterior angles are a pair of angles formed by two lines and a transversal. The angles must both lie in the exterior of the figures,must lie on alternate sides of the transversal, and must have different vertices, Definition # 39 Corresponding Angles are a pair of angles, formed by two lines and a transversal. One angle must lie in the interior of the figure, and the other must lie in the exterior. The angles must lie on the same side of the transversal but have different vertices. line a a b exterior interior line b º exterior Find the measures of angle 1 angle 4 angle 2 angle 3 angle 5 angle 6 angle 7 angle 8 = 37º Alt. Interior angles Alt. Exterior angles Corresponding angles Vertical angles

11 What definition makes these angles congruent? Clearly identify the transversal and find the parallel lines B 6 5 A C D F E BCA and DFE are BCD and EFA are A 7 B 8 D C Find a pair of alternate interior angles that are congruent. Can you find a pair of alternate interior angles formed by lines AD and BC with transversal AC. Definition #40 Parallel lines are two coplanar lines that do not intersect. ( symbol for parallel lines ( ll ) AB ll CD Draw three different arrangements of parallel lines. Name the lines and write the notation for the lines being parallel.

12 12 Homework # 3 A ( 7,14) a) Find the coordinates of M, the midpoint of AB. b) Find the coordinates of N, the midpoint of AC. C ( 11,6) C) Draw line MN. What appears to be true about line MN and line BC? B ( 1,2 ) D) What appears to be true about AMN and ABC? d) Name a pair of corresponding angles formed by MN and BC with transversal AC. Homework #4 J 2 O K M a) For which pair of lines are angles 1 and 4 a pair of alternate interior angles? b) For which pair of lines are angles 2 and 3 a pair of alternate interior angles? c) How many transversals of JO and KM are shown? 5) Locate the following points on a graph: ( x 1, y 1 ) = (0,0), ( x 2,, y 2 ) = ( 4,5), ( x 3, y 3 ) = ( 0,3) and ( x 4, y 4) = ( 4,8) a) Find y 2 y 1 x 2 x 1 b) Find y 4 y 3 x 4 x 3 c) Draw a line through the first two points and a line through the second two points. What appears to be true about these lines?

13 4.6 Slope 13 Definition # 41 The slope (m) of a nonvertical line, segment or ray contains ( x 1, y 1) and ( x 2, y 2 ) is defined by the formula m= y 2 y 1 x 2 x 1 or m= y 1 y 2 x 1 x 2 or m = Δ y Δ x Explain in words what the slope formula means a)use the points ( -2, 3) and ( 6, 5 ) to find slope two different ways. m= b) Find slope with ( 6,12) and 6, 2) m= What is unique about this slope? Describe the line with this slope. Give two characteristics of this slope c) Find slope with ( -2, 4) and ( 5, 4) m= What is unique about his slope? Describe the line with this slope. Give two characteristics of this slope

14 14 Parallel and Perpendicular Lines: Theorem # 26 If two nonvertical lines are parallel, then their slopes are equal. Graph the following lines using a slope that will fit on these graphs. Given AB ll CD How can you prove your lines have the same slope? Theorem # 27 Theorem # 28 Theorem # 29 Graph: AB CD How can you prove your lines are perpendicular?

An Approach to Geometry (stolen in part from Moise and Downs: Geometry)

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