Mth 97 Fall 2013 Chapter 4

Transcription

1 4.1 Reasoning and Proof in Geometry Direct reasoning or reasoning is used to draw a conclusion from a series of statements. Conditional statements, if p, then q, play a central role in deductive reasoning. If p, the q is written, and can also be read p q or p only if q. p is the and q is the What can you deduce from the following statements? If ABC is an equilateral triangle, then it is also an equiangular triangle. ABC has 3 congruent sides. Therefore (symbol, a 3 dot triangle), ABC is an. Not all statements are written in conditional form (if then), but we can write most statements in this form. Rewrite the following statements as conditional statements. The only polyhedron with four vertices is a triangular pyramid. A cylinder is not a polyhedron. Law of Detachment If p q is a true conditional statement and p is true, then q is true. Example: 1. If an angle is obtuse, its measure is greater than 90 and less than 180. obtuse. Bے is Law of Syllogism If p q and q r are true conditional statements, then p r is also true. Example: 1. If M is the midpoint of AB, then AM = MB. 2. If the measures of two segments are equal, then they are congruent. 3. Other forms of the conditional If p, then q. Converse of p q Inverse of p q Contrapositive of p q q p not p not q not q not p 1

2 Write the converse, inverse and contrapositive of If ABCD is a square, it has four right angles. Converse Inverse Contrapositive A biconditional statement exists only when the conditional statement and its converse are both If p q and q p are true, then, read p if and only if q. (Shorthand for this is p iff q.) Example: Write the converse of the statement below. If both the statement and its converse are true, write the biconditional statement. If three sides of a triangle are congruent, then the three angles of the triangle are congruent. Converse: Biconditional: Do ICA 6 Vertical Angles are formed by a pair of intersecting lines and are opposite each other. In the sketch below, the pair 1 ے and 3 ے and the pair 2 ے and 4 ے are vertical angles. 2 Theorem 4.1 If two angles are a pair of vertical angles, 4 ےm = 2 ےm 3 ےm and = 1 ےm 1 3 their measures are equal.. 4 See proof on top of page 191 Proof A proof is a convincing mathematical argument. The two kinds of proof used extensively in Geometry are paragraph proof and statement-reason proof (2 column proof). If two angles are congruent and supplementary, then they are right angles. Given: 1 2 and 1 and 2 are supplementary 1 2 Prove: 1 and 2 are right angles Statement Reason (Other samples of this proof are on page 191.) ; m 1+ m 2 = m 1= m m 2 + m 2 =180; 2(m 2 ) = m 2 = m 1= and 2 are right angles 6. 2

3 4.2 Triangular Congruence Conditions Definition of Congruent Triangles: ABC DEF, if and only if 1. All 3 pairs of corresponding angles are congruent. 2. All 3 pairs of corresponding sides are congruent. Reflexive Property Something is congruent or equal to Examples: DEF DEF or 15 = 15 Symmetric Property It doesn t matter which side of the = or you are on. Examples: If ABC DEF, then DEF ABC. If = 7, then 7 = Transitive Property If two thing are or = to the same thing, then they are or = to each other. Examples: If ABC DEF and DEF GHI, then If ABC GHI. or If = 8 and 8 = (2)(4), then = SAS Postulate LL Theorem HL Theorem ASA Postulate SSS Postulate Postulates and Theorems used to prove Two Triangles are Congruent If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent. If two legs of one right triangle are congruent respectively to two legs of another right triangle, then the two triangles are congruent. If the hypotenuse and a leg of one right triangle are congruent respectively to the hypotenuse and the leg of another right triangle, then the two triangles are congruent. If two angles and the included side of one triangle are congruent respectively to two angles and the included side of another triangle, then the two triangles are congruent. If three sides of one triangle are congruent respectively to three sides of another triangle, then the two triangles are congruent. 3

4 Theorem 4.4 Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the square of the third side, then the triangle is a right triangle. For each pair of triangles decide whether they are necessarily congruent. If so, write and appropriate congruence statement and specify which congruence principle applies. F a) A B b) D C I H G c) H d) T U O M P A Y X W V Proof using triangular congruence N R Given: RA TA, PA CA, RP CT C A Prove : R T P Statement T Subgoal: Reason 1. RA TA, PA CA, RP CT RAP R T 3. 4

5 Section 4.3 Problem Solving Using Triangle Congruence Isosceles Triangles Theorem 4.5 In an isosceles triangle, the angles opposite the congruent sides are congruent. A Given: ABC with AB AC Prove: B C C B Subgoal to be prove first: ABC ACB Statements Reasons 1. AB AC AC AB A A ABC ACB B C 5. Corallary 4.6 Every equilateral triangle is equiangular. If then Theorem 4.7 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. A Given: ABC with B C Prove: AB AC C B Subgoal to be prove first: ABC ACB Statements Reasons 1. B C C B BC BC ABC ACB AB AC 5. Corallary 4.8 Every equiangular triangle is equilateral. If then 5

6 Theorem 4.7 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. A A If then If, then B C B C Corollary 4.8 Every equiangular triangle is equilateral. If, then Perpendicular Bisector of a Segment Definitions: 1) To means to divide in half. 2) An of a line segment is a line, line segment or ray that divides the angle into two congruent angles. 3) The of a line segment is a line, line segment or ray that passes through the midpoint of the segment and is perpendicular to the line segment. 4) A of a triangle is a segment whose endpoints are the vertex of an angle of the triangle and the midpoint of the side opposite that angle. Theorem 4.9 In an isosceles triangle, the ray that bisects the vertex angle bisects the base and is perpendicular to it. A A If then B C B D C See proof on page 210. Theorem 4.10 Perpendicular Bisector Theorem A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of the segment. P P See proof on page 211. A B A B P is on the perpendicular bisector of AB if and only if An angle of a triangle is formed by on side of a triangle and an extension of an adjacent side. 6

7 Theorem 4.11 Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles. 2 and Proof is on pages Section 4.4 Basic Geometric Constructions with a Compass and Strait Edge Construction 1: Copy a line segment a) Draw a line segment longer than the one you wish to copy. b) Open your compass to match the length of you wish to copy. c) Without changing the compass setting, place the tip of the compass at one end of the new line segment and swing an arc to intersect the segment. Copy each line segment. A B C D Constuct a line segment twice as long ascd. Construction 2: Copy an Angle a) Draw a line segment or a ray. This will be one side of the new angle. b) Place the tip of the compass at the vertex of the angle you wish to copy and swing an arc that intersects the sides of the angle. c) Without changing the compass setting, place the tip of your compass on an endpoint of your copy and swing a similar arc that intersects your copy. d) Open your compass to match the distance between the intersection points of the arc and the sides of the angle you are copying. e) Without changing the compass setting, copy this length on the arc beginning where the arc intersected the side or your new angle. f) Draw a segment from the endpoint of your copy (the new vertex) through the intersection of your arcs. This is the second side of your angle. 7

8 Copy the angle. Construction 3: Bisect an Angle a. Place the compass point on the vertex of the angle and swing an arc that intersects both sides of the angle. b. Set your compass for any opening that over half the distance between the points where your arc intersected the sides of your angle. Your will use this setting to draw the next two arcs. Place your compass point on each of these intersections and draw arcs that interest in the interior of your angle. c. Draw a segment or ray from the vertex of your angle through the intersection point of the last two rays you drew. Constructing Perpendiculars Construction 4: Construct a perpendicular to a point on a line by bisecting the straight angle. 8

9 Construction 5: Construct a perpendicular to a line from a point not on the line a. Place the compass on the point not on the line and swing an arc that intersects the line in two places. b. Use the same compass setting to draw the next two arcs. Using each intersection point of the arc with your line from the first step as a center draw arcs that intersect either above or below your line. c. Draw a segment from the point not on your line through the intersection point of your last two arcs that intersects your line. Construct a 45 angle by bisecting one of the right angles above. Construction 6: Construct the Perpendicular Bisector of a Segment a. Open your compass to more than half the length of the segment to be bisected. Use this compass setting for this steps a and b. Place your compass on one endpoint and swing an arc above and below this segment. b. Place the compass point on the other end of the segment and swing arcs above and below the segment. c. Draw a line through the intersection points of the arc. 9

10 Construction 7: Construct and equilateral triangle give the length of one side a. Open the compass to the length of the segment to be copied. Use this compass setting for both arcs in the next step. b. Place the compass on each endpoint and swing an arc either, both arcs above or both below. c. The intersection point of the two arcs is the third point of your triangle. Use your straight edge to help you draw the triangle. Construction 8: Methods of constructing a right triangle See pages First construct an equilateral triangle. 1. First construct the perpendicular bisector of a 2. Bisect one of the angles. segment 2. Copy the length of your line segment using one endpoint as a center so that the second endpoint of your copy lies on the perpendicular bisector. 10

11 Construction 9: Construct a triangle given the lengths of two sides and the included angle a. Copy the angle. b. Copy the length of one side of the triangle to a side of the angle. c. Copy the length of the other side of the triangle to the second side of the angle. d. Draw a segment connecting the endpoints of the segments you just drew. Construction 10: Construct a triangle give two angles and the included side a. Copy the segment. b. Using one endpoint as a center, copy one of the angles. c. Using the other endpoint as a center copy the other angle. The intersection point of the two new sides of the angles you copied is the third vertex of your triangle. 11