Geometry: general information, symbols, and theorems

Transcription

1 Foundations of Mathematics 11 Chapter 2 Geometry seems like it s full of rules. They do need to be in your head, but not really by memorizing them. It s more a case of understanding them. In this section, we ll list a great many of these rules (technically known as theorems) and you should endeavour to learn them in an integrated, holistic way learn what each one really has to say. We ll also show you some example questions and lead you in to doing geometric proofs. Below is the Ministry of Education s curriculum statement for what you must be able to do for this section. It will account for about 9% of your overall mark. Geometry: general information, symbols, and theorems Symbols and markings congruent Not equal, not congruent Angle Perpendicular or right angle Triangle ~ < or

2 Parallel lines or marked by arrowheads Equal segments Equal angles - small hash marks across the angle arc or the segment, or dots, Xs, or other symbols may show equality Triangles 1. Triangles are named for their vertices (corners), in order. ABC is has corresponding parts on the other triangle, DEF. 2. Angle sum is 180 degrees. 3. Equilateral triangle all <s equal at 60 deg. All sides equal. 4. Isosceles triangle two <s equal, and their opposite sides equal. 5. Scalene triangle no equal sides or angles. 6. Exterior angle is equal to sum of the opposite two angles. 7. Right triangles have one 90-degree angle 8. Pythagorus theorem applies to right triangles. 9. Similar triangles have the same angle measurements; their sides are proportional. 10. Congruent triangles are similar triangles with identical measurements for sides. This can be proven by one of three theorems: a. Angle-Side-Angle (ASA) means that the two triangles each have and angle of the same measurement, next to a side of the same measurement, next to an angle of the same measurement. The corresponding sides must be between the respective angles. b. Side-Angle-Side (SAS) means that each triangle has an equal pattern of a matching angle between two matching sides. c. Side-Side-Side (SSS) means that each side of one triangle matches

3 Angles and Lines 1. Complementary angles add to 90 degrees 2. Supplementary angles add to 180 degrees 3. Angles on a line add to 180 degrees 4. Angles at a point add to 360 degrees 5. Vertically Opposite angles are equal ( Xrule ) 6. Alternate interior angles between parallels are equal ( Z-rule ) 7. Corresponding angles are equal ( F-rule ) 8. Interior angles on the same side of the transversal add to 180 degrees ( C-rule, or IASST) 9. Any point on an angle bisector is equidistant from the arms of the angle. 10. Any line segment is congruent to itself. Quadrilaterals 1. Named for their vertices, in order. 2. Angle sum is 360 degrees. 3. Trapezoids have one set of parallel sides. Opposite angles add to 180 degrees 4. Parallelograms are trapezoids, have two sets of opposite sides parallel, and equal. Diagonals are equal and bisect each other. 5. Rhombuses are parallelograms with all sides equal. Diagonals bisect angles and each other, at right angles. 6. Rectangles are parallelograms with 90 degree angles. 7. Squares are rectangles that are also rhombuses i.e. all sides equal. Circles 1. A chord is a segment through a circle 2. A chord through the center is a diameter 3. Radii of a circle are all equal Important definitions: A theorem is a statement that has already been proven to be true. It can be used as part of a proof in order to prove another statement. Statements that haven t been proven yet are called propositions, or conjectures. Proofs A proof is a structured series of statements. It usually begins with some given information, either written or marked on a diagram. This is the information the problem s creator has provided to you for free. You can use the given statements in your proof.

4 For example, a problem might say Given the diagram below, <PRQ=70 o and <ABC=20 o The problem will also include a statement that says Prove something. In this way, they set the stage for the problem. You continue the process by writing further statements, and giving reasons (theorems) for those statements. Your statements must lead logically and inevitably to a final statement, the one they asked you to prove. Examine the example below. A Geometric proof Here s a typical geometric proof. It has two given statements and a diagram. As you read each line of the proof, mark the diagram with appropriate numbers and symbols so you understand it completely. Given: <eba = 40 <bcg = 140 Prove: ab=bc (in other words, prove that the segment from a to b is the same length as the segment from b to c.) Strategy and thinking: We can prove those two segments equal if we can prove that triangle abc is isosceles. We can prove the triangle s isosceles if we can prove that <bac and <bca are equal. (This backwards analysis is often a useful way to develop a strategy before embarking on a geometric proof. One usually uses a diagram, marks the statement to be proven on it, marks something that would lead to that statement, and then works backwards from there to the given statements.) Notice what happens in the proof at the right: We must give a reason for each statement we make. Each reason is either a given (an assumption) or a theorem or definition from the list above. Notice as well: nothing is considered too obvious to be mentioned in the proof. We pointed out and gave a reason for the two angles were equal: both 40 degrees. This level of detail is expected of senior students in geometric proofs. Proof: Statement <eba=40 <bac=40 <bcg=140 <bca=40 <bca=<bac triangle abc is isosceles ab=ac Reason given alternate interior angles given supplementary angles both 40 degrees two equal angles (definition of isosceles) sides of isosceles triangle Note the last statement in the proof: it s the statement we were told to prove. Do you understand exactly where each rule or reason in the right-hand column came from? Do you understand the strategy that was employed to get from the given statements to the final line?

5 Example questions 2.1, Prove XY is parallel to QZ in two or more ways We could show that interior angles on the same side of the transvesal add to 180 degrees ( ), or We could show that there is a 33 o angle on a line AB, next to the 147 o angle. This gives us alternate interior angles that are the same. 2. Identify the equal angles in the diagram. There are several other ways to prove the two lines are parallel. Find a method that uses corresponding angles on your own. < AKI = <COM = <MKL = <GOH < JLD = <KLN = <FPH = <NPO <BMO = <KMN These are three examples. There are other equal angles in the diagram, and you can use the theorems to prove them equal. Note that angles originating on line BE cannot be proven equal to angles from other lines; it s not parallel to any others. We don t know if IG and JH are parallel or not. 3. Given the diagram, and AB CD, prove that <AED is 95 degrees. Proof: Statement Reason < BAE = 45 deg. Given AB CD <DCE = 45 deg. <CDE = 50 deg. <CED = 85 deg. Therefore <AED = 95 deg. Given Alternate Interior angles Given Angle sum of a triangle = 180 angles on a line = 180

6 Example questions 2.3, Find the measure of <x and <y in the diagram below. <x must be 33 degrees since it s vertically opposite to a 33-degree angle. The 3 rd angle of the triangle, then, must be 180 (33+95), using the angle sum of a triangle rule, a value of 52 degree. Angle y is , using angles on a line. 2. Prove that quadrilateral ABCD is a parallelogram Statement 3. Find the measures for interior angles of an 11-sided regular polygon a 3-sided regular polygon (an equilateral triangle) has an < sum of 180 and interior angles of = 60 a 4-sided = 90 a 5-sided = continue the pattern. Can you create a formula that uses n as the number of sides, and calculates the interior angle measurement for any regular polygon? 4. Draw an example of a convex polygon and a non-convex polygon Reason <C= 80 deg. angle sum of triangle <BDA = 70 angle sum of triangle <ABD=<CDB given DC AB Alt. interior <s are 30deg. AD BC Alt. interior <s are 80deg. ABCD is a parallelogram opposite sides are parallel (defn. of a parallelogram) Note: abbreviations are used in some of our proofs you can do the same! Just make it clear what you said

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