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10/15/2014 Learning Scale 4 3 2 1 I can apply what I know about the relationships and properties of quadrilaterals consistently and could easily prove them in real world applications and feel confident I could even use them in proving broader relationships in other polygons. I can apply what I know about the relationships and properties of quadrilaterals consistently and can consistently create correct proofs for those relationships and properties. I can sometimes create correct proofs for those relationships and properties of quadrilaterals, but still need help to have a complete proof(missing 1 or 2 steps) I use the properties to set up the problem right but don’t always get the right answer yet. I am beginning to see the relationships and properties of quadrilaterals, but I get stuck just past the given information in my proofs and I am usually missing more than 2 steps in my proofs on my homework and/or I need help setting up the problems applying properties. 5.1 Indirect Proof From the given information we can prove the triangles congruent by ASA, which makes AC ≅ DF But wait! This is not possible, since the given information states that AC ≅ DF Therefore, our assumption was false and we can now say ∠B ≅ ∠E , because this is the only other possibility. Indirect Proof Procedure… 1. List the possibilities for the conclusion. 2. Assume that the negation of the desired conclusion is correct. 3. Write a chain of reasons until you reach an impossibility. This will be a contradiction of either: a. Given information or b. A theorem, definition, or other known fact. 4. State the remaining possibility as the desired conclusion. Objective: After studying this section, you will be able to write indirect proofs. An indirect proof is useful when a direct proof is difficult to apply. Remember to start by looking at the conclusion! P Let’s take a S A B D C E Given: ∠A ≅ ∠D, AB ≅ DE , AC ≅ DF Prove: ∠B ≅ ∠E Proof: Either ∠B ≅ ∠E or ∠B ≅ ∠E Let’s assume ∠B ≅ ∠E Given: RS ⊥ PQ F R Q PR ≅ QR JJJG Prove: RS does not bisect ∠PRQ 1 10/15/2014 P Summary S R Q JJJG JJJG Proof: Either RS does bisect ∠PRQ or RS does not bisect ∠PRQ JJJG Assume RS bisects ∠PRQ. Then we can say ∠PRS ≅ ∠QRS. In your own words describe how to write an indirect proof. Since RS ⊥ PQ , we know that ∠PSR ≅ ∠QSR , thus, triangle PSR is congruent to triangle QSR by ASA (Since SR ≅ SR ). This means PR ≅ QR by CPCTC, but this contradicts the given information that PR ≅ QR therefore JJJ the G assumption must be false, leaving the only other possibility: RS does not bisect ∠PRQ Hang in there, one last example… C B A Given: O O Homework Worksheet 5.1 AB ≅ BC Prove: ∠AOB ≅ ∠COB 2 10/15/2014 Theorem: If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel. 5.2 Proving That Lines Are Parallel (short form: Alt. int. ∠s ≅⇒ ll lines.) Given: ∠3 ≅ ∠6 Prove: a ll b Objective: After studying this section, you will be able to apply the exterior angle inequality theorem and use various methods to prove lines are parallel. Theorem: If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel. (short form: Alt. ext.∠s ≅⇒ ll lines.) F Given: ∠1 ≅ ∠8 Prove: a ll b Remote Interior angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Given: exterior angle BCD Prove: m∠BCD>m∠B B m∠BCD > m∠BAC P M A C Locate the midpoint, M, of BC. Draw AP so that AM=MP Draw CP Triangles AMB and PMC are congruent because of SAS This makes angle B and angle MCP congruent (CPCTC) This also proves that angle BCD is greater than angle B Extend BC, creating a vertical angle to angle BCD The following results: b Use an indirect proof to prove that a ll b. D Exterior angle E 3 6 An 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 Interior angle a D a 1 b 8 This can be proved by use of Alt. int. ∠s ≅⇒ ll lines. Theorem: If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel. (short form: corr.∠s ≅⇒ ll lines.) Given: ∠2 ≅ ∠6 Prove: a ll b a 2 b 6 This can be proved by use of Alt. int. ∠s ≅⇒ ll lines. 1 10/15/2014 a Theorem: If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel. Given: ∠4 supplementary to ∠6 Prove: a ll b Given: a ⊥ c and b ⊥ c Prove: a ll b b 1 2 c a 4 b 6 This can be proved by use of Alt. int. ∠s ≅⇒ ll lines. Given: ∠1 ≅ ∠2 Theorem: If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are parallel. ∠MAT ≅ ∠THM Prove: MATH is a parallelogram A 1 Given: ∠2 supplementary to ∠8 Prove: a ll b a T 3 2 4 M H 2 b 8 This can be proved by use of Alt. int. ∠s ≅⇒ ll lines. Theorem: If two coplanar lines are perpendicular to a third line, they are parallel. Given: ∠2 ≅ ∠3 ∠1 is supplementary ∠3 Prove: FROM is a TRAPEZOID S a b R Given: a ⊥ c and b ⊥ c Prove: a ll b F 2 3 O 1 M c This can be proved by use of corr. ∠s ≅⇒ ll lines. 2 10/15/2014 Summary Name the different ways to prove lines are parallel. Homework: worksheet 3 10/15/2014 5.3 Congruent Angles Associated With Parallel Lines Theorem: If two parallel lines are cut by a transversal, then any pair of angles formed are either congruent or supplementary. Objective: After studying this section, you will be able to: a. apply the parallel postulate, b. identify the pairs of angles formed by a transversal cutting parallel lines, and c. apply six theorems about parallel lines. x Theorem: If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent. Postulate: Through a point not on a line there is exactly one parallel to the given line. (short form: ll lines ⇒ alt. ext. ∠s ≅ .) Given: a ll b Prove: ∠1 ≅ ∠8 P a 1 b 8 Theorem: If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. (short form: ll lines ⇒ alt. int. ∠s ≅ .) 1 2 Theorem: If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. (short form: ll lines ⇒ corr. ∠s ≅ .) Given: a ll b Prove: ∠1 ≅ ∠5 a 1 b 5 1 10/15/2014 Theorem: If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary. Theorem: If two lines are parallel to a third line, they are parallel to each other. (Transitive property of Parallel Lines) a Given: a ll b Prove: ∠4 supplementary to ∠6 4 b a Given: a ll b, b ll c Prove: a ll c b 6 c Theorem: If two parallel lines are cut by a transversal each pair of exterior angles on the same side of the transversal are supplementary. If c ll d, find m ∠1 a Given: a ll b Prove: ∠2 supplementary to ∠8 2 b c 1 d 2x + 10 3x + 5 8 Theorem: In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. a b Given: a ll b a⊥c Prove: b ⊥ c Given: FA & DE FA ≅ DE AB ≅ CD Prove:∠F ≅ ∠E F A B C D E c 2 10/15/2014 Given: g&h Prove: ∠1 supplementary ∠2 1 g 2 h If a ll b, find m ∠1 a 3 100 Hint: use the parallel postulate to start 1 b C 40 Summary If lines are parallel, name the different ways angles are congruent or supplementary. Homework: worksheet 3 10/15/2014 So, polygons must have: 5.4 Four Sided Polygons Segments Consecutive sides intersect only at endpoints Nonconsecutive sides should not intersect Objective: a. b. c. d. e. After studying this section, you will be able to: Recognize polygons, Understand how polygons are named, Recognize convex polygons, Recognize diagonals of polygons, and Identify special types of quadrilaterals. Each vertex must belong to only two sides Segments must meet at a vertex Consecutive sides must be noncollinear Polygons are plane figures. To name a polygon begin at any vertex and proceed either clockwise or counterclockwise around the figure. A B C F E D Definition: A convex polygon is a polygon in which each interior angle has a measure less than 180. Non-polygons Convex a e b Not convex (concave) c d 1 10/15/2014 Definition: A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon. What is a Sketch Parallelogram A quad with 2 sets of parallel sides. Rhombus A parallelogram in which at least two consecutive sides are congruent. What is a quadrilateral? Sketch What is a It is a four-sided polygon. Special Quadrilaterals What is a … Trapezoid A quadrilateral with exactly 1 pair of parallel sides (bases). Isosceles Trapezoid Kite A trapezoid in which the non‐ parallel sides (legs) are congruent. (lower and upper base angles) A quad with 2 disjoint pairs of consecutive congruent sides. Rectangle A parallelogram with at least one right angle. Square A parallelogram that is both a rectangle and a rhombus. Quadrilateral Sketch Parallelogram Trapezoid Isosceles Trapezoid Square Rhombus Rectangle Kite a. Trapezoid b. Rectangle e. Rhombus f. Isosceles Trapezoid C. Square D. Quadrilateral g. Parallelogram h. Kite 2 10/15/2014 Summary Describe the difference between convex and concave. Explain how you will remember the special quadrilaterals. Homework: worksheet 3 10/15/2014 Properties of Kites 5.5 Properties of Quadrilaterals Objective: After studying this section, you will be able to identify some properties of: parallelograms, rectangles, kites, rhombuses, squares, and isosceles triangles. a. b. c. d. e. f. Properties of Parallelograms 2 1 3 1. Two disjoint pair of consecutive sides are congruent by definition 2. Diagonals are perpendicular 3. One diagonal is the perpendicular bisector of the other 4. One of the diagonals bisects a pair of opposite angles 5. One pair of opposite angles are congruent Properties of Rhombuses 4 1. Opposite sides are parallel by definition 2. Opposite sides are congruent 3. Opposite angles are congruent angles 1 and 3 are congruent; angles 2 and 4 are congruent 4. Diagonals bisect each other 5. Any pair of consecutive angles are supplementary 1. All properties of a parallelogram apply by definition 2. All properties of a kite apply 3. All sides are congruent (equilateral) 4. Diagonals bisect the angles 5. Diagonals are perpendicular bisectors of each other 6. Diagonals divide the rhombus into four congruent right triangles Properties of Rectangles Properties of Squares 1. All the properties of a parallelogram apply by definition. 2. All angles are right angles 1. All the properties of a rectangle apply by definition 2. All the properties of a rhombus apply by definition 3. Diagonals form four isosceles right triangles 3. Diagonals are congruent 1 10/15/2014 Properties of Isosceles Trapezoid A Given: ABCD is a parallelogram 3 2 D E Prove: AC and BD bisect each other 1 B 1. Legs are congruent by definition 2. Bases are parallel by definition 3. Lower base angles are congruent 4. Upper base angles are congruent 5. Diagonals are congruent 6. Any lower base angle is supplementary to any upper base angle E D C Given: ABCD is a parallelogram ∠GHA ≅ ∠FEC HB ≅ DE Conclusion: A 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. H 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. C Summary Draw each of the figures and apply the properties in a foldable. B Homework: worksheet A Given: VRZA is a parallelogram AV = 2x - 4 VR = 3y + 5 RZ = 1/2x + 8 ZA = y + 12 Find: The perimeter of VRZA 1. F G GH ≅ EF 1. 4 V Z R 2 10/15/2014 E F D Given: ACDF is a parallelogram ∠AFB ≅ ∠ECD 5.6 Proving That a Quadrilateral is a Parallelogram Prove: FBCE is a parallelogram A Objective: After studying this section, you will be able to prove that a quadrilateral is a parallelogram. Methods to prove quadrilateral ABCD is a parallelogram A D 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. C Given: +CAR is isosceles, with base CR AC ≅ BK ∠C ≅ ∠K B 1. If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram (reverse of the definition). 2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram (converse of a property). 3. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram. 4. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram (converse of a property). 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. B A C Prove: BARK is a parallelogram C B R K ) (x ) Q ⎡3 x ⎣ Given: Quadrilateral QUAD with angles as shown Show that QUAD is a parallelogram ( (x ⎡ x ⎢⎣ 2 ) D − 5 x ⎤⎦ 2 5 ⎤ ⎥⎦ (3 x 10 3 − 15x 2 ) A U 5. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram (converse of a property). 1 10/15/2014 Given: NRTW is a parallelogram V W NX ≅ TS WV ≅ PR T X S Prove: XPSV is a parallelogram N 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. P R Summary Using one of the methods to prove quadrilaterals are parallelograms, create your own problem and show how it is a parallelogram. Homework: worksheet 2 10/15/2014 Proving that a quadrilateral is a rhombus J O K 5.7 Proving That Figures Are Special Quadrilaterals M Show that the quadrilateral is a parallelogram first then apply either of the following methods. Objective: 1. If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus (reverse of the definition). After studying this section, you will be able to prove that a quadrilateral is: a. A rectangle b. A kite c. A rhombus d. A square e. An isosceles triangle 2. If either diagonals of a parallelogram bisects two angles of the parallelogram, then it is a rhombus. You can prove that a quadrilateral is a rhombus without first showing that it is a parallelogram 3. If the diagonals of a quadrilateral are perpendicular bisectors of each other, then the quadrilateral is a rhombus. Proving that a quadrilateral is a rectangle E H F G Show that the quadrilateral is a parallelogram first then use one of the methods to complete the proof. 1. If a parallelogram contains at least one right angle, then it is a rectangle (reverse of the definition). Proving that a quadrilateral is a square N S P R The following method can be used to prove the NPRS is a square. 1. If a quadrilateral is both a rectangle and a rhombus, then it is a square (reverse of the definition). 2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. You can prove that a quadrilateral is a rectangle without first showing that it is a parallelogram 3. If all four angles of a quadrilateral are right angles, then it is a rectangle. Proving that a quadrilateral is a kite K E I T To prove that a quadrilateral is a kite, either of the following methods can be used. 1. If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it is a kite (reverse of the definition). 2. If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. Proving that a trapezoid is an isosceles A D B C 1. If the nonparallel sides of a trapezoid are congruent, then it is isosceles (reverse of the definition). 2. If the lower or the upper base angles of a trapezoid are congruent, then it is isosceles. 3. If the diagonals of a trapezoid are congruent, then it is isosceles. 1 10/15/2014 Summary What is the most descriptive name for quadrilateral ABCD with vertices A = (-3, -7), B = (-9, 1), C = (3, 9), and D = (9, 1)? Write a description of each of three special quadrilaterals without using the names of the quadrilaterals. Each description should include sufficient properties to establish the quadrilateral’s identity. Homework: worksheet A Given: ABCD is a parallelogram D BD bisects ∠ADC and ∠ABC B Prove: ABCD is a rhombus Given: GJMO is a parallelogram C M O OH ⊥ GK MK is an altitude of +MKJ Prove: OHKM is a rectangle G H J K 2