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Quadrilaterals Eleanor Roosevelt High School Chin-Sung Lin ERHS Math Geometry Definitions of the Quadrilaterals Mr. Chin-Sung Lin ERHS Math Geometry Quadrilaterals A quadrilateral is a polygon with four sides Mr. Chin-Sung Lin ERHS Math Geometry Parts & Properties of the Quadrilaterals Mr. Chin-Sung Lin ERHS Math Geometry Consecutive (Adjacent) Vertices Consecutive vertices or adjacent vertices are vertices that are endpoints of the same side P and Q, Q and R, R and S, S and P Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Consecutive (Adjacent) Sides Consecutive sides or adjacent sides are sides that have a common endpoint PQ and QR, QR and RS, RS and SP, SP and PQ Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Opposite Sides Opposite sides of a quadrilateral are sides that do not have a common endpoint PQ and RS, SP and QR Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Consecutive angles Consecutive angles of a quadrilateral are angles whose vertices are consecutive P and Q, Q and R, R and S, S and P Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Opposite Angles Opposite angles of a quadrilateral are angles whose vertices are not consecutive P and R, Q and S Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Diagonals A diagonal of a quadrilateral is a line segment whose endpoints are two nonadjacent vertices of the quadrilateral PR and QS Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Sum of the Measures of Angles The sum of the measures of the angles of a quadrilateral is 360 degrees m P+m Q+m R+m S = 360 Q P S R Mr. Chin-Sung Lin ERHS Math Geometry Parallelograms Mr. Chin-Sung Lin ERHS Math Geometry A B Parallelogram D C A parallelogram is a quadrilateral in which two pairs of opposite sides are parallel AB || CD, AD || BC A parallelogram can be denoted by the symbol ABCD The use of arrowheads, pointing in the same direction, to show sides that are parallel in the figure Mr. Chin-Sung Lin ERHS Math Geometry Theorems of Parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Theorems of Parallelogram Theorem of Dividing Diagonals Theorem of Opposite Sides Theorem of Opposite Angles Theorem of Bisecting Diagonals Theorem of Consecutive Angles Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Dividing Diagonals A diagonal divides a parallelogram into two congruent triangles A If ABCD is a parallelogram, then ∆ ABD ∆ CDB D B C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Dividing Diagonals A B 1 3 4 D 2 Statements C Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. 1 2 and 3 4 3. Alternate interior angles 4. BD BD 4. Reflexive property 5. ∆ ABD ∆ CDB 5. ASA postulate Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Opposite Sides Opposite sides of a parallelogram are congruent If ABCD is a parallelogram, then AB CD, and BC DA A D B C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Opposite Sides A B 1 3 4 D 2 C Statements 1. 2. 3. 4. 5. 6. 7. ABCD is a parallelogram Connect BD AB || DC and AD || BC 1 2 and 3 4 BD BD ∆ ABD ∆ CDB AB CD and BC DA Reasons 1. 2. 3. 4. 5. 6. 7. Given Form two triangles Definition of parallelogram Alternate interior angles Reflexive property ASA postulate CPCTC Mr. Chin-Sung Lin ERHS Math Geometry Application Example 1 ABCD is a parallelogram, what’s the perimeter of ABCD ? A B 15 10 D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 1 ABCD is a parallelogram, what’s the perimeter of ABCD ? A B 15 perimeter = 50 10 D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 2 ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x A B x-20 10 D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 2 ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x A B x-20 x = 50 10 D C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Opposite Angles Opposite angles of a parallelogram are congruent If ABCD is a parallelogram, then A C, and B D A D B C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Opposite Angles A Statements D 1. ABCD is a parallelogram 2. AB || DC and AD || BC 3. A and B are supplementary A and D are supplementary C and B are supplementary 4. A C B D B C Reasons 1. Given 2. Definition of parallelogram 3. Same side interior angles 4. Supplementary angle theorem ERHS Math Geometry Application Example 3 ABCD is a parallelogram, what are the values of x and y? A B 120o y D 60o x C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 3 ABCD is a parallelogram, what are the values of x and y? A x = 120o y = 60o B 120o y D 60o x C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 4 ABCD is a parallelogram, what are the values of x and y? A X+20 180 - y D B y - 20 2x - 60 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 4 ABCD is a parallelogram, what are the values of x and y? A x = 80o y = 100o X+20 180 - y D B y - 20 2x - 60 C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Bisecting Diagonals The diagonals of a parallelogram bisect each other If ABCD is a parallelogram, then AC and BD bisect each other at O A B O D C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Bisecting Diagonals A B 1 3 O 4 Statements 2 D C Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC 2. Definition of parallelogram 3. 1 2 and 3 4 3. Alternate interior angles 4. AB DC 4. Opposite sides congruent 5. ∆ AOB ∆ COD 5. ASA postulate 6. AO = OC and BO = OD 6. CPCTC 7. AC and BD bisect each other 7. Definition of segment bisector Mr. Chin-Sung Lin ERHS Math Geometry Application Example 5 ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? A B 6 3 4 O D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 5 ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? A B 6 AC + BD = 14 3 4 O D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 6 ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y A B x+4 O y+2 D 2y-6 3x-4 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 6 ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y A x=4 y=8 B x+4 O y+2 D 2y-6 3x-4 C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Consecutive Angles The consecutive angles of a parallelogram are supplementary If ABCD is A and C and A and B and a parallelogram, then B are supplementary D are supplementary D are supplementary C are supplementary A D B C Mr. Chin-Sung Lin ERHS Math Geometry Theorem of Consecutive Angles A Statements D B C Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. A and B, C and D 3. Same-side interior angles A and D, B and C are supplementary are supplementary Mr. Chin-Sung Lin ERHS Math Geometry Application Example 7 ABCD is a parallelogram, what are the values of x, y and z? A B 120o z D x y C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 7 ABCD is a parallelogram, what are the values of x, y and z? A x = 60o y = 120o z = 60o B 120o z D x y C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 8 ABCD is a parallelogram, what are the values of x and y? A B X+30 X-30 Y+20 D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 8 ABCD is a parallelogram, what are the values of x and y? A x = 90o y = 100o B X+30 X-30 Y+20 D C Mr. Chin-Sung Lin ERHS Math Geometry Group Work Mr. Chin-Sung Lin ERHS Math Geometry Question 1 ABCD is a parallelogram, calculate the perimeter of ABCD A B x+30 2y-10 D y+10 2x-10 C Mr. Chin-Sung Lin ERHS Math Geometry Question 1 ABCD is a parallelogram, calculate the perimeter of ABCD perimeter = 200 A B x+30 2y-10 D y+10 2x-10 C Mr. Chin-Sung Lin ERHS Math Geometry Question 2 ABCD is a parallelogram, solve for x A B X+30 X-10 O X+10 D 2X C Mr. Chin-Sung Lin ERHS Math Geometry Question 2 ABCD is a parallelogram, solve for x A x = 30 B X+30 X-10 O X+10 D 2X C Mr. Chin-Sung Lin ERHS Math Geometry Question 3 Given: ABCD is a parallelogram Prove: XO YO A X B O D Y C Mr. Chin-Sung Lin ERHS Math Geometry Question 4 Given: ABCD is a parallelogram, BO OD Prove: EO OF A E B O D F C Mr. Chin-Sung Lin ERHS Math Geometry Question 5 Given: ABCD is a parallelogram, AF || CE Prove: FAB ECD A B E F D C Mr. Chin-Sung Lin ERHS Math Geometry Review: Theorems of Parallelogram Theorem of Dividing Diagonals Theorem of Opposite Sides Theorem of Opposite Angles Theorem of Bisecting Diagonals Theorem of Consecutive Angles Mr. Chin-Sung Lin ERHS Math Geometry Prove Quadrilaterals are Parallelograms Mr. Chin-Sung Lin ERHS Math Geometry Criteria for Proving Parallelograms Parallel opposite sides Congruent opposite sides Congruent & parallel opposite sides Congruent opposite angles Supplementary consecutive angles Bisecting diagonals Mr. Chin-Sung Lin ERHS Math Geometry Parallel Opposite Sides If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram A AB || CD, and BC || DA then, ABCD is a parallelogram D B If C Mr. Chin-Sung Lin ERHS Math Geometry Parallel Opposite Sides A D Statements B C Reasons 1. AB || CD and BC || DA 1. Given 2. ABCD is a parallelogram 2. Definition of parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Application Example 1 If m1 = m2 = m3, then ABCD is a parallelogram A B 1 2 3 D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 2 ABCD is a quadrilateral as shown below, solve for x A 60o D B 3x-20 50o 50o 2x+10 60o C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 2 ABCD is a quadrilateral as shown below, solve for x A x = 30 60o D B 3x-20 50o 50o 2x+10 60o C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Opposite Sides If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram A AB CD, and BC DA then, ABCD is a parallelogram If D B C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Opposite Sides If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram A AB CD, and BC DA then, ABCD is a parallelogram If D B C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Opposite Sides A B 1 3 4 Statements 1. 2. 3. 4. 5. 6. D 2 Connect BD AB CD and BC DA BD BD ∆ ABD ∆ CDB 1 2 and 3 4 AB || DC and AD || BC 7. ABCD is a parallelogram C Reasons 1. Form two triangles 2. Given 3. Reflexive property 4. SSS postulate 5. CPCTC 6. Converse of alternate interior angles theorem 7. Definition of parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Application Example 3 ABCD is a quadrilateral, solve for x A B 15 X+50 10 10 2x-30 D 15 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 3 ABCD is a quadrilateral, solve for x A x = 80 B 15 X+50 10 10 2x-30 D 15 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 4 ABCD is a parallelogram, if DF = BE, then AECF is also a parallelogram A D E F B C Mr. Chin-Sung Lin ERHS Math Geometry Congruent & Parallel Opposite Sides If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram AB CD, and AB || CD then, ABCD is a parallelogram A If D B C Mr. Chin-Sung Lin ERHS Math Geometry Congruent & Parallel Opposite Sides A B 1 3 4 Statements 1. 2. 3. 4. 5. 6. 7. D 2 Connect BD AB CD and AB || CD BD BD 1 2 ∆ ABD ∆ CDB 3 4 AD || BC C Reasons 1. Form two triangles 2. Given 3. Reflexive property 4. Alternate interior angles 5. SAS postulate 6. CPCTC 7. Converse of alternate interior angles theorem 8. ABCD is a parallelogram 8. Definition of parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Application Example 5 ABCD is a quadrilateral, solve for x and y A B X+5 y+50o 30o 10 30o D 10 2y-20o C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 5 ABCD is a quadrilateral, solve for x and y A x=5 y = 70o B X+5 y+50o 30o 10 30o D 10 2y-20o C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 6 ABCD is a parallelogram, if m1 = m2, then AECF is also a parallelogram A E B 1 2 D F C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Opposite Angles If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram A A C, and B D Then, ABCD is a parallelogram D B If C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Opposite Angles A B 1 3 4 Statements D 2 1. Connect BD 2. m1 +m4 + mA 180 m2 +m3 + mC 180 3. m1 +m4 + mA + m2 +m3 + mC 360 4. m1 +m3 = mB m4 +m2 = mD 5. mA +mB + mC + mD = 360 C Reasons 1. Form two triangles 2. Triangle angle-sum theorem 3. Addition property 4. Partition property 5. Substitution property Mr. Chin-Sung Lin ERHS Math Geometry Congruent Opposite Angles A B 1 3 4 Statements D 2 6. A C and B D 7. 2mA + 2mB = 360 2mA + 2mD = 360 8. mA + mB = 180 mA + mD = 180 9. AD || BC, AB || DC 10. ABCD is a parallelogram C Reasons 6. Given 7. Substitution property 8. Division property 9. Converse of same-side interior angles 10. Definition of parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Application Example 7 ABCD is a quadrilateral, solve for x A X+30 130o 50o D B 50o 130o 2x-40 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 7 ABCD is a quadrilateral, solve for x A x = 70 X+30 130o 50o D B 50o 130o 2x-40 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 8 if m1 = m2, m3 = m4, then ABCD is a parallelogram 1 A 4 B 3 D C 2 Mr. Chin-Sung Lin ERHS Math Geometry Bisecting Diagonals If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram If AC and BD bisect each other at O, then, ABCD is a parallelogram A B O D C Mr. Chin-Sung Lin ERHS Math Geometry Bisecting Diagonals A B 1 O 4 Statements 3 2 D C Reasons 1. AC and BD bisect at O 1. Given 2. AO CO and BO DO 2. Def. of segment bisector 3. 4. 5. 6. 3. 4. 5. 6. AOB COD, AOD COB ∆AOB ∆COD, ∆AOD ∆COB 1 2 and 3 4 AB || DC and AD || BC 7. ABCD is a parallelogram Vertical angles SAS postulate CPCTC Converse of alternate interior angles theorem 7. Definition of parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Application Example 9 ∆ AOB ∆ COD, then ABCD is a parallelogram A B O D C Mr. Chin-Sung Lin ERHS Math Geometry Supplementary Consecutive Angles If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram If A and B are supplementary A and D are supplementary then, ABCD is a parallelogram A D B C Mr. Chin-Sung Lin ERHS Math Geometry Supplementary Consecutive Angles A D Statements 1. A and B, A and D B C Reasons 1. Given are supplementary 2. AB || DC and AD || BC 2. Converse of same-side interior angles theorem 3. ABCD is a parallelogram 3. Definition of parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Application Example 10 ABCD is a quadrilateral, solve for x A 2x+80 3x D B 100-2x 2(x+45)-10 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example 10 ABCD is a quadrilateral, solve for x A 2x+80 x = 20 3x D B 100-2x 2(x+45)-10 C Mr. Chin-Sung Lin ERHS Math Geometry Review: Proving Parallelograms Parallel opposite sides Congruent opposite sides Congruent & parallel opposite sides Congruent opposite angles Supplementary consecutive angles Bisecting diagonals Mr. Chin-Sung Lin ERHS Math Geometry Rectangles Mr. Chin-Sung Lin ERHS Math Geometry Rectangles A rectangle is a parallelogram containing one right angle A B D C Mr. Chin-Sung Lin ERHS Math Geometry All Angles Are Right Angles All angles of a rectangle are right angles Given: ABCD is a rectangle with A = 90o Prove: B = 90o, C = 90o, D = 90o A B D C Mr. Chin-Sung Lin ERHS Math Geometry All Angles Are Right Angles Statements A B D C Reasons 1. ABCD is a rectangle & A = 90o 1. Given 2. C = 90o 2. Opposite angles 3. mA + mD = 180 mA + mB = 180 4. 90 + mD = 180 90 + mB = 180 5. mB = 90, mD = 90 6. B = 90o, D = 90o 3. Consecutive angles 4. Substitution 5. Subtraction 6. Def. of measurement of angles Mr. Chin-Sung Lin ERHS Math Geometry Diagonals Are Congruent The diagonals of a rectangle are congruent Given: ABCD is a rectangle Prove: AC BD A B D C Mr. Chin-Sung Lin ERHS Math Geometry All Angles Are Right Angles Statements A B D C Reasons 1. ABCD is a rectangle 1. Given 2. C = 90o, D = 90o 2. All angles are right angles 3. 4. 5. 6. 7. 3. 4. 5. 6. 7. C D DC DC AD BC ∆ADC ∆BCD AC BD Substitution Reflexive Opposite sides SAS postulate CPCTC Mr. Chin-Sung Lin ERHS Math Geometry Properties of Rectangle The properties of a rectangle All the properties of a parallelogram Four right angles (equiangular) A B D C Congruent diagonals Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangles Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangles To show that a quadrilateral is a rectangle, by showing that the quadrilateral is equiangular or a parallelogram that contains a right angle, or with congruent diagonals If a parallelogram does not contain a right angle, or doesn’t have congruent diagonals, then it is not a rectangle Mr. Chin-Sung Lin ERHS Math Geometry A B D C Proving Rectangles If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle Given: ABCD is a parallelogram and mA = 90 Prove: ABCD is a rectangle Mr. Chin-Sung Lin ERHS Math Geometry A B D C Proving Rectangles If a quadrilateral is equiangular, it is a rectangle Given: ABCD is a quadrangular & mA = mB = mC = mD Prove: ABCD is a rectangle Mr. Chin-Sung Lin ERHS Math Geometry B A Proving Rectangles O D C The diagonals of a parallelogram are congruent Given: AC BD Prove: ABCD is a rectangle Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a parallelogram, mA = 6x - 30 and mC = 4x + 10. Show that ABCD is a rectangle A B D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a parallelogram, mA = 6x - 30 and mC = 4x + 10. Show that ABCD is a rectangle A B D C x =20 mA = 90 ABCD is a rectangle Mr. Chin-Sung Lin ERHS Math Geometry Rhombuses Mr. Chin-Sung Lin ERHS Math Geometry Rhombus A rhombus is a parallelogram that has two congruent consecutive sides A D B C Mr. Chin-Sung Lin ERHS Math Geometry All Sides Are Congruent All sides of a rhombus are congruent Given: ABCD is a rhombus with AB Prove: AB BC CD DA A D DA B C Mr. Chin-Sung Lin ERHS Math Geometry All Sides Are Congruent A D B C Statements Reasons 1. ABCD is a rhombus w. AB 2. AB DC, AD congruent 3. AB BC CD BC DA DA 1. Given 2. Opposite sides are 3. Transitive Mr. Chin-Sung Lin ERHS Math Geometry Perpendicular Diagonals The diagonals of a rhombus are perpendicular to each other A Given: ABCD is a rhombus Prove: AC BD D O B C Mr. Chin-Sung Lin A ERHS Math Geometry Perpendicular Diagonals Statements O D Reasons 1. ABCD is a rhombus 1. Given 2. AO AO 2. Reflexive 3. AD AB 4. BO DO 5. ∆AOD ∆AOB 6. AOD AOB 7. mAOD + mAOB = 180 8. 2mAOD = 180 9. AOD = 90o 10. AC BD 3. 4. 5. 6. 7. 8. B C Congruent sides Bisecting diagonals SSS postulate CPCTC Supplementary angles Substitution 9. Division postulate 10. Definition of perpendicular ERHS Math Geometry Diagonals Bisecting Angles The diagonals of a rhombus bisect its angles Given: ABCD is a rhombus A Prove: AC bisects DAB and DCB DB bisects CDA and CBA D B C Mr. Chin-Sung Lin A ERHS Math Geometry Diagonals Bisecting Angles Statements D Reasons 1. ABCD is a rhombus 1. Given 2. AD AB, DC BC 2. Congruent sides B C AD DC, AB BC 3. AC AC, DB DB 4. ∆ACD ∆ACB, ∆BAD ∆BCD 5. DAC BAC, DCA BCA ADB CDB, ABD CBD 6. AC bisects DAB and DCB DB bisects CDA and CBA 3. Reflexive postulate 4. SSS postulate 5. CPCTC 6. Definition of angle bisector Mr. Chin-Sung Lin ERHS Math Geometry Properties of Rhombus A The properties of a rhombus All the properties of a parallelogram D B Four congruent sides (equilateral) Perpendicular diagonals C Diagonals that bisect opposite pairs of angles Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombus Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombus To show that a quadrilateral is a rhombus, by showing that the quadrilateral is equilateral or a parallelogram that contains two congruent consecutive sides with perpendicular diagonals, or with diagonals bisecting opposite angles If a parallelogram does not contain two congruent consecutive sides, or doesn’t have perpendicular diagonals, then it is not a rectangle Mr. Chin-Sung Lin A ERHS Math Geometry Proving Rhombus D B If a parallelogram has two congruent consecutive C sides, then the parallelogram is a rhombus Given: ABCD is a parallelogram and AB Prove: ABCD is a rhombus DA Mr. Chin-Sung Lin A ERHS Math Geometry Proving Rhombus D If a quadrilateral is equilateral, it is a rhombus B C Given: ABCD is a parallelogram and AB BC CD DA Prove: ABCD is a rhombus Mr. Chin-Sung Lin ERHS Math Geometry A Proving Rhombus D B The diagonals of a parallelogram are perpendicular C Given: AC BD Prove: ABCD is a rhombus Mr. Chin-Sung Lin ERHS Math Geometry A 1 2 Proving Rhombus D B 3 4 If each diagonal of a parallelogram bisects two opposite angles, then it is a rhombus C Given: AC bisects DAB and DCB BD bisects ADC and ABC Prove: ABCD is a rhombus Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13 Prove: ABCD is a rhombus A B 2x+1 x+13 D 3x-11 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13 Prove: ABCD is a rhombus A x = 12 AB = AD = 25 ABCD is a rhombus B 2x+1 x+13 D 3x-11 C Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus A D B C Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus A x=4 AB = BC = 10 D B ABCD is a rhombus C Mr. Chin-Sung Lin ERHS Math Geometry Squares Mr. Chin-Sung Lin ERHS Math Geometry Squares A square is a rectangle that has two congruent consecutive sides A B D C Mr. Chin-Sung Lin ERHS Math Geometry Squares A square is a rectangle with four congruent sides (an equilateral rectangle) A B D C Mr. Chin-Sung Lin ERHS Math Geometry Squares A square is a rhombus with four right angles (an equiangular rhombus) A B D C Mr. Chin-Sung Lin ERHS Math Geometry Squares A square is an equilateral quadrilateral A square is an equiangular quadrilateral A B D C Mr. Chin-Sung Lin ERHS Math Geometry Squares A square is a rhombus A square is a rectangle A B D C Mr. Chin-Sung Lin ERHS Math Geometry Properties of Square The properties of a square All the properties of a parallelogram All the properties of a rectangle A B D C All the properties of a rhombus Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares If a rectangle has two congruent consecutive sides, then the rectangle is a square A B D C Given: ABCD is a rectangle and AB DA Prove: ABCD is a square Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares If one of the angles of a rhombus is a right angle, then the rhombus is a square Given: ABCD is a rhombus and A = 90o Prove: ABCD is a square A B D C Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares To show that a quadrilateral is a square, by showing that the quadrilateral is a rectangle with a pair of congruent consecutive sides, or a rhombus that contains a right angle Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y A B D C Mr. Chin-Sung Lin ERHS Math Geometry Application Example ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y A B D C 4x – 30 = 90 x = 30 y = 25 Mr. Chin-Sung Lin ERHS Math Geometry Review Questions Mr. Chin-Sung Lin ERHS Math Geometry Question 1 A parallelogram where all angles are right angles (90o) is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 1 Answer A parallelogram where all angles are right angles (90o) is a _________? Rectangle Mr. Chin-Sung Lin ERHS Math Geometry Question 2 A parallelogram where all sides are congruent is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 2 Answer A parallelogram where all sides are congruent is a _________? Rhombus Mr. Chin-Sung Lin ERHS Math Geometry Question 3 A rectangle with four congruent sides is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 3 Answer A rectangle with four congruent sides is a _________? Square Mr. Chin-Sung Lin ERHS Math Geometry Question 4 A rhombus with four right angles is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 4 Answer A rhombus with four right angles is a _________? Square Mr. Chin-Sung Lin ERHS Math Geometry Question 5 A parallelogram with congruent diagonals is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 5 Answer A parallelogram with congruent diagonals is a _________? Rectangle Mr. Chin-Sung Lin ERHS Math Geometry Question 6 A parallelogram where all angles are right angles and all sides are congruent is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 6 Answer A parallelogram where all angles are right angles and all sides are congruent is a _________? Square Mr. Chin-Sung Lin ERHS Math Geometry Question 7 A parallelogram with perpendicular diagonals is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 7 Answer A parallelogram with perpendicular diagonals is a _________? Rhombus Mr. Chin-Sung Lin ERHS Math Geometry Question 8 A parallelogram whose diagonals bisect opposite pairs of angles is a ______? Mr. Chin-Sung Lin ERHS Math Geometry Question 8 Answer A parallelogram whose diagonals bisect opposite pairs of angles is a ______? Rhombus Mr. Chin-Sung Lin ERHS Math Geometry Question 9 A quadrilateral which is both rectangle and rhombus is a _________? Mr. Chin-Sung Lin ERHS Math Geometry Question 9 Answer A quadrilateral which is both rectangle and rhombus is a _________? Square Mr. Chin-Sung Lin ERHS Math Geometry Question 10 Choose the right answer(s): 1. 2. 3. A parallelogram is a rhombus A rectangle is a square A rhombus is a parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Question 10 Answer Choose the right answer(s): 1. A parallelogram is a rhombus 2. A rectangle is a square 3. A rhombus is a parallelogram Mr. Chin-Sung Lin ERHS Math Geometry Question 11 Choose the right answer(s): 1. 2. 3. A quadrilateral is a parallelogram A square is a rhombus A rectangle is a rhombus Mr. Chin-Sung Lin ERHS Math Geometry Question 11 Answer Choose the right answer(s): 1. A quadrilateral is a parallelogram 2. A square is a rhombus 3. A rectangle is a rhombus Mr. Chin-Sung Lin ERHS Math Geometry Question 12 Choose the right answer(s): 1. 2. 3. A rectangle is a parallelogram A square is a rectangle A rhombus is a square Mr. Chin-Sung Lin ERHS Math Geometry Question 12 Answer Choose the right answer(s): 1. A rectangle is a parallelogram 2. A square is a rectangle 3. A rhombus is a square Mr. Chin-Sung Lin ERHS Math Geometry Trapezoids Mr. Chin-Sung Lin ERHS Math Geometry Definitions of Trapezoids Mr. Chin-Sung Lin ERHS Math Geometry Trapezoids A trapezoid is a quadrilateral that has exactly one pair of parallel sides The parallel sides of a trapezoid are called bases. The nonparallel sides of a trapezoid are the legs A Upper base Leg D B Leg Lower base C Mr. Chin-Sung Lin ERHS Math Geometry Isosceles Trapezoids A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid A Upper base Leg D B Leg Lower base C Mr. Chin-Sung Lin ERHS Math Geometry Median of a Trapezoid The median of a trapezoid is the line segment connecting the midpoints of the nonparallel sides A Upper base B Median D Lower base C Mr. Chin-Sung Lin ERHS Math Geometry Examples of Trapezoids B A D A 80o 100o 100o 80o 70o 135o 110o D 120o 90o 90o C A B 60o D B A C 70o 45o C 60o B 110o 120o D C Mr. Chin-Sung Lin ERHS Math Geometry Exercise - Trapezoids Which one is a trapezoid? Why? A D B 75o 130o C 110o D 45o C 105o A 75o B Mr. Chin-Sung Lin ERHS Math Geometry Exercise - Trapezoids Which one is a trapezoid? Why? A D B 75o 130o C 110o D 45o C 105o A 75o B Mr. Chin-Sung Lin ERHS Math Geometry Exercise - Trapezoids Which one is a trapezoid? D 80o 100o 90o C 90o A B Mr. Chin-Sung Lin ERHS Math Geometry Exercise - Trapezoids Which one is a trapezoid? D 80o 100o 90o C 90o A B Mr. Chin-Sung Lin ERHS Math Geometry Properties of Isosceles Trapezoids Mr. Chin-Sung Lin ERHS Math Geometry Properties of Isosceles Trapezoids The properties of a isosceles trapezoid Base angles are congruent Diagonals are congruent The property of a trapezoid Median is parallel to and average of the bases Mr. Chin-Sung Lin ERHS Math Geometry Congruent Base Angles In an isosceles trapezoid the two angles whose vertices are the endpoints of either base are congruent The upper and lower base angles are congruent Given: Isosceles trapezoid ABCD AB || CD and AD BC A B Prove: A B; C D D C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Base Angles Given: Isosceles trapezoid ABCD AB || CD and AD BC Prove: A B; C D B A D E A C D B C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Diagonals The diagonals of an isosceles trapezoid are congruent Given: Isosceles trapezoid ABCD AB || CD and AD BC A B Prove: AC BD D C Mr. Chin-Sung Lin ERHS Math Geometry Congruent Diagonals Given: Isosceles trapezoid ABCD AB || CD and AD BC A B Prove: AC BD D C Mr. Chin-Sung Lin ERHS Math Geometry Parallel and Average Median The median of a trapezoid is parallel to the bases, and its length is half the sum of the lengths of the bases Given: Isosceles trapezoid ABCD AB || CD and median EF Prove: AB || EF , CD || EF and EF = (1/2)(AB + CD) A E D B F C Mr. Chin-Sung Lin ERHS Math Geometry Parallel and Average Median Given: Isosceles trapezoid ABCD AB || CD and median EF Prove: AB || EF , CD || EF and EF = (1/2)(AB + CD) A E D B F C G H Mr. Chin-Sung Lin ERHS Math Geometry Proving Trapezoids Mr. Chin-Sung Lin ERHS Math Geometry Proving Trapezoids To prove that a quadrilateral is a trapezoid, show that two sides are parallel and the other two sides are not parallel To prove that a quadrilateral is not a trapezoid, show that both pairs of opposite sides are parallel or that both pairs of opposite sides are not parallel Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids To prove that a trapezoid is an isosceles trapezoid, show that one of the following statements is true: The legs are congruent The lower/upper base angles are congruent The diagonals are congruent Mr. Chin-Sung Lin ERHS Math Geometry Application Examples Mr. Chin-Sung Lin ERHS Math Geometry Numeric Example of Trapezoids Isosceles Trapezoid ABCD, AB || CD and AD BC Solve for x and y A D xo 2xo B 3yo C Mr. Chin-Sung Lin ERHS Math Geometry Numeric Example of Trapezoids Isosceles Trapezoid ABCD, AB || CD and AD BC Solve for x and y A x = 60 y = 20 D xo 2xo B 3yo C Mr. Chin-Sung Lin ERHS Math Geometry Numeric Example of Trapezoids Trapezoid ABCD, AB || CD and median EF Solve for x A E D 2x 2x + 4 3x + 2 B F C Mr. Chin-Sung Lin ERHS Math Geometry Numeric Example of Trapezoids Trapezoid ABCD, AB || CD and median EF Solve for x A E x=6 D 2x 2x + 4 3x + 2 B F C Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Given: Trapezoid ABCD and A B Prove: ABCD is an isosceles trapezoid A D B C Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Given: Trapezoid ABCD and AC BD Prove: ABCD is an isosceles trapezoid B A O D C Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Given: Trapezoid ABCD, AB || CD and AE BE Prove: ABCD is an isosceles trapezoid E A D B C Mr. Chin-Sung Lin ERHS Math Geometry Summary of Quadrilaterals Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 1 Properties Cong. Oppo. Sides (1 P) Cong. Oppo. Sides (2 P) Cong. Four Sides Parallel Oppo. Sides (1P) Parallel Oppo. Sides (2P) Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 1 Properties Cong. Oppo. Sides (1 P) Cong. Oppo. Sides (2 P) Cong. Four Sides Parallel Oppo. Sides (1P) Parallel Oppo. Sides (2P) Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 1 Properties Cong. Oppo. Sides (1 P) Cong. Oppo. Sides (2 P) Cong. Four Sides Parallel Oppo. Sides (1P) Parallel Oppo. Sides (2P) Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 1 Properties Cong. Oppo. Sides (1 P) Cong. Oppo. Sides (2 P) Cong. Four Sides Parallel Oppo. Sides (1P) Parallel Oppo. Sides (2P) Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 1 Properties Cong. Oppo. Sides (1 P) Cong. Oppo. Sides (2 P) Cong. Four Sides Parallel Oppo. Sides (1P) Parallel Oppo. Sides (2P) Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 1 Properties Cong. Oppo. Sides (1 P) Cong. Oppo. Sides (2 P) Cong. Four Sides Parallel Oppo. Sides (1P) Parallel Oppo. Sides (2P) Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 3 Properties Cong. Adj. Angles (1 P) Cong. Adj. Angles (2 P) Cong. Four Right Angles Diagonals Bisect Angles Non-Parallel Oppo. Sides Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 3 Properties Cong. Adj. Angles (1 P) Cong. Adj. Angles (2 P) Cong. Four Right Angles Diagonals Bisect Angles Non-Parallel Oppo. Sides Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 3 Properties Cong. Adj. Angles (1 P) Cong. Adj. Angles (2 P) Cong. Four Right Angles Diagonals Bisect Angles Non-Parallel Oppo. Sides Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 3 Properties Cong. Adj. Angles (1 P) Cong. Adj. Angles (2 P) Cong. Four Right Angles Diagonals Bisect Angles Non-Parallel Oppo. Sides Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 3 Properties Cong. Adj. Angles (1 P) Cong. Adj. Angles (2 P) Cong. Four Right Angles Diagonals Bisect Angles Non-Parallel Oppo. Sides Mr. Chin-Sung Lin ERHS Math Geometry Properties of Quadrilaterals - 3 Properties Cong. Adj. Angles (1 P) Cong. Adj. Angles (2 P) Cong. Four Right Angles Diagonals Bisect Angles Non-Parallel Oppo. Sides Mr. Chin-Sung Lin ERHS Math Geometry Quadrilaterals and Proofs Mr. Chin-Sung Lin ERHS Math Geometry Quadrilaterals and Proofs Given: Isosceles trapezoid ABCD AB || CD and AD BC A B 1 2 Prove: 1 2 D C Mr. Chin-Sung Lin ERHS Math Geometry Quadrilaterals and Proofs Given: Parallelogram ABCD and ABDE Prove: EAD DBC E A D B C Mr. Chin-Sung Lin ERHS Math Geometry Quadrilaterals and Proofs Given: ABC is a right , O is the midpoint of AC Prove: 1 2 A O 1 B 2 C Mr. Chin-Sung Lin ERHS Math Geometry Quadrilaterals and Proofs Given: ABCD is a rhombus, DBFE is an isosceles trapezoid A Prove: CE CF D E B C F Mr. Chin-Sung Lin ERHS Math Geometry Coordinate Geometry and Quadrilaterals Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangles To show that a quadrilateral is a rectangle, by showing that the quadrilateral is a parallelogram that contains a right angle, or with congruent diagonals Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangles Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a rectangle Can be done by ……. (in terms of coordinate geometry) Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangles Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a rectangle Can be done by proving a parallelogram and the product of the slopes of adjacent sides is equal to -1 the diagonals have the same lengths Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangle - Parallelogram with a Right Angle ABCD is a quadrilateral, where A (1, 1), B(7, 5), C(9, 2) and D(3, -2) prove ABCD is a rectangle by proving that ABCD is a parallelogram with a right angle Mr. Chin-Sung Lin ERHS Math Geometry Proving Rectangle - Parallelogram with Congruent Diagonals ABCD is a quadrilateral, where A (1, 1), B(7, 5), C(9, 2) and D(3, -2) prove ABCD is a rectangle by proving that ABCD is a parallelogram with congruent diagonals Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombuses To show that a quadrilateral is a rhombus, by showing that the quadrilateral has four congruent sides, or is a parallelogram: a pair of adjacent sides are congruent the diagonals intersect at right angles, or the opposite angles are bisected by the diagonals Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombuses Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a rhombus Can be done by ……. (in terms of coordinate geometry) Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombuses Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a rhombus Can be done by proving All four sides have the same lengths A parallelogram and the adjacent sides have the same lengths A parallelogram with the product of the slopes of the diagonals is equal to -1 Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombus - Quadrilateral with Four Congruent Sides ABCD is a quadrilateral, where A (3, 7), B(5, 3), C(3, -1) and D(1, 3) prove ABCD is a rhombus by proving that ABCD is a quadrilateral with four congruent sides Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombus - Parallelogram with Congruent Adjacent Sides ABCD is a quadrilateral, where A (3, 7), B(5, 3), C(3, -1) and D(1, 3) prove ABCD is a rhombus by proving that ABCD is a parallelogram with a pair of congruent adjacent sides Mr. Chin-Sung Lin ERHS Math Geometry Proving Rhombus - Parallelogram with Perpendicular Diagonals ABCD is a quadrilateral, where A (3, 7), B(5, 3), C(3, -1) and D(1, 3) prove ABCD is a rhombus by proving that ABCD is a parallelogram with perpendicular diagonals Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares To show that a quadrilateral is a square, by showing that the quadrilateral is a a rhombus that contains a right angle, or a rectangle with a pair of congruent adjacent sides Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a square Can be done by ……. (in terms of coordinate geometry) Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a square Can be done by proving A rhombus and the product of the slopes of adjacent sides is equal to -1 A rectangle and two adjacent sides have the same lengths Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares - Rhombus with a Right Angle ABCD is a quadrilateral, where A (0, 4), B(3, 5), C(4, 2) and D(1, 1) prove ABCD is a square by proving that ABCD is a rhombus with a right angle Mr. Chin-Sung Lin ERHS Math Geometry Proving Squares - Rectangle with Congruent Adjacent Sides ABCD is a quadrilateral, where A (0, 4), B(3, 5), C(4, 2) and D(1, 1) prove ABCD is a square by proving that ABCD is a rectangle with congruent adjacent sides Mr. Chin-Sung Lin ERHS Math Geometry Proving Trapezoids To prove that a quadrilateral is a trapezoid, show that two sides are parallel and the other two sides are not parallel Mr. Chin-Sung Lin ERHS Math Geometry Proving Trapezoids Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a trapezoid Can be done by ……. (in terms of coordinate geometry) Mr. Chin-Sung Lin ERHS Math Geometry Proving Trapezoids Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is a trapezoid Can be done by proving the slopes of one pair of opposite sides are equal while the slopes of the other pair of opposite sides are not equal Mr. Chin-Sung Lin ERHS Math Geometry Proving Trapezoids - Parallel Bases and Non-Parallel Legs ABCD is a quadrilateral, where A (-3, 5), B(4, 5), C(6, 1) and D(-5, 1) prove ABCD is a trapezoid by proving that there are two parallel bases and two non-parallel legs Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids To prove that a trapezoid is an isosceles trapezoid, show that one of the following statements is true: The legs are congruent The lower/upper base angles are congruent The diagonals are congruent Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is an isosceles trapezoid Can be done by ……. (in terms of coordinate geometry) Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Given: The coordinates of the vertices of a quadrilateral Prove: A given quadrilateral is an isosceles trapezoid Can be done by proving A trapezoid whose two legs have the same lengths A trapezoid whose two diagonals have the same lengths Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Trapezoid with Congruent Legs ABCD is a quadrilateral, where A (-3, 5), B(4, 5), C(6, 1) and D(-5, 1) prove ABCD is an isosceles trapezoid by proving that ABCD is a trapezoid with congruent legs Mr. Chin-Sung Lin ERHS Math Geometry Proving Isosceles Trapezoids Trapezoid w. Congruent Diagonals ABCD is a quadrilateral, where A (-3, 5), B(4, 5), C(6, 1) and D(-5, 1) prove ABCD is an isosceles trapezoid by proving that ABCD is a trapezoid with congruent diagonals Mr. Chin-Sung Lin ERHS Math Geometry Application Example Mr. Chin-Sung Lin ERHS Math Geometry Finding the Type of Quadrilateral Given ABCD is a quadrilateral, where A (3, 6), B(7, 0), C(1, -4), D(-3, 2) Find the type of quadrilateral ABCD Mr. Chin-Sung Lin ERHS Math Geometry Areas of Polygons Mr. Chin-Sung Lin ERHS Math Geometry Areas of Polygons The area of a polygon is the unique real number assigned to any polygon that indicates the number of nonoverlapping square units contained in the polygon’s interior Mr. Chin-Sung Lin ERHS Math Geometry Areas of Quadrilaterals The area of a quadrilateral is the product of the length of the base and the length of the altitude (height) B A altitude D base C Mr. Chin-Sung Lin ERHS Math Geometry Areas of Parallelograms The area of a parallelogram is the product of the length of the base and the length of the altitude (height) B A altitude D base C Mr. Chin-Sung Lin ERHS Math Geometry Q&A Mr. Chin-Sung Lin ERHS Math Geometry The End Mr. Chin-Sung Lin

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