L14_Properties of a Parallelogram

L14_Properties of a Parallelogram
Quadrilaterals Eleanor Roosevelt High School Chin-Sung Lin

Definitions of the Quadrilaterals
L14_Properties of a Parallelogram 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

Parts & Properties of the Quadrilaterals
L14_Properties of a Parallelogram ERHS Math Geometry Parts & Properties of the Quadrilaterals Mr. Chin-Sung Lin

Consecutive (Adjacent) Vertices
L14_Properties of a Parallelogram 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 P Q S R Mr. Chin-Sung Lin

Consecutive (Adjacent) Sides
L14_Properties of a Parallelogram 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 P Q 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 P Q 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 P Q 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 P Q 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 P Q S R Mr. Chin-Sung Lin

Sum of the Measures of Angles
L14_Properties of a Parallelogram 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 P Q S R Mr. Chin-Sung Lin

ERHS Math Geometry Parallelograms Mr. Chin-Sung Lin

ERHS Math Geometry A B D C Parallelogram 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

Theorems of Parallelogram
L14_Properties of a Parallelogram ERHS Math Geometry Theorems of Parallelogram Mr. Chin-Sung Lin

Theorems of Parallelogram
L14_Properties of a Parallelogram 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

Theorem of Dividing Diagonals
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Dividing Diagonals A diagonal divides a parallelogram into two congruent triangles If ABCD is a parallelogram, then ∆ ABD  ∆ CDB A B D C Mr. Chin-Sung Lin

Theorem of Dividing Diagonals
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Dividing Diagonals 1 2 3 4 A B D C Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. 1  2 and 3   Alternate interior angles 4. BD  BD Reflexive property 5. ∆ ABD  ∆ CDB 5. ASA postulate Mr. Chin-Sung Lin

Theorem of Opposite Sides
L14_Properties of a Parallelogram 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 B D C Mr. Chin-Sung Lin

Theorem of Opposite Sides
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Opposite Sides 1 2 3 4 A B D C Statements Reasons 1. ABCD is a parallelogram 1. Given 2. Connect BD 2. Form two triangles 3. AB || DC and AD || BC 3. Definition of parallelogram 4. 1  2 and 3   Alternate interior angles 5. BD  BD Reflexive property 6. ∆ ABD  ∆ CDB 6. ASA postulate 7. AB  CD and BC  DA 7. 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 ? perimeter = 50 A B 15 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 x = 50 A B x-20 10 D C Mr. Chin-Sung Lin

Theorem of Opposite Angles
L14_Properties of a Parallelogram 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 B D C Mr. Chin-Sung Lin

Theorem of Opposite Angles
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Opposite Angles A B D C Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. A and B are supplementary 3. Same side interior angles A and D are supplementary C and B are supplementary 4. A  C Supplementary angle theorem B  D

ERHS Math Geometry Application Example 3 ABCD is a parallelogram, what are the values of x and y? A B 120o 60o y x D C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 3 ABCD is a parallelogram, what are the values of x and y? x = 120o y = 60o A B 120o 60o y x D C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 4 ABCD is a parallelogram, what are the values of x and y? A B X+20 y - 20 180 - y 2x - 60 D C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 4 ABCD is a parallelogram, what are the values of x and y? x = 80o y = 100o A B X+20 y - 20 180 - y 2x - 60 D C Mr. Chin-Sung Lin

Theorem of Bisecting Diagonals
L14_Properties of a Parallelogram 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 D C O Mr. Chin-Sung Lin

Theorem of Bisecting Diagonals
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Bisecting Diagonals 1 2 3 4 A B D C O Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC Definition of parallelogram 3. 1  2 and 3   Alternate interior angles 4. AB  DC 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 = ? AC + BD = 24 A B 6 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 2y-6 O y+2 3x-4 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 x = 4 y = 8 A B x+4 2y-6 O y+2 3x-4 D C Mr. Chin-Sung Lin

Theorem of Consecutive Angles
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Consecutive Angles The consecutive angles of a parallelogram are supplementary If ABCD is a parallelogram, then A and B are supplementary C and D are supplementary A and D are supplementary B and C are supplementary A B D C Mr. Chin-Sung Lin

Theorem of Consecutive Angles
L14_Properties of a Parallelogram ERHS Math Geometry Theorem of Consecutive Angles A B Statements 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 Same-side interior angles A and D, B and C are supplementary are supplementary D 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 B 120o x z y D C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 7 ABCD is a parallelogram, what are the values of x, y and z? x = 60o y = 120o z = 60o A B 120o x z y 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 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? x = 90o y = 100o A 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 y+10 D 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 y+10 D 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 2X D C Mr. Chin-Sung Lin

ERHS Math Geometry Question 2 ABCD is a parallelogram, solve for x x = 30 A B X+30 X-10 O X+10 2X D C Mr. Chin-Sung Lin

ERHS Math Geometry Question 3 Given: ABCD is a parallelogram Prove: XO  YO A B D C O Y X 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

Review: Theorems of Parallelogram
L14_Properties of a Parallelogram 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

Prove Quadrilaterals are Parallelograms
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Prove Quadrilaterals are Parallelograms Mr. Chin-Sung Lin

Criteria for Proving Parallelograms
L15_Proving Quadrilaterals Are Parallelograms 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

Parallel Opposite Sides
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Parallel Opposite Sides If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram If AB || CD, and BC || DA then, ABCD is a parallelogram A B D C Mr. Chin-Sung Lin

Parallel Opposite Sides
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Parallel Opposite Sides A B D C Statements Reasons 1. AB || CD and BC || DA 1. Given 2. ABCD is a parallelogram 2. Definition of parallelogram Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms
ERHS Math Geometry Application Example 1 If m1 = m2 = m3, 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 B 3x-20 50o 60o 60o 50o D 2x+10 C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 2 ABCD is a quadrilateral as shown below, solve for x x = 30 A B 3x-20 50o 60o 60o 50o D 2x+10 C Mr. Chin-Sung Lin

Congruent Opposite Sides
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Congruent Opposite Sides If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram If AB  CD, and BC  DA then, ABCD is a parallelogram A B D C Mr. Chin-Sung Lin

Congruent Opposite Sides
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Congruent Opposite Sides 1 2 3 4 A B D C Statements Reasons 1. Connect BD 1. Form two triangles 2. AB  CD and BC  DA 2. Given 3. BD  BD Reflexive property 4. ∆ ABD  ∆ CDB 4. SSS postulate 5. 1  2 and 3   CPCTC 6. AB || DC and AD || BC 6. Converse of alternate interior angles theorem 7. ABCD is a parallelogram 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 C 15 Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 3 ABCD is a quadrilateral, solve for x x = 80 A B 15 X+50 10 10 2x-30 D C 15 Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 4 ABCD is a parallelogram, if DF = BE, then AECF is also a parallelogram A B D C E F Mr. Chin-Sung Lin

Congruent & Parallel Opposite Sides
L15_Proving Quadrilaterals Are Parallelograms 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 If AB  CD, and AB || CD then, ABCD is a parallelogram A B D C Mr. Chin-Sung Lin

Congruent & Parallel Opposite Sides
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Congruent & Parallel Opposite Sides 1 2 3 4 A B D C Statements Reasons 1. Connect BD 1. Form two triangles 2. AB  CD and AB || CD 2. Given 3. BD  BD Reflexive property 4. 1   Alternate interior angles 5. ∆ ABD  ∆ CDB 5. SAS postulate 6. 3   CPCTC 7. AD || BC 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+50 30o 10 10 30o 2y-20 D C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 5 ABCD is a quadrilateral, solve for x and y x = 5 y = 70o A B X+5 y+50o 30o 10 10 30o 2y-20o D C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 6 ABCD is a parallelogram, if m1 = m2, then AECF is also a parallelogram A B D C E F 1 2 Mr. Chin-Sung Lin

Congruent Opposite Angles
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Congruent Opposite Angles If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram If A  C, and B  D Then, ABCD is a parallelogram A B D C Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Congruent Opposite Angles 1 2 3 4 A B D C Statements Reasons 1. Connect BD 1. Form two triangles 2. m1 +m4 + mA  Triangle angle-sum theorem m2 +m3 + mB  180 3. m1 +m4 + mA Addition property m2 +m3 + mC  360 4. m1 +m3 = mB 4. Partition property m4 +m2 = mD 5. mA +mB + mC + mD Substitution property = 360 Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Congruent Opposite Angles 1 2 3 4 A B D C Statements Reasons 6. A  C and B  D Given 7. 2mA + 2mB = Substitution property 2mA + 2mD = 360 8. mA + mB = Division property mA + mD = 180 9. AD || BC, AB || DC 9. Converse of same-side interior angles 10. ABCD is a parallelogram 10. Definition of parallelogram Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 7 ABCD is a quadrilateral, solve for x A B X+30 130o 50o 50o 130o D 2x-40 C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 7 ABCD is a quadrilateral, solve for x x = 70 A B X+30 130o 50o 50o 130o D 2x-40 C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 8 if m1 = m2, m3 = m4, then ABCD is a parallelogram A B 1 4 3 C D 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 D C O Mr. Chin-Sung Lin

ERHS Math Geometry Bisecting Diagonals 1 2 3 4 A B D C O Statements Reasons 1. AC and BD bisect at O 1. Given 2. AO  CO and BO  DO 2. Def. of segment bisector 3. AOB  COD, AOD  COB Vertical angles 4. ∆AOB  ∆COD, ∆AOD  ∆COB SAS postulate 5. 1  2 and 3   CPCTC 6. AB || DC and AD || BC 6. Converse of alternate interior angles theorem 7. ABCD is a parallelogram 7. Definition of parallelogram Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 9 ∆ AOB  ∆ COD, then ABCD is a parallelogram A B D C O Mr. Chin-Sung Lin

Supplementary Consecutive Angles
L15_Proving Quadrilaterals Are Parallelograms 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 B D C Mr. Chin-Sung Lin

Supplementary Consecutive Angles
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Supplementary Consecutive Angles A B D C Statements Reasons 1. A and B, A and D Given are supplementary 2. AB || DC and AD || BC Converse of same-side interior angles theorem 3. ABCD is a parallelogram Definition of parallelogram Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 10 ABCD is a quadrilateral, solve for x A B D C 2x+80 2(x+45)-10 100-2x 3x Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 10 ABCD is a quadrilateral, solve for x x = 20 A B D C 2x+80 2(x+45)-10 100-2x 3x Mr. Chin-Sung Lin

Review: Proving Parallelograms
L15_Proving Quadrilaterals Are Parallelograms 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

L17_Rectangles Rhombuses and Squares
ERHS Math Geometry Rectangles Mr. Chin-Sung Lin

ERHS Math Geometry Rectangles A rectangle is a parallelogram containing one right angle A B C D Mr. Chin-Sung Lin

All Angles Are Right Angles
L17_Rectangles Rhombuses and Squares 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 C D Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry All Angles Are Right Angles A B C D Statements Reasons 1. ABCD is a rectangle & A = 90o 1. Given 2. C = 90o Opposite angles 3. mA + mD = Consecutive angles mA + mB = 180 mD = Substitution 90 + mB = 180 5. mB = 90, mD = Subtraction 6. B = 90o, D = 90o 6. Def. of measurement of angles Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry All Angles Are Right Angles The diagonals of a rectangle are congruent Given: ABCD is a rectangle Prove: AC  BD A B C D Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry All Angles Are Right Angles A B C D Statements Reasons 1. ABCD is a rectangle 1. Given 2. C = 90o, D = 90o 2. All angles are right angles 3. C  D Substitution 4. DC  DC Reflexive 5. AD  BC Opposite sides 6. ∆ADC  ∆BCD 6. SAS postulate 7. AC  BD CPCTC Mr. Chin-Sung Lin

Properties of Rectangle
L17_Rectangles Rhombuses and Squares ERHS Math Geometry Properties of Rectangle The properties of a rectangle All the properties of a parallelogram Four right angles (equiangular) Congruent diagonals A B C D 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 C D Proving Rectangles If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle Given: ABCD is a parallelogram and mA = 90 Prove: ABCD is a rectangle Mr. Chin-Sung Lin

ERHS Math Geometry A B C D Proving Rectangles If a quadrilateral is equiangular, it is a rectangle Given: ABCD is a quadrangular & mA = mB = mC = mD Prove: ABCD is a rectangle Mr. Chin-Sung Lin

ERHS Math Geometry A B C D O Proving Rectangles 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, mA = 6x - 30 and mC = 4x Show that ABCD is a rectangle A B C D Mr. Chin-Sung Lin

ERHS Math Geometry Application Example ABCD is a parallelogram, mA = 6x - 30 and mC = 4x Show that ABCD is a rectangle x =20 mA = 90 ABCD is a rectangle A B C D 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 B C D Mr. Chin-Sung Lin

All Sides Are Congruent
L17_Rectangles Rhombuses and Squares ERHS Math Geometry All Sides Are Congruent All sides of a rhombus are congruent Given: ABCD is a rhombus with AB  DA Prove: AB  BC  CD  DA A B C D Mr. Chin-Sung Lin

All Sides Are Congruent
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry All Sides Are Congruent A B C D Statements Reasons 1. ABCD is a rhombus w. AB  DA 1. Given 2. AB  DC, AD  BC 2. Opposite sides are congruent 3. AB  BC  CD  DA 3. Transitive Mr. Chin-Sung Lin

Perpendicular Diagonals
L17_Rectangles Rhombuses and Squares ERHS Math Geometry Perpendicular Diagonals The diagonals of a rhombus are perpendicular to each other Given: ABCD is a rhombus Prove: AC  BD A B C D O Mr. Chin-Sung Lin

Perpendicular Diagonals
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry A B C D O Perpendicular Diagonals Statements Reasons 1. ABCD is a rhombus 1. Given 2. AO  AO Reflexive 3. AD  AB Congruent sides 4. BO  DO Bisecting diagonals 5. ∆AOD  ∆AOB 5. SSS postulate 6. AOD  AOB 6. CPCTC 7. mAOD + mAOB = Supplementary angles 8. 2mAOD = Substitution 9. AOD = 90o Division pustulate 10. AC  BD Definition of perpendicular

Diagonals Bisecting Angles
L17_Rectangles Rhombuses and Squares ERHS Math Geometry Diagonals Bisecting Angles The diagonals of a rhombus bisect its angles Given: ABCD is a rhombus Prove: AC bisects DAB and DCB DB bisects CDA and CBA A B C D Mr. Chin-Sung Lin

Diagonals Bisecting Angles
L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry A B C D Diagonals Bisecting Angles Statements Reasons 1. ABCD is a rhombus 1. Given 2. AD  AB, DC  BC 2. Congruent sides AD  DC, AB  BC 3. AC  AC, DB  DB 3. Reflexive postulate 4. ∆ACD  ∆ACB, ∆BAD  ∆BCD 4. SSS postulate 5. DAC  BAC, DCA  BCA 5. CPCTC ADB  CDB, ABD  CBD 6. AC bisects DAB and DCB 6. Definition of angle bisector DB bisects CDA and CBA Mr. Chin-Sung Lin

ERHS Math Geometry Properties of Rhombus A B C D The properties of a rhombus All the properties of a parallelogram Four congruent sides (equilateral) Perpendicular diagonals 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

ERHS Math Geometry A B C D Proving Rhombus If a parallelogram has two congruent consecutive sides, then the parallelogram is a rhombus Given: ABCD is a parallelogram and AB  DA Prove: ABCD is a rhombus Mr. Chin-Sung Lin

ERHS Math Geometry A B C D Proving Rhombus If a quadrilateral is equilateral, it is a rhombus Given: ABCD is a parallelogram and AB  BC  CD  DA Prove: ABCD is a rhombus Mr. Chin-Sung Lin

ERHS Math Geometry A B C D Proving Rhombus The diagonals of a parallelogram are perpendicular Given: AC  BD Prove: ABCD is a rhombus Mr. Chin-Sung Lin

ERHS Math Geometry A B C D 1 2 3 4 Proving Rhombus Each diagonal of a rhombus bisects two angles of the rhombus Given: AC bisects DAB and DCB 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 x = 12 AB = AD = 25 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 = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus A B C D 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 x = 4 AB = BC = 10 ABCD is a rhombus A B C D 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 C D Mr. Chin-Sung Lin

ERHS Math Geometry Squares A square is a rectangle with four congruent sides (an equilateral rectangle) A B C D Mr. Chin-Sung Lin

ERHS Math Geometry Squares A square is a rhombus with four right angles (an equiangular rhombus) A B C D Mr. Chin-Sung Lin

ERHS Math Geometry Squares A square is an equilateral quadrilateral A square is an equiangular quadrilateral A B C D Mr. Chin-Sung Lin

ERHS Math Geometry Squares A square is a rhombus A square is a rectangle A B C D 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 All the properties of a rhombus A B C D Mr. Chin-Sung Lin

ERHS Math Geometry Proving Squares Mr. Chin-Sung Lin

ERHS Math Geometry Proving Squares A B C D If a rectangle has two congruent consecutive sides, then the rectangle is a square Given: ABCD is a rectangle and AB  DA Prove: ABCD is a square Mr. Chin-Sung Lin

ERHS Math Geometry Proving Squares A B C D 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 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, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y A B C D Mr. Chin-Sung Lin

ERHS Math Geometry Application Example ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y 4x – 30 = 90 x = 30 y = 25 A B C D 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

Rectangle Question 1 Answer
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

Rhombus Question 2 Answer
ERHS Math Geometry Question 2 Answer A parallelogram where all sides are congruent is a _________? Rhombus Mr. Chin-Sung Lin

Question 3 A rectangle with four congruent sides is a _________?
ERHS Math Geometry Question 3 A rectangle with four congruent sides is a _________? Mr. Chin-Sung Lin

Square Question 3 Answer
ERHS Math Geometry Question 3 Answer A rectangle with four congruent sides is a _________? Square Mr. Chin-Sung Lin

Question 4 A rhombus with four right angles is a _________?
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

Question 5 A parallelogram with congruent diagonals is a _________?
ERHS Math Geometry Question 5 A parallelogram with congruent diagonals is a _________? Mr. Chin-Sung Lin

Rectangle Question 5 Answer
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

Question 10 Choose the right answer(s): A parallelogram is a rhombus
ERHS Math Geometry Question 10 Choose the right answer(s): A parallelogram is a rhombus A rectangle is a square A rhombus is a parallelogram Mr. Chin-Sung Lin

Question 10 Answer Choose the right answer(s):
ERHS Math Geometry Question 10 Answer Choose the right answer(s): A parallelogram is a rhombus A rectangle is a square A rhombus is a parallelogram Mr. Chin-Sung Lin

Question 11 Choose the right answer(s):
ERHS Math Geometry Question 11 Choose the right answer(s): 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): A quadrilateral is a parallelogram A square is a rhombus A rectangle is a rhombus Mr. Chin-Sung Lin

Question 12 Choose the right answer(s): A rectangle is a parallelogram
ERHS Math Geometry Question 12 Choose the right answer(s): 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): A rectangle is a parallelogram A square is a rectangle A rhombus is a square Mr. Chin-Sung Lin

ERHS Math Geometry Trapezoids Mr. Chin-Sung Lin

Definitions of Trapezoids
L18_Trapezoids ERHS Math Geometry Definitions of Trapezoids Mr. Chin-Sung Lin

L18_Trapezoids 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 B C D Upper base Lower base Leg Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Isosceles Trapezoids A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid A B C D Upper base Lower base Leg Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Median of a Trapezoid The median of a trapezoid is the line segment connecting the midpoints of the nonparallel sides A B C D Upper base Lower base Median Mr. Chin-Sung Lin

Examples of Trapezoids
L18_Trapezoids ERHS Math Geometry Examples of Trapezoids C A B D 100o 80o 120o 60o 90o D C B A 110o 70o 45o 135o D C B A 110o 70o 120o 60o D C B A Mr. Chin-Sung Lin

Exercise - Trapezoids Which one is a trapezoid? Why? B D A A C C D B
L18_Trapezoids ERHS Math Geometry Exercise - Trapezoids Which one is a trapezoid? Why? 110o 75o 45o 130o D C B A 105o 75o D C B A Mr. Chin-Sung Lin

Exercise - Trapezoids Which one is a trapezoid? A D B A B C D C
L18_Trapezoids ERHS Math Geometry Exercise - Trapezoids Which one is a trapezoid? 110o 65o 120o D C B A C A B D 90o 80o 100o Mr. Chin-Sung Lin

Properties of Isosceles Trapezoids
L18_Trapezoids ERHS Math Geometry Properties of Isosceles Trapezoids Mr. Chin-Sung Lin

Properties of Isosceles Trapezoids
L18_Trapezoids 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

L18_Trapezoids 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 Prove: A  B; C  D A B C D Mr. Chin-Sung Lin

Congruent Base Angles Given: Isosceles trapezoid ABCD
L18_Trapezoids ERHS Math Geometry Congruent Base Angles Given: Isosceles trapezoid ABCD AB || CD and AD  BC Prove: A  B; C  D E A B C D A B C D Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Congruent Diagonals The diagonals of an isosceles trapezoid are congruent Given: Isosceles trapezoid ABCD AB || CD and AD  BC Prove: AC  BD A B C D Mr. Chin-Sung Lin

Congruent Diagonals Given: Isosceles trapezoid ABCD
L18_Trapezoids ERHS Math Geometry Congruent Diagonals Given: Isosceles trapezoid ABCD AB || CD and AD  BC Prove: AC  BD A B C D Mr. Chin-Sung Lin

Parallel and Average Median
L18_Trapezoids 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 B C D E F Mr. Chin-Sung Lin

Parallel and Average Median
L18_Trapezoids 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 B C D H F E G Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Proving Trapezoids Mr. Chin-Sung Lin

L18_Trapezoids 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

Proving Isosceles Trapezoids
L18_Trapezoids 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

Application Examples ERHS Math Geometry Mr. Chin-Sung Lin
L18_Trapezoids ERHS Math Geometry Application Examples Mr. Chin-Sung Lin

Numeric Example of Trapezoids
L18_Trapezoids ERHS Math Geometry Numeric Example of Trapezoids Isosceles Trapezoid ABCD, AB || CD and AD  BC Solve for x and y A B C D 2xo xo 3yo Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Numeric Example of Trapezoids Isosceles Trapezoid ABCD, AB || CD and AD  BC Solve for x and y x = 60 y = 20 A B C D 2xo xo 3yo Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Numeric Example of Trapezoids Trapezoid ABCD, AB || CD and median EF Solve for x A B C D E F 2x 2x + 4 3x + 2 Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Numeric Example of Trapezoids Trapezoid ABCD, AB || CD and median EF Solve for x x = 6 A B C D E F 2x 2x + 4 3x + 2 Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Proving Isosceles Trapezoids Given: Trapezoid ABCD and A  B Prove: ABCD is an isosceles trapezoid A B C D Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Proving Isosceles Trapezoids Given: Trapezoid ABCD and AC  BD Prove: ABCD is an isosceles trapezoid A B C D O Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Proving Isosceles Trapezoids Given: Trapezoid ABCD, AB || CD and AE  BE Prove: ABCD is an isosceles trapezoid A B C D E Mr. Chin-Sung Lin

Summary of Quadrilaterals
L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Summary of Quadrilaterals Mr. Chin-Sung Lin

Properties of Quadrilaterals - 1
L20_Geometric Proofs Involving Quadrilaterals 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

L20_Geometric Proofs Involving Quadrilaterals 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

L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Properties of Quadrilaterals - 2 Properties Cong. Diagonals  Bisecting Diagonals Perpendicular Diagonals Cong. Opposite Angles Supp. Opposite Angles Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals 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

L20_Geometric Proofs Involving Quadrilaterals 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

Quadrilaterals and Proofs
L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Quadrilaterals and Proofs Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Quadrilaterals and Proofs Given: Isosceles trapezoid ABCD AB || CD and AD  BC Prove: 1  2 A B C D 1 2 Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Quadrilaterals and Proofs Given: Parallelogram ABCD and ABDE Prove:  EAD   DBC A B D C E Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Quadrilaterals and Proofs Given: ABC is a right , O is the midpoint of AC Prove: 1  2 A C B O 1 2 Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals ERHS Math Geometry Quadrilaterals and Proofs Given: ABCD is a rhombus, DBFE is an isosceles trapezoid Prove: CE  CF E A B C D F Mr. Chin-Sung Lin

Coordinate Geometry and Quadrilaterals
L19_Coordinate Geometry and Quadrilaterals ERHS Math Geometry Coordinate Geometry and Quadrilaterals Mr. Chin-Sung Lin

L19_Coordinate Geometry and Quadrilaterals
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

Proving Rectangle - Parallelogram with a Right Angle
L19_Coordinate Geometry and Quadrilaterals 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

Proving Rectangle - Parallelogram with Congruent Diagonals
L19_Coordinate Geometry and Quadrilaterals 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

Proving Rhombus - Quadrilateral with Four Congruent Sides
L19_Coordinate Geometry and Quadrilaterals 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

Proving Rhombus - Parallelogram with Congruent Adjacent Sides
L19_Coordinate Geometry and Quadrilaterals 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

Proving Rhombus - Parallelogram with Perpendicular Diagonals
L19_Coordinate Geometry and Quadrilaterals 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

Proving Squares - Rhombus with a Right Angle
L19_Coordinate Geometry and Quadrilaterals 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

Proving Squares - Rectangle with Congruent Adjacent Sides
L19_Coordinate Geometry and Quadrilaterals 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

Proving Trapezoids - Parallel Bases and Non-Parallel Legs
L19_Coordinate Geometry and Quadrilaterals 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

L19_Coordinate Geometry and Quadrilaterals 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

L19_Coordinate Geometry and Quadrilaterals 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

L19_Coordinate Geometry and Quadrilaterals 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

Proving Isosceles Trapezoids - Trapezoid with Congruent Legs
L19_Coordinate Geometry and Quadrilaterals 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

Proving Isosceles Trapezoids - Trapezoid w. Congruent Diagonals
L19_Coordinate Geometry and Quadrilaterals 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

Finding the Type of Quadrilateral
L19_Coordinate Geometry and Quadrilaterals 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 non-overlapping square units contained in the polygon’s interior Mr. Chin-Sung Lin

Areas of Quadrilaterals
L19_Coordinate Geometry and Quadrilaterals 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) A B C D base altitude Mr. Chin-Sung Lin

Areas of Parallelograms
L19_Coordinate Geometry and Quadrilaterals 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) A B C D altitude base Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals
ERHS Math Geometry Q & A Mr. Chin-Sung Lin

L20_Geometric Proofs Involving Quadrilaterals
ERHS Math Geometry The End Mr. Chin-Sung Lin

L14_Properties of a Parallelogram

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L14_Properties of a Parallelogram

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