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## Quadrilaterals

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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 R S 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 R S 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 R S 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 R S 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 R S 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 R S 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 mP + mQ + mR + mS = 360 Q P R S Mr. Chin-Sung Lin**ERHS Math Geometry**Parallelograms Mr. Chin-Sung Lin**A**B D C ERHS Math Geometry 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 Parallelogram 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**A**B D C ERHS Math Geometry • A diagonal divides a parallelogram into two congruent triangles • If ABCD is a parallelogram, then • ∆ ABD∆ CDB Theorem of Dividing Diagonals Mr. Chin-Sung Lin**1**3 4 2 A B D C ERHS Math Geometry Theorem of Dividing Diagonals Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. 12 and 34 3. Alternate interior angles 4. BD BD 4. Reflexive property 5. ∆ ABD ∆ CDB 5. ASA postulate Mr. Chin-Sung Lin**A**B D C ERHS Math Geometry • Opposite sides of a parallelogram are congruent • If ABCD is a parallelogram, then • ABCD, and • BCDA Theorem of Opposite Sides Mr. Chin-Sung Lin**A**B 1 3 4 2 D C ERHS Math Geometry Theorem of Opposite Sides 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. 12 and 344. Alternate interior angles 5. BD BD 5. Reflexive property 6. ∆ ABD ∆ CDB 6. ASA postulate 7. AB CD and BC DA 7. CPCTC Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what’s the perimeter of ABCD ? Application Example 1 A B 15 10 D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what’s the perimeter of ABCD ? perimeter = 50 Application Example 1 A B 15 10 D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x Application Example 2 A B x-20 10 D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x x = 50 Application Example 2 A B x-20 10 D C Mr. Chin-Sung Lin**A**B D C ERHS Math Geometry • Opposite angles of a parallelogram are congruent • If ABCD is a parallelogram, then • AC, and • BD Theorem of Opposite Angles Mr. Chin-Sung Lin**A**B D C ERHS Math Geometry Theorem of Opposite Angles 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. AC 4. Supplementary angle theorem BD**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x and y? Application Example 3 A B 120o 60o y x D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x and y? x = 120o y = 60o Application Example 3 A B 120o 60o y x D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x and y? Application Example 4 A B X+20 y - 20 180 - y 2x - 60 D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x and y? x = 80o y = 100o Application Example 4 A B X+20 y - 20 180 - y 2x - 60 D C Mr. Chin-Sung Lin**A**B O D C ERHS Math Geometry • The diagonalsof a parallelogram bisect each other • If ABCD is a parallelogram, then • ACandBD bisect each other at O Theorem of Bisecting Diagonals Mr. Chin-Sung Lin**1**3 4 2 A B O D C ERHS Math Geometry Theorem of Bisecting Diagonals Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC 2. Definition of parallelogram 3. 12 and 34 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**ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? Application Example 5 A B 6 3 4 O D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? AC + BD = 24 Application Example 5 A B 6 3 4 O D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y Application Example 6 A B x+4 2y-6 O y+2 3x-4 D C Mr. Chin-Sung Lin**ERHS Math Geometry**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 Application Example 6 A B x+4 2y-6 O y+2 3x-4 D C Mr. Chin-Sung Lin**A**B D C ERHS Math Geometry • The consecutive angles of a parallelogram are supplementary • If ABCD is a parallelogram, then • A and Bare supplementary • C and Dare supplementary • A and Dare supplementary • B and Care supplementary Theorem of Consecutive Angles Mr. Chin-Sung Lin**ERHS Math Geometry**A B Theorem of Consecutive Angles Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. AandB,CandD 3. Same-side interior angles AandD,BandC are supplementary are supplementary D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x, y and z? Application Example 7 A B 120o x z y D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x, y and z? x = 60o y = 120o z = 60o Application Example 7 A B 120o x z y D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x and y? Application Example 8 A B X+30 X-30 Y+20 D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, what are the values of x and y? x = 90o y = 100o Application Example 8 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**ABCD is a parallelogram, calculate the perimeter of ABCD Question 1 A B x+30 2y-10 y+10 D C 2x-10 Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, calculate the perimeter of ABCD perimeter = 200 Question 1 A B x+30 2y-10 y+10 D C 2x-10 Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, solve for x Question 2 A B X+30 X-10 O X+10 2X D C Mr. Chin-Sung Lin**ERHS Math Geometry**ABCD is a parallelogram, solve for x x = 30 Question 2 A B X+30 X-10 O X+10 2X D C Mr. Chin-Sung Lin**A**X B O D C Y ERHS Math Geometry Given: ABCD is a parallelogram Prove: XO YO Question 3 Mr. Chin-Sung Lin**ERHS Math Geometry**Given: ABCD is a parallelogram, BOOD Prove: EOOF Question 4 A E B O D C F Mr. Chin-Sung Lin**ERHS Math Geometry**Given: ABCD is a parallelogram, AF || CE Prove: FABECD Question 5 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