Parallelograms.ppt
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The document defines properties of parallelograms and provides examples demonstrating these properties: - Opposite sides and angles of a parallelogram are congruent - Consecutive angles of a parallelogram are supplementary - Diagonals of a parallelogram bisect each other Examples show using these properties to find unknown lengths and angle measures. The last example proves that opposite sides and opposite angles of a parallelogram are congruent by drawing the diagonal and using properties of parallelograms and triangle congruence.
Parallelograms.ppt
- 1. Theorems about parallelograms • If a quadrilateral is a parallelogram, then its opposite sides are congruent. ►PQ≅RS and SP≅QR P Q R S
- 2. Theorems about parallelograms • If a quadrilateral is a parallelogram, then its opposite angles are congruent. P ≅ R and Q ≅ S P Q R S
- 3. Theorems about parallelograms • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary (add up to 180°). mP +mQ = 180°, mQ +mR = 180°, mR + mS = 180°, mS + mP = 180° P Q R S
- 4. Theorems about parallelograms • If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM ≅ SM and PM ≅ RM P Q R S
- 5. Ex. 1: Using properties of Parallelograms • FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. a. JH b. JK F G J H K 5 3 b.
- 6. Ex. 1: Using properties of Parallelograms • FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. a. JH b. JK SOLUTION: a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. F G J H K 5 3 b.
- 7. Ex. 1: Using properties of Parallelograms • FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. a. JH b. JK SOLUTION: a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. F G J H K 5 3 b. b. JK = GK Diagonals of a bisect each other. JK = 3 Substitute 3 for GK
- 8. TRY YOURSELF : 9/29/15 PQRS is a parallelogram. Find the angle measure. a. mR b. mQ P R Q 70° S
- 9. PQRS is a parallelogram. Find the angle measure. a. mR b. mQ a. mR = mP Opposite angles of a are ≅. mR = 70° Substitute 70° for mP. P R Q 70° S
- 10. PQRS is a parallelogram. Find the angle measure. a. mR b. mQ a. mR = mP Opposite angles of a are ≅. mR = 70° Substitute 70° for mP. b. mQ + mP = 180° Consecutive s of a are supplementary. mQ + 70° = 180° Substitute 70° for mP. mQ = 110° Subtract 70° from each side. P R Q 70° S
- 11. Ex. 3: Using Algebra with Parallelograms PQRS is a parallelogram. Find the value of x. mS + mR = 180° 3x + 120 = 180 3x = 60 x = 20 Consecutive s of a □ are supplementary. Substitute 3x for mS and 120 for mR. Subtract 120 from each side. Divide each side by 3. S Q P R 3x° 120°
- 12. TOTD
- 13. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given A D B C
- 14. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given 2. Through any two points, there exists exactly one line. A D B C
- 15. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given 2. Through any two points, there exists exactly one line. 3. Definition of a parallelogram A D B C
- 16. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given 2. Through any two points, there exists exactly one line. 3. Definition of a parallelogram 4. Alternate Interior s Thm. A D B C
- 17. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given 2. Through any two points, there exists exactly one line. 3. Definition of a parallelogram 4. Alternate Interior s Thm. 5. Reflexive property of congruence A D B C
- 18. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given 2. Through any two points, there exists exactly one line. 3. Definition of a parallelogram 4. Alternate Interior s Thm. 5. Reflexive property of congruence 6. ASA Congruence Postulate A D B C
- 19. Ex. 5: Proving Theorem 6.2 Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. 1. ABCD is a . 2. Draw BD. 3. AB ║CD, AD ║ CB. 4. ABD ≅ CDB, ADB ≅ CBD 5. DB ≅ DB 6. ∆ADB ≅ ∆CBD 7. AB ≅ CD, AD ≅ CB 1. Given 2. Through any two points, there exists exactly one line. 3. Definition of a parallelogram 4. Alternate Interior s Thm. 5. Reflexive property of congruence 6. ASA Congruence Postulate 7. CPCTC A D B C