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Theorems about parallelograms

If a quadrilateral is a parallelogram, then its opposite sides are congruent. ► PQ≅RS and SP≅QR. Theorems about parallelograms. Q. R. P. S. If a quadrilateral is a parallelogram, then its opposite angles are congruent. P ≅ R and Q ≅ S. Theorems about parallelograms. Q. R. P.

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Theorems about parallelograms

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  1. If a quadrilateral is a parallelogram, then its opposite sides are congruent. ►PQ≅RS and SP≅QR Theorems about parallelograms Q R P S

  2. If a quadrilateral is a parallelogram, then its opposite angles are congruent. P ≅ R and Q ≅ S Theorems about parallelograms Q R P S

  3. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary (add up to 180°). mP +mQ = 180°, mQ +mR = 180°, mR + mS = 180°, mS + mP = 180° Theorems about parallelograms Q R P S

  4. If a quadrilateral is a parallelogram, then its diagonals bisect each other. QM ≅ SM and PM ≅ RM Theorems about parallelograms Q R P S

  5. FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK Ex. 1: Using properties of Parallelograms 5 G F 3 K H J b.

  6. FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK SOLUTION: a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. Ex. 1: Using properties of Parallelograms 5 G F 3 K H J b.

  7. FGHJ is a parallelogram. Find the unknown length. Explain your reasoning. JH JK SOLUTION: a. JH = FG Opposite sides of a are ≅. JH = 5 Substitute 5 for FG. Ex. 1: Using properties of Parallelograms 5 G F 3 K H J b. • JK = GK Diagonals of a bisect each other. • JK = 3 Substitute 3 for GK

  8. PQRS is a parallelogram. Find the angle measure. mR mQ TRY YOURSELF : 9/29/15 Q R 70° P S

  9. PQRS is a parallelogram. Find the angle measure. mR mQ a. mR = mP Opposite angles of a are ≅. mR = 70° Substitute 70° for mP. Q R 70° P S

  10. PQRS is a parallelogram. Find the angle measure. mR mQ a. mR = mP Opposite angles of a are ≅. mR = 70° Substitute 70° for mP. mQ + mP = 180° Consecutive s of a are supplementary. mQ + 70° = 180° Substitute 70° for mP. mQ = 110° Subtract 70° from each side. Q R 70° P S

  11. PQRS is a parallelogram. Find the value of x. mS + mR = 180° 3x + 120 = 180 3x = 60 x = 20 Consecutive s of a □ are supplementary. Substitute 3x for mS and 120 for mR. Subtract 120 from each side. Divide each side by 3. Ex. 3: Using Algebra with Parallelograms P Q 3x° 120° S R

  12. TOTD

  13. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Ex. 5: Proving Theorem 6.2

  14. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Through any two points, there exists exactly one line. Ex. 5: Proving Theorem 6.2

  15. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Through any two points, there exists exactly one line. Definition of a parallelogram Ex. 5: Proving Theorem 6.2

  16. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Through any two points, there exists exactly one line. Definition of a parallelogram Alternate Interior s Thm. Ex. 5: Proving Theorem 6.2

  17. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Through any two points, there exists exactly one line. Definition of a parallelogram Alternate Interior s Thm. Reflexive property of congruence Ex. 5: Proving Theorem 6.2

  18. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Through any two points, there exists exactly one line. Definition of a parallelogram Alternate Interior s Thm. Reflexive property of congruence ASA Congruence Postulate Ex. 5: Proving Theorem 6.2

  19. Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB. ABCD is a . Draw BD. AB ║CD, AD ║ CB. ABD ≅ CDB, ADB ≅  CBD DB ≅ DB ∆ADB ≅ ∆CBD AB ≅ CD, AD ≅ CB Given Through any two points, there exists exactly one line. Definition of a parallelogram Alternate Interior s Thm. Reflexive property of congruence ASA Congruence Postulate CPCTC Ex. 5: Proving Theorem 6.2

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