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Properties of Parallelograms and Theorems: Understanding Quadrilaterals

Learn the theorems and properties of parallelograms related to sides, angles, diagonals, and supplementary angles. Understand proofs and solve unknown lengths or angles in parallelograms. Improve your geometry skills now!

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Properties of Parallelograms and Theorems: Understanding Quadrilaterals

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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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