Question

theoram : opposite sides and opposite angles of a parallelogram are congruent . prove ​

Answer 1

Step-by-step explanation:

Problem:--------

ABCD is a parallelogram, AD||BC and AB||DC. Prove that ∠BAD ≅ ∠DCB and that ∠ADC ≅ ∠CBA.

Strategy:-----

There are two ways to go about this. The first is to use congruent triangles to show the corresponding angles are congruent, the other is to use the Alternate Interior Angles Theorem and apply it twice.

Let’s use congruent triangles first because it requires less additional lines. Draw the diagonal BD, and we will show that ΔABD and ΔCDB are congruent.

To show these two triangles are congruent we’ll use the fact that this is a parallelogram, and as a result, the two opposite sides are parallel, and the diagonal acts as a transversal line.

The diagonal is a common side, and it is also a transversal that intersects both pairs of opposite sides of the parallelogram – creating two pairs of congruent alternate interior angles.

The triangles ΔABD and ΔCDB are congruent based on the angle-side-angle postulate, and we can show that the opposite angles of the parallelogram are congruent as corresponding angles (using the angle addition theorem for one of the pairs).

Proof:

In Parallelograms, opposite angles are congruent

(1) ABCD is a parallelogram //Given

(2) AD || BC //From the definition of a parallelogram

(3) ∠ADB ≅ ∠CBD //Alternate Interior Angles Theorem

(4) AB || DC //From the definition of a parallelogram

(5) ∠DBA ≅ ∠CDB //Alternate Interior Angles Theorem

(6) BD= BD // Common side, reflexive property of equality

(7) ΔABD ≅ ΔCDB // (3), (6), (5) Angle-Side-Angle postulate

(8) ∠BAD ≅ ∠DCB // Corresponding angles in congruent triangles (CPCTC)

(9) m∠ADC = m∠ADB+m∠BDC //angle addition theorem

(10) m∠CBA = m∠CBD+m∠DBA //angle addition theorem

(11) m∠ADC = m∠CBD+m∠BDC //(3), (7), substitution

(12) m∠ADC = m∠CBD+m∠DBA //(5), (9), substitution

(13) m∠CBA = m∠ADC //(10), (8), transitive property of equality

(14) ∠CBA ≅ ∠ADC //(11), definition of congruent angles

Answer 2

Given : opposite sides and opposite angles of a parallelogram are congruent

To Find : Prove

Solution:

Parallelogram  , Quadrilateral in which opposite sides are parallel

Take ABCD as parallelogram

AB || CD  and BC || AD

Join AC

Check Triangle

ΔABC  and ΔCDA

∠BAC  = ∠DCA  ( ∵ AB || DC and AC is transversal  , alternate interior angles)

∠BCA  = ∠DAC  ( ∵ AD || BC and AC is transversal  , alternate interior angles)

AC ≅ AC   reflexive property

Hence ΔABC ≅ ΔCDA   using ASA congruence

Corresponding sides and angles of congruent triangles are congruent

AB ≅ CD   and BC ≅ AD  so opposite sides are congruent

∠ABC ≅  ∠CDA  => ∠B ≅   ∠D  opposite angles are  congruent

Similarly can be shown ∠A ≅ ∠C  by joining BD

QED

Hence proved

opposite sides and opposite angles of a parallelogram are congruent

Additional Info:

Diagonals of parallelogram bisect each other

Adjacent angles are supplementary. ( adds up to 180°)

Properties of angles formed by transversal line  with two parallel lines :

• Corresponding angles are congruent. ( Equal in Measure)

• Alternate angles are congruent.  ( Interiors & Exterior  both )  

• Co-Interior angles are supplementary. ( adds up to 180°)

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