$transitive postulate definition geometry$

(14) ∠CBA ≅ ∠ADC //(11), definition of congruent angles Another way to prove this (13) m∠CBA = m∠ADC //(10), (8), transitive property of equality (10) m∠CBA = m∠CBD+m∠DBA // angle addition theorem (9) m∠ADC = m∠ADB+m∠BDC // angle addition theorem (8) ∠BAD ≅ ∠DCB // Corresponding angles in congruent triangles (CPCTC) (6) BD= BD // Common side, reflexive property of equality (5) ∠DBA ≅ ∠CDB //Alternate Interior Angles Theorem

(4) AB || DC //From the definition of a parallelogram (3) ∠ADB ≅ ∠CBD // Alternate Interior Angles Theorem (2) AD || BC //From the definition of a parallelogram Here's how you prove that in parallelograms, opposite angles are congruent: 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). 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. 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. Draw the diagonal BD, and we will show that ΔABD and ΔCDB are congruent. Let's use congruent triangles first because it requires less additional lines. 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. Prove that ∠BAD ≅ ∠DCB and that ∠ADC ≅ ∠CBA Strategy Since this a property of any parallelogram, it is also true of any special parallelogram like a rectangle, a square, or a rhombus, ProblemĪBCD is a parallelogram, AD||BC and AB||DC. One of the properties of parallelograms is that the opposite angles are congruent, as we will now show. A parallelogram is defined as a quadrilateral where the two opposite sides are parallel.

Transitive postulate definition geometry