Question

Більша діагональ трапеції є бісектисою тупого кута. Довести що більша основа трапеції дорівнює одній з бічних сторін Поможіть пожалуста

Answer 1

АВСД трапеция, АВ и СД боковые стороны. пусть ВД - наибольшая диагональ. т.к. у трапеции ВС и АД параллельны ⇒ угол СВД= углу АВД, как накрест лежащие и угол АВД = углк СВД (ВД - биссектриса) ⇒ треугольник АВД - равнобедренный, т.е. АВ=АД, ⇒что и требовалось доказать

Answer 2

The Relationship Between the Diagonal and the Bases of a Trapezoid

To prove that the longer diagonal of a trapezoid is the bisector of the obtuse angle and that the longer base of the trapezoid is equal to one of the lateral sides, we can use the properties of trapezoids and the concept of diagonals.

Let's consider a trapezoid ABCD, where AB is the longer base, CD is the shorter base, and AD and BC are the lateral sides. Let's also assume that AC is the longer diagonal and BD is the shorter diagonal.

To prove that AC is the bisector of the obtuse angle, we need to show that the angles formed by AC with the longer base AB and the shorter base CD are equal.

Proof: AC is the Bisector of the Obtuse Angle

1. Draw a line segment EF parallel to AB and CD, passing through the point C. This creates two triangles, ACE and CDF, which are congruent by the Side-Angle-Side (SAS) congruence criterion. 2. Since the triangles ACE and CDF are congruent, their corresponding angles are equal. Therefore, angle ACE is equal to angle CDF. 3. Since angle ACE is an exterior angle of triangle ACD, it is equal to the sum of the two remote interior angles, which are angles CAD and ACD. 4. Similarly, angle CDF is an exterior angle of triangle BCD and is equal to the sum of angles CBD and BCD. 5. Since angles CAD and CBD are corresponding angles, and angles ACD and BCD are corresponding angles, they are equal. 6. Therefore, angle ACE is equal to angle CDF, which means that AC is the bisector of the obtuse angle formed by the longer and shorter bases of the trapezoid.

Proof: AB is Equal to One of the Lateral Sides

To prove that the longer base AB is equal to one of the lateral sides, we can use the concept of diagonals and the properties of trapezoids.

1. Draw the diagonals AC and BD of the trapezoid ABCD. 2. Since AC is the longer diagonal, it divides the trapezoid into two triangles, ABC and ACD. 3. By the Side-Angle-Side (SAS) congruence criterion, triangle ABC is congruent to triangle ACD. 4. Therefore, the corresponding sides of the congruent triangles are equal. In particular, side AB is equal to side AD. 5. Since AB is equal to AD, AB is equal to one of the lateral sides of the trapezoid.

In conclusion, we have proved that the longer diagonal of a trapezoid is the bisector of the obtuse angle formed by the longer and shorter bases, and the longer base is equal to one of the lateral sides of the trapezoid.

Please let me know if you have any further questions!