Question

ДАЮ 30 БАЛЛОВ.Угол ADB=CDB, AD=DC. Докажите, что угол BAC=углу BCA и BD перпендикулярно AC. Пожалуйста помогите решить.

Answer 1

Рассмотрим треугольник ABD и треугольник BDC:
по условию AD=DC.
угол BDA=углу BDC.
cторона BD-общая.
значит треугольник ABD=треугольнику BDC(по двум сторонам и углу между ними)
значит угол BAD=углу BCD( в равных треугольниках,против равных сторон лежат равные углы)

Рассмотрим треугольник ADC:
треугольник ADC  равнобедренный,т.к. AD=DC.значит угол DAC=углу DCA
т.к. в равнобедренном треугольнике углы при основании равны)

угол BAC=угол BAD+угол DAC
угол BCA=угол BCD+угол DCA.
мы доказали,что угол BAD=углу BCD,а угол DAC=углу DCA,значит:
угол BAC=углу BCA

Answer 2

Given Information

We are given the following information: - Angle ADB is equal to angle CDB. - AD is equal to DC. - We need to prove that angle BAC is equal to angle BCA and that BD is perpendicular to AC.

Proof

To prove that angle BAC is equal to angle BCA, we can use the fact that angle ADB is equal to angle CDB. Since AD is equal to DC, we can conclude that triangle ADB is congruent to triangle CDB by the Side-Angle-Side (SAS) congruence criterion.

By the congruence of triangles ADB and CDB, we can say that angle BAD is equal to angle BCD. Since angle BAC is equal to angle BAD and angle BCA is equal to angle BCD, we can conclude that angle BAC is equal to angle BCA.

To prove that BD is perpendicular to AC, we need to show that angle BDA is a right angle. Since triangle ADB is congruent to triangle CDB, we can say that angle ADB is equal to angle CDB. Since angle ADB is equal to angle CDB and angle ADB is a right angle (since BD is perpendicular to AC), we can conclude that angle CDB is also a right angle. Therefore, angle BDA is a right angle, which means that BD is perpendicular to AC.

Hence, we have proved that angle BAC is equal to angle BCA and that BD is perpendicular to AC.

Note: The specific lengths of the sides or angles are not provided in the given information. Therefore, we cannot determine the exact values of the angles or sides in this proof.