Question

Одна из диагоналей трапеции перпендикулярна боковой стороне и образует с большим основанием угол в 40 градусов,а другая боковая сторона равная меньшему основанию.Найдите углы трапеции.

Answer 1

Рассмотрим трапецию АВСД(АД -большее основание,ВС меньшее),диагональ ДВ перпендикулярна стороне АВ, угол ВДА 40°,ВС=СД. Треугольник АВД прямоугольный,т.к. ДВ перпендик. АВ,Угол АДВ 40°(по условию) , следовательно по теореме о сумме углов треугольника угол ВАД=180°-90°-40°= 50° . Т.к. АД параллельна ВС, то угол ВАД=СВО= 50°,как соответствнные углы, следоват. угол СВД =180°-90°-50°=30°, угол АВС=90°+40°=130°. Т.к. ВС=СД по условию, то треугольник ВСД равнобедренный, следовательно по теореме о равнобедренном треугольнике углы СВД=СДВ=40°. По теореме о сумме углов в треугольнике угол ВСД=180°-(40°+40°)= 100° .Угол АДС= 40°+40°= 80°.
Ответ: ВАД=50°, АВС=130°, ВСА=100°,АДС=80°
Надеюсь подробно расписал и без чертежа понятно:)

Answer 2

Finding the Angles of the Trapezoid

To find the angles of the trapezoid, let's analyze the given information:

- One of the diagonals of the trapezoid is perpendicular to one of the lateral sides. - The diagonal forms a 40-degree angle with the longer base. - One of the lateral sides is equal to the shorter base.

Let's denote the longer base as AB, the shorter base as CD, and the diagonal as AC. The angle formed between the diagonal AC and the longer base AB is 40 degrees.

To find the angles of the trapezoid, we need to consider the properties of the trapezoid and use the given information.

Properties of a Trapezoid

A trapezoid is a quadrilateral with one pair of parallel sides. In a trapezoid, the sum of the interior angles is always equal to 360 degrees.

Let's denote the angles of the trapezoid as follows:

- Angle A is the angle between the longer base AB and the diagonal AC. - Angle B is the angle between the longer base AB and the lateral side BC. - Angle C is the angle between the shorter base CD and the lateral side BC. - Angle D is the angle between the shorter base CD and the diagonal AC.

Using the properties of a trapezoid, we can determine the relationships between these angles.

Relationships Between the Angles

1. The sum of angles A and B is equal to 180 degrees, as they are adjacent angles on a straight line. 2. The sum of angles C and D is equal to 180 degrees, as they are adjacent angles on a straight line. 3. The sum of angles B and C is equal to 180 degrees, as they are opposite angles formed by the parallel sides of the trapezoid.

Now, let's use the given information to find the values of these angles.

Calculation

From the given information, we know that angle A is 40 degrees.

Using the relationships between the angles mentioned above, we can determine the values of the other angles.

1. Angle B = 180 degrees - angle A = 180 degrees - 40 degrees = 140 degrees. 2. Angle C = 180 degrees - angle B = 180 degrees - 140 degrees = 40 degrees. 3. Angle D = 180 degrees - angle C = 180 degrees - 40 degrees = 140 degrees.

Therefore, the angles of the trapezoid are as follows:

- Angle A = 40 degrees - Angle B = 140 degrees - Angle C = 40 degrees - Angle D = 140 degrees

Please let me know if you need any further assistance!