Question

дано четырехугльник АБСД АБ=СД БС=АД угол А = 30градусов точка Е принадлежит БС угол СДЕ = 60 градусов Доказать АБЕД прямоугольная трапеция

Answer 1

Дан четырехугольник ABCD

AB=CD

BC=AD

угол A = 30⁰

E ∋ BC

угол CDE = 60⁰

 

Доказать. ABED - прямоугольная трапеция.

 

Доказательство.

 

Рассм. ABCD. угол A = 30⁰ ⇒ угол С = 30⁰

угол В = углу D = (360⁰ - 30⁰ - 30⁰)/2 = 300⁰/2 = 150⁰

угол ADE = угол ADC - угол CDE

т.к. угол ADC 150⁰, a по условию угол CDE = 60⁰, то угол AED = 150⁰ - 60⁰ = 90⁰

 

Опеределения:

- трапецией называется четырехугольник, у которого две противолежащие стороны параллельны, а две другие не параллельны.

 - трапеция, один из углов которой прямой, называется прямоугольной

 

Рассмотрим ABED - четырехугольник. 

BE||AD,

AB не параллельно ED (т.к. ED перпендикуляр к AD)

угол EDA - 90⁰

 

След-но ABED - прямоугольная трапеция.

 

Answer 2

Given Information:

We are given a quadrilateral ABCD, where AB = CD, BS = AD, angle A = 30 degrees, point E lies on BS, and angle SDE = 60 degrees. We need to prove that ABCD is a right trapezoid.

Proof:

To prove that ABCD is a right trapezoid, we need to show that one pair of opposite angles is right angles and that the non-parallel sides are congruent.

Let's start by examining the given information and the angles in the quadrilateral:

1. Angle A = 30 degrees (given) 2. Angle SDE = 60 degrees (given)

Since angle A is given as 30 degrees, we can conclude that angle B is also 30 degrees because opposite angles in a quadrilateral are congruent.

Now, let's examine the angles in triangle SDE:

3. Angle SDE = 60 degrees (given) 4. Angle SED = 180 - 60 = 120 degrees (sum of angles in a triangle)

Since angle SED is 120 degrees, we can conclude that angle SDE is also 120 degrees because opposite angles in a triangle are congruent.

Now, let's examine the angles in triangle ABE:

5. Angle ABE = 180 - 30 - 120 = 30 degrees (sum of angles in a triangle)

Since angle ABE is 30 degrees, we can conclude that angle BED is also 30 degrees because opposite angles in a triangle are congruent.

Now, let's examine the angles in triangle BCD:

6. Angle BCD = 180 - 30 - 30 = 120 degrees (sum of angles in a triangle)

Since angle BCD is 120 degrees, we can conclude that angle BDC is also 120 degrees because opposite angles in a triangle are congruent.

Now, let's examine the angles in triangle BAC:

7. Angle BAC = 180 - 30 - 30 = 120 degrees (sum of angles in a triangle)

Since angle BAC is 120 degrees, we can conclude that angle BCA is also 120 degrees because opposite angles in a triangle are congruent.

From the above information, we can conclude that angle BCD = angle BDC = 120 degrees and angle BAC = angle BCA = 120 degrees. Therefore, opposite angles in the quadrilateral ABCD are congruent.

Now, let's examine the sides of the quadrilateral:

8. AB = CD (given) 9. BS = AD (given)

Since AB = CD and BS = AD, we can conclude that opposite sides in the quadrilateral ABCD are congruent.

Based on the above information, we can conclude that ABCD is a right trapezoid because one pair of opposite angles (BCD and BDC) is right angles and the non-parallel sides (AB and CD) are congruent.

Therefore, we have proved that ABCD is a right trapezoid.

Note: The given information and the proof provided are based on the given angles and sides. If there are any additional constraints or information, please let me know, and I will be happy to assist you further.