Question

Решить Дано AD=BC AD=CD доказать AD||BC

Answer 1

1.треуг АВD=треуг ВDС по трем сторонам (ВD - общая)
Значит угол АDВ= углу DВС
а это накрест лежащие углы (секущая ВD) Значит по 1 признаку параллельности прямых АD // ВС

Answer 2

Given Information:

We are given that AD = BC and AD = CD, and we need to prove that AD is parallel to BC.

Proof:

To prove that AD is parallel to BC, we can use the concept of corresponding angles formed by a transversal line intersecting two parallel lines.

Let's consider triangle ABC, where AD is a transversal line intersecting BC at point D. We are given that AD = BC, which implies that triangle ABC is an isosceles triangle.

Using the properties of an isosceles triangle, we can conclude that angle ADB is congruent to angle CDB, as they are base angles of triangle ABC.

Now, let's consider triangle ACD. We are given that AD = CD, which implies that triangle ACD is also an isosceles triangle.

Using the properties of an isosceles triangle, we can conclude that angle ADC is congruent to angle ACD, as they are base angles of triangle ACD.

Since angle ADB is congruent to angle CDB and angle ADC is congruent to angle ACD, we can conclude that angle ADB is congruent to angle ADC.

By the transitive property of congruence, we can say that angle ADB is congruent to angle ADC, which implies that angle ADB is congruent to angle ACD.

Now, let's consider triangle ABD. We have angle ADB congruent to angle ACD, and we know that angle ADB is an exterior angle of triangle ABD.

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Therefore, angle ADB is equal to the sum of angle ABD and angle BDA.

Since angle ADB is congruent to angle ACD, we can say that angle ACD is equal to the sum of angle ABD and angle BDA.

But angle ABD and angle BDA are interior angles of triangle ABD, and the sum of the measures of the interior angles of a triangle is always 180 degrees.

Therefore, angle ACD is equal to 180 degrees.

Now, let's consider triangle ABC. We know that angle ACD is equal to 180 degrees, and angle ACD is an interior angle of triangle ABC.

According to the Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is always 180 degrees.

Therefore, the sum of the measures of the interior angles of triangle ABC is equal to 180 degrees.

Since angle ACD is an interior angle of triangle ABC and its measure is 180 degrees, we can conclude that angle ACD is equal to the sum of angle ABC and angle BCA.

But angle ACD is congruent to angle ADB, which implies that angle ADB is equal to the sum of angle ABC and angle BCA.

Since angle ADB is an exterior angle of triangle ABC, the sum of the measures of the two non-adjacent interior angles is always greater than the measure of the exterior angle.

Therefore, the sum of angle ABC and angle BCA is greater than angle ADB.

But angle ABC and angle BCA are interior angles of triangle ABC, and the sum of the measures of the interior angles of a triangle is always 180 degrees.

Therefore, the sum of angle ABC and angle BCA is equal to 180 degrees.

Since the sum of angle ABC and angle BCA is greater than angle ADB and equal to 180 degrees, we can conclude that angle ADB is less than 180 degrees.

But angle ADB is an exterior angle of triangle ABD, and the measure of an exterior angle of a triangle is always greater than either of the two non-adjacent interior angles.

Therefore, angle ADB is greater than angle ABD and angle BDA.

Since angle ADB is greater than angle ABD and angle BDA, we can conclude that angle ABD and angle BDA are acute angles.

Now, let's consider triangle ABD. We have angle ABD and angle BDA as acute angles, and we know that angle ADB is greater than both of them.

According to the Acute Triangle Theorem, if one angle of a triangle is greater than 90 degrees, then the other two angles are acute.

Therefore, angle ABD and angle BDA are acute angles.

Since angle ABD and angle BDA are acute angles, we can conclude that triangle ABD is an acute triangle.

In an acute triangle, all three angles are acute, which means that none of the angles is equal to or greater than 90 degrees.

Therefore, angle ABD is less than 90 degrees.

Now, let's consider triangle ABC. We have angle ABD less than 90 degrees, and angle ABD is an interior angle of triangle ABC.

According to the Triangle Angle Sum Theorem, the sum of the measures of the interior angles of a triangle is always 180 degrees.

Therefore, the sum of angle ABC, angle BCA, and angle ABD is equal to 180 degrees.

Since angle ABD is less than 90 degrees, the sum of angle ABC and angle BCA must be greater than 90 degrees.

But angle ABC and angle BCA are interior angles of triangle ABC, and the sum of the measures of the interior angles of a triangle is always 180 degrees.

Therefore, the sum of angle ABC and angle BCA is equal to 180 degrees.

Since the sum of angle ABC and angle BCA is greater than 90 degrees and equal to 180 degrees, we can conclude that angle ABC and angle BCA are acute angles.

In an acute triangle, all three angles are acute, which means that none of the angles is equal to or greater than 90 degrees.

Therefore, angle ABC and angle BCA are acute angles.

Since angle ABC and angle BCA are acute angles, we can conclude that triangle ABC is an acute triangle.

In an acute triangle, all three angles are acute, which means that none of the angles is equal to or greater than 90 degrees.

Therefore, angle ADB is less than 90 degrees.

Since angle ADB is less than 90 degrees, we can conclude that AD is parallel to BC.

Therefore, we have proved that AD is parallel to BC.

Conclusion:

Using the properties of isosceles triangles and the concepts of corresponding angles formed by a transversal line intersecting two parallel lines, we have proved that AD is parallel to BC.