Question

Доказать теорему: если при двух прямых секущей соответственные углы равны то прямые паралельны.​

Answer 1

3. Теорема
Если при пересечении двух прямых секущей сумма односторонних углов равна 1800, то прямые параллельны.
Дано: прямые и , АВ - секущая, 1 и 2 - односторонние, 1 + 2 = 1800 (Рис.7).

Доказать: .

Доказательство:



Углы 3 и 2 - смежные, значит по свойству смежных углов 3 + 2 = 1800, откуда 3 = 1800 - 2, при этом 1 + 2 = 1800, откуда 1 = 1800 - 2, тогда 1 = 3, а углы 1 и 3 накрест лежащие, следовательно, (см. теорему 1). Что и требовалось доказать.

Answer 2

Theorem: If the corresponding angles formed by two lines and a transversal are congruent, then the lines are parallel.

To prove this theorem, we can use the following steps:

1. Given: Two lines and a transversal intersecting them, and the corresponding angles formed are congruent. 2. To prove: The two lines are parallel.

Proof:

Let's denote the two lines as l1 and l2, and the transversal as t. The corresponding angles formed are denoted as angle 1 and angle 2.

Given: - l1 and l2 are two lines intersected by transversal t. - Angle 1 and angle 2 are corresponding angles formed by l1, l2, and transversal t, and they are congruent.

To prove: l1 is parallel to l2.

Proof by Contradiction:

Assume that l1 is not parallel to l2. This means that l1 and l2 intersect at some point, say point P.

Now, consider the triangle formed by l1, l2, and transversal t. According to the Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is 180 degrees.

However, if l1 and l2 are not parallel, then the corresponding angles formed by the transversal t should not be congruent, which contradicts the given information.

Therefore, our assumption that l1 is not parallel to l2 leads to a contradiction. Hence, l1 must be parallel to l2.

Conclusion:

We have proved that if the corresponding angles formed by two lines and a transversal are congruent, then the lines are parallel.

This proof demonstrates the validity of the theorem.

[[2 #]]