Question

Две параллельные прямые пересечены секущей. Докажите , что биссектрисы односторонних углов перпендикулярны

Answer 1

Я думаю к этому вопросу нет ответа лично у меня

Answer 2

Proof that the Bisectors of Alternate Angles are Perpendicular

To prove that the bisectors of alternate angles formed by two intersecting lines are perpendicular, we can use the properties of parallel lines and angles. Let's break down the proof step by step:

1. Given: Two parallel lines intersected by a transversal line. 2. We need to prove: The bisectors of the alternate angles formed are perpendicular.

Proof:

Step 1: Let's consider two parallel lines, line m and line n, intersected by a transversal line t. This forms eight angles, four on the left side and four on the right side of the transversal.

Step 2: We will focus on the alternate angles, which are the angles that are on opposite sides of the transversal and are not adjacent to each other. Let's denote these angles as angle 1, angle 2, angle 3, and angle 4.

Step 3: Now, let's consider the bisectors of angle 1 and angle 3. The bisector of an angle is a line that divides the angle into two equal parts.

Step 4: Since lines m and n are parallel, the alternate angles formed by the transversal are congruent (equal in measure) according to the alternate interior angles theorem.

Step 5: Let's assume that the bisectors of angle 1 and angle 3 intersect at point P.

Step 6: Now, we will prove that angle 1 and angle 3 are right angles (90 degrees) by showing that they are supplementary to each other.

Step 7: Since angle 1 and angle 2 are congruent, and angle 2 is a straight angle (180 degrees), angle 1 and angle 2 together form a straight angle.

Step 8: Similarly, angle 3 and angle 4 are congruent, and angle 4 is a straight angle. Therefore, angle 3 and angle 4 together form a straight angle.

Step 9: Since angle 1 and angle 2 form a straight angle, their measures add up to 180 degrees. Similarly, angle 3 and angle 4 also add up to 180 degrees.

Step 10: Since angle 1 and angle 3 are congruent, and their measures add up to 180 degrees, each of them must be 90 degrees. Therefore, angle 1 and angle 3 are right angles.

Step 11: Since the bisectors of angle 1 and angle 3 intersect at point P and angle 1 and angle 3 are right angles, we can conclude that the bisectors are perpendicular to each other.

Therefore, we have proved that the bisectors of alternate angles formed by two intersecting lines are perpendicular.