Explanation:
A) When both parallel lines are secant to the circle:
Let's consider two parallel lines, line l and line m, that intersect a circle at four points. Let A and B be two of the points where line l intersects the circle, and let C and D be two of the points where line m intersects the circle, as shown below:
```
C D
o-----o
/ \
/ \
/ \
/ \
A B
o------------------------o
```
Since line l is parallel to line m, we know that ∠A = ∠C and ∠B = ∠D, because they are corresponding angles. We also know that the sum of opposite angles of a cyclic quadrilateral is 180 degrees. Therefore, we have:
∠ACB + ∠ADB = 180° (opposite angles of cyclic quadrilateral ABCD)
∠A + ∠C + ∠B + ∠D = 180° (sum of angles in quadrilateral ABCD)
Substituting ∠A = ∠C and ∠B = ∠D, we get:
∠ACB + ∠ADB = 180°
2∠A + 2∠B = 180°
∠A + ∠B = 90°
This means that ∠ACB and ∠ADB are complementary angles. Since the intercepted arcs are defined by the central angles ∠ACB and ∠ADB, and these angles are complementary, it follows that the intercepted arcs are congruent.
B) When both parallel lines are tangent to the circle:
Let's consider two parallel lines, line l and line m, that are tangent to a circle at points A and B, as shown below:
```
o
/ | \
/ | \
/ | \
/ | \
A--------o--------B
\ | /
\ | /
\ | /
\ | /
o
```
Since line l is parallel to line m, we know that ∠A = ∠B, because they are alternate interior angles. We also know that the angle between a tangent and a chord that intersects the tangent is equal to the intercepted arc. Therefore, we have:
∠ALB = 2∠A (angle between tangent line l and chord AB)
∠BLA = 2∠B (angle between tangent line m and chord AB)
Substituting ∠A = ∠B, we get:
∠ALB = 2∠A
∠BLA = 2∠A
This means that the intercepted arcs AL and BL are congruent, because they are defined by the central angles ∠ALB and ∠BLA, which are equal.
Therefore, we have shown that in both cases, when two parallel lines intercept a circle, they intercept congruent arcs on the same circle.