Sometimes, Always or NEVER (Geometry)?

Question

1. If the measure of an arc in one circle is equal to the measure of an arc in another circle, then the chords determined by these arcs are congruent.

2. The external angle of a circle formed by two tangents is supplementary to the angle formed by the radii drawn to the points of tangency,

Answer 1

1. Sometimes. Only if the circles are congruent (i.e. have the same radius)

2. Always. The two tangents and the two radii form a quadrilateral. The tangent-radius sides are 90 degrees. Two of these make 180. The other 180 is divided between the angle formed by the radii and the external angle formed by the tangents. Therefore, they are always supplementary.

Answer 2

1. Sometimes - If the circles are have the same measure of degree but the size of the circles are different, the chords will be different.
2. Always. An external angle made by two tangents is supplementary to the measure of the arc it cuts off. If you draw radii to the points of tangency, you get a central angle and the measure of a central angle is congruent to the arc it cuts off. So since the the external angle is supplementary to the arc, and the arc is congruent to the central angle formed by the radii drawn to the points of tangency, using transitive property will prove that it is supplementary to the central angle.

Answer 3

1. sometimes -- iff circles are congruent (have equal radii)

2. always -- the tangents and the radii form a quadrilateral, sum of angles 360, and the tangents and radii are perpendicular, sum of angles 180, so the sum of the other 2 must also be 180, which makes them supplementary.

Answer 4

1 sometimes, because it can be the same chord in congruent circles but not the same cord in non-congruent circles.

2. Never

Answer 5

1 always i think
2. never