Theorem 7-5
In the same circle or congruent circles if the chords are equal, then the arcs are equal.
Theorem 7-6
In the same circle or in congruent circles if the arcs are equal, then the chords are equal.
Theorem 7-7
If a diameter is perpendicular to a chord, then it bisects the chord and its two
arcs.
Theorem 7-8
In the same circle or in congruent circles if the chords are equidistant from the
center, then their lengths are equal.
Theorem 7-9
In the same circle or in congruent circles if the chords have the same length, then
they are equidistant from the center.
Theorem 7-1
A radius drawn to a point of tangency is perpendicular to the tangent.
Theorem 7-2
A line in the plane of a circle and perpendicular to a radius at its
outer endpoint is tangent to the circle.
Postulate 16
If the intersection of arc AB and arc BC of a circle is the
single point B, then m arc AB + m arc BC = m arc AC (Arc addition postulate)
Theorem 7-3
If in the same circle or congruent circles two central angles are equal, then their
arcs are equal.
Theorem 7-4
If in the same circle or congruent circles two minor arcs are equal, then their central angles are equal.
inscribed angle
an angle with sides containing the endpoints of an arc and with a vertex that is a point of the arc other than an endpoint of the arc
intercepted arc
an angle intercepts an arc if the endpoints of the arc lie on the sides of the angle and all points of the arc except the endpoints lie in the interior of the angle
Theorem 7-10
The measure of an inscribed angle is equal to half the measure of its intercepted arc.
secant ray
a ray that lies on a secant line and contains both points of intersection with the circle
tangent ray
a ray that lies on a tangent line and contains the point of tangency
Theorem 7-11
The measure of an angle formed by a secant ray and a tangent ray drawn from a point on a circle is equal to half the measure of its intercepted arc.
Theorem 7-12
The measure of an angle formed by two secants that intersect inside the circle is equal to half the sum of the intercepted arcs
When chords intersect in a circle, the vertical angles formed intercept congruent arcs.
Sometimes
Theorem 7-13
The measure of the angle formed by two secants intersecting outside the circle equals half the difference of the intercepted arcs.
Theorem 7-14
The measure of the angle formed by a tangent ray and a secant ray intersecting outside the circle is equal to half the difference of the intercepted arcs.
Theorem 7-15
The measure of the angle formed by two tangent rays intersecting outside the circle is half the difference of the intercepted arcs.
Theorem 7-16
If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the other chord.
Theorem 7-17
If two secants intersect at a point outside a circle, the length of one secant times the length of its external part is equal to the length of the other secant times the length of its external part.
Theorem 7-18
If a tangent segment and a secant segment intersect outside a circle, the length of the tangent segment is the geometric mean between the secant segment and the external part of the secant segment.