Geometry: Triangle Proportions, Circle Theorems, and Arc Measures

Theorem 7-5 Side Splitter Theorem

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Theorem 7-6 Triangle Midsegment

If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long.

Corollary to the Side-Splitter Theorem

If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

Triangle Angle Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments such that the ratio between the segments is the same as the ratio between the sides adjacent to each segment.

Central Angle

An angle formed by radii with the vertex at the center of the circle.

Intercepted Arc

The part of a circle that lies between two segments, rays, or lines that intersect the circle.

Arc Measure

The measure of an arc is equal to the measure of its corresponding central angle.

Arc Length to Circumference

The measure of an arc is a fraction of 360; the arc length is a fraction of the circumference.

Arc Length Formula

s = n/360 x 2πr, where s is arc length, r is radius, and n is the number of degrees.

Area of a Circle

A = n/360 x πr^2, where A is the area, n is the angle in degrees, and r is the radius.

Theorem 10-8 Inscribed Angles Theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Corollary 1 to Inscribed Angles Theorem

Two inscribed angles that intercept the same arc are congruent.

Corollary 2 to Inscribed Angles Theorem

An angle inscribed in a semicircle is a right angle.

Corollary 3 to Inscribed Angles Theorem

The opposite angles of an inscribed quadrilateral are supplementary.

Theorem 10-9

The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc.