Geometry Chapter 9 Theorems Flashcards

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Questions and Answers

What does Theorem 9-1 state?

Tangents to a circle from a point are diameters.

The line is tangent to the circle.

The line is perpendicular to the radius at the endpoint.

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. (correct)

What is the Corollary for Theorem 9-1?

Tangents to a circle from a point are congruent. (correct)

The tangent to a circle is always longer than the radius.

A diameter bisects the tangent at the point of contact.

All tangents to a circle intersect at one point.

What does Theorem 9-2 indicate?

A line is always perpendicular to the radius if it meets the circle.

Chords can be tangent to the circle.

All lines in the plane of a circle intersect at two points.

The line is tangent to the circle if it is perpendicular to the radius. (correct)

What does the Arc Addition Postulate state?

The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs. Signup and view all the answers

Theorem 9-3 states that:

If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle. Signup and view all the answers

What does Theorem 9-4 cover?

Congruent arcs have congruent chords, and vice versa. Signup and view all the answers

According to Theorem 9-5, two minor arcs in the same or congruent circles are congruent if:

Their central angles are congruent. Signup and view all the answers

Theorem 9-6 indicates that:

Chords at equal distances from the center are congruent. Signup and view all the answers

What does Theorem 9-7 state about inscribed angles?

They are equal to half the measure of their intercepted arcs. Signup and view all the answers

Corollary 1 for Theorem 9-7 explains that:

If two inscribed angles intercept the same arc, then the angles are congruent. Signup and view all the answers

Corollary 2 for Theorem 9-7 states that:

An angle inscribed in a semicircle is a right angle. Signup and view all the answers

Corollary 3 for Theorem 9-7 indicates that:

Opposite angles in a quadrilateral inscribed in a circle are supplementary. Signup and view all the answers

According to Theorem 9-8, the measure of an angle formed by a chord and a tangent is:

Equal to half the measure of the intercepted arc. Signup and view all the answers

Theorem 9-9 specifies that the measure of an angle formed by two chords that intercept inside a circle is:

Equal to half the sum of the measures of the intercepted arcs. Signup and view all the answers

What does Theorem 9-10 state?

The measure of an angle formed by two secants from an external point is equal to half the difference of the intercepted arcs. Signup and view all the answers

Theorem 9-11 states that when two chords intersect inside a circle:

The product of the segments of one chord equals the product of the segments of the other chord. Signup and view all the answers

According to Theorem 9-12:

When two secant segments are drawn to a circle from an external point, their products equal. Signup and view all the answers

Theorem 9-13 indicates that when a secant and a tangent segment are drawn to a circle from an external point:

The product of the secant segment and its external segment is equal to the square of the tangent segment. Signup and view all the answers

Study Notes

Tangents and Circles

Theorem 9-1: A tangent line to a circle is perpendicular to the radius at the point where it touches the circle.
Corollary for Theorem 9-1: Tangents drawn from a common external point are congruent.
Theorem 9-2: A line that is perpendicular to the radius at its outer endpoint is tangent to the circle.

Arc Properties

Arc Addition Postulate: The measure of an arc formed by two adjacent arcs equals the sum of their individual measures.
Theorem 9-3: Two minor arcs are congruent in the same circle (or congruent circles) if and only if their central angles are congruent.
Theorem 9-4: Congruent arcs imply congruent chords, and vice versa, within the same circle or congruent circles.

Chord and Diameter Relationships

Theorem 9-5: A diameter perpendicular to a chord bisects both the chord and its corresponding arc.
Theorem 9-6: Chords that are equidistant from the circle's center are congruent, while congruent chords maintain equal distance from the center.

Inscribed Angles

Theorem 9-7: The measure of an inscribed angle is half the measure of its intercepted arc.
Corollary 1 for Theorem 9-7: Inscribed angles that intercept the same arc are congruent.
Corollary 2 for Theorem 9-7: An angle inscribed in a semicircle is always a right angle.
Corollary 3 for Theorem 9-7: The opposite angles of a quadrilateral inscribed in a circle are supplementary.

Angles Formed by Chords and Tangents

Theorem 9-8: The measure of an angle formed by a chord and a tangent is half the measure of the intercepted arc.
Theorem 9-9: The angle formed by two intersecting chords inside a circle equals half the sum of the measures of the intercepted arcs.
Theorem 9-10: The angle formed by two secants, two tangents, or a secant and tangent from an external point is half the difference of the intercepted arcs' measures.

Segment Products

Theorem 9-11: When two chords intersect within a circle, the product of the segments of one chord equals the product of the segments of the other chord.
Theorem 9-12: The product of the entire secant segment and its external segment equals the product of the other secant segment and its external segment for segments drawn from an external point.
Theorem 9-13: The product of a secant segment with its external segment equals the square of the tangent segment drawn from the same external point.

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Test your knowledge on the theorems presented in Geometry Chapter 9. These flashcards cover important theorems and corollaries related to tangents and circles. Perfect for quick review and memorization before exams.