Understanding Secant Angles: A Guide to Their Types

Step-by-step Guide: Secant Angles and Their Types

Definition of a Secant:
A secant is a line that intersects a circle at two distinct points. When a secant segment is drawn from a point outside the circle, the part of the segment that lies outside the circle is often termed the “external segment,” while the part within the circle is the “internal segment.”

Types of Secant Angles:

• Angle Formed by Two Secants: An angle is formed at an external point by two secant lines intersecting outside the circle. The measure of this angle is half the difference of the measures of the intercepted arcs.
• Angle Formed by a Secant and a Tangent: When a secant and tangent line meet at an external point, the measure of the angle they form is half the measure of its intercepted arc.
• Angle Formed by Two Intersecting Chords: When two secants (in the form of chords) intersect inside a circle, the angle between them is half the sum of the measures of their intercepted arcs.

Properties of Secant Angles:

• Secant-Secant Power Theorem: For two secants drawn from an external point, the product of the lengths of one secant and its external segment equals the product of the lengths of the other secant and its external segment.
• Secant-Tangent Power Theorem: For a tangent and a secant drawn from a common external point, the square of the length of the tangent is equal to the product of the length of the secant and its external segment.

Examples

Example 1:
Two secants, \( AC \) and \( AB \), are drawn to a circle from an external point \( A \). If the intercepted arcs are \( 80^\circ \) and \( 140^\circ \) respectively, find the measure of angle \( CAB \).

Solution:
Using the formula for the angle formed by two secants:
\( \text{Angle} = \frac{\text{difference of intercepted arcs}}{2} \)
\( \angle CAB = \frac{140^\circ – 80^\circ}{2} = \frac{60^\circ}{2} = 30^\circ \)

Example 2:
A secant \( AB \) and a tangent \( AC \) meet outside the circle at point \( A \). If the intercepted arc by \( AB \) is \( 100^\circ \), determine the measure of angle \( CAA’ \).

Solution:
Using the formula for the angle formed by a secant and a tangent:
\( \text{Angle} = \frac{\text{intercepted arc}}{2} \)
\( \angle CAA’ = \frac{100^\circ}{2} = 50^\circ \)

Practice Questions:

1. Two secants meet outside a circle forming an angle of \( 50^\circ \). If one intercepted arc measures \( 120^\circ \), what’s the measure of the other intercepted arc?
2. A secant and a tangent meet outside a circle forming an angle of \( 40^\circ \). Determine the measure of the intercepted arc by the secant.
3. Two chords intersect inside a circle forming an angle of \( 35^\circ \). If one of the intercepted arcs measures \( 50^\circ \), determine the measure of the other intercepted arc.

Answers:

1. \( 180^\circ \)
2. \( 80^\circ \)
3. \( 40^\circ \)

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Frequently Asked Questions

What is a secant angle?

A secant angle is an angle formed by two secants, or by a secant and a tangent, that meet at a vertex related to a circle. Its measure depends on the arcs the two lines cut off and on where the vertex sits: on the circle, inside it, or outside it. For example, two secants meeting outside a circle and cutting off arcs of 120° and 40° form a 40° angle.

What is the tangent-secant angle theorem?

The tangent-secant angle theorem says that when a tangent and a secant meet outside a circle, the angle equals half the difference of the intercepted arcs: (far arc − near arc) ÷ 2. For example, if the far arc is 160° and the near arc is 60°, the angle is (160 − 60) ÷ 2 = 50°. If the tangent and secant meet on the circle, the angle is half the intercepted arc.

How do you find secant angles in a circle?

First decide where the vertex is. If two secants or chords cross inside the circle, add the two intercepted arcs and divide by 2. For instance, arcs of 70° and 90° give (70 + 90) ÷ 2 = 80°. If the vertex is on the circle, take half the one intercepted arc. If the vertex is outside, subtract the near arc from the far arc and halve the result.

What is the formula for secant angles?

There are three formulas, one for each vertex location. Vertex on the circle: angle = ½ × intercepted arc. Vertex inside: angle = ½ × (arc₁ + arc₂). Vertex outside: angle = ½ × (far arc − near arc). To find a missing arc, work backward. If an outside angle is 35° and the near arc is 50°, then the far arc = 2 × 35 + 50 = 120°.

Is a secant the same as a tangent?

No. A secant is a line that crosses a circle at two points, while a tangent touches the circle at exactly one point and meets the radius there at a right angle. They are related, though. A tangent-secant angle and a tangent-tangent angle both use the same half-the-difference rule as two secants meeting outside the circle.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• How to find the area of a quarter circle
• Area and circumference of circles (resource currently unavailable)
• Angles and angle measure
• How to use the Pythagorean theorem
• How to solve angle-measurement word problems