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Tangent-Chord Angle
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06.15 • Circle Geometry

Tangent-Chord Angle

Follow the angle formed by a tangent and a chord at the point of tangency and connect it to the intercepted arc it opens toward.

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◯ Tangent-Chord Angle
Interactive diagram

Tangent-Chord Angle Diagram

Move the tangency point and chord endpoint while watching the angle stay anchored at the point where the tangent meets the circle.

Use the movable diagram to see what defines tangent-chord angle, how the labels relate to the figure, and what stays true as the board changes.

Definition: A tangent-chord angle is formed by a tangent and a chord through the point of tangency.
Detailed definition

Understanding Tangent-Chord Angle

A tangent-chord angle is formed by a tangent line and a chord that meet at the point of tangency. The vertex is therefore on the circle, but one side is a tangent rather than a second chord.

Its key theorem says that the measure of the angle equals half the measure of the intercepted arc. That makes the angle a close relative of the inscribed angle, but the construction of the angle is different.

This page keeps the touch point and intercepted arc visible together so you can see why the tangent-chord angle must be read from the arc opposite the opening, not from the small corner alone.

Key facts

Important ideas to remember

  • A tangent-chord angle is formed by a tangent and a chord through the point of tangency.
  • The vertex of a tangent-chord angle is the point of tangency.
  • Its measure is half the measure of its intercepted arc.
  • One side is tangent to the circle and the other side is a chord through the tangency point.
Where it is used

Where tangent-chord angle shows up

  • Use tangent-chord angles in circle theorems that mix tangency with intercepted-arc reasoning.
  • Use them when solving for unknown angles formed on the edge of a circle by a tangent line.
  • Use the concept in proof problems where a tangent introduces a half-arc measure relationship.
Common mistakes

What to watch out for

  • Do not use the wrong arc; the angle depends on the intercepted arc opened by the tangent and chord together.
  • Do not treat the figure as an inscribed angle if one side is clearly tangent rather than a chord.
  • Do not move the vertex away from the point of tangency and keep the same name.
Worked examples

Tangent-Chord Angle examples

Use these worked examples to see the idea in a clean diagram first, then in the kind of reasoning students usually need for classwork, homework, or test practice.

Example 1

Example 1: Building tangent-chord angle from the intercepted arc

Start with the points on the circle, then read the angle from the arc they determine.

  • Locate the arc first.
  • Identify where the angle's sides meet the circle or tangent.
  • Connect the angle measure to the intercepted arc.

Result: The angle is understood as part of a circle relationship, not as an isolated opening.

Example 2

Example 2: Using tangent-chord angle in a theorem step

Treat the diagram type as the reason a circle-angle rule can be used next.

  • Name the angle type correctly.
  • Recall the matching circle theorem.
  • Apply it to the arc or measure shown on the board.

Result: The theorem step is justified by the structure of the circle diagram.

For

Why this page helps

This page helps because tangent-chord angle combines two different circle objects in one theorem setting. Students need to see the tangent, the chord, the tangency point, and the intercepted arc all at once to read the figure correctly.

Do

What you can do here

  • Adjust the tangent and chord at one point of contact and read the resulting angle from the arc.
  • Compare the tangent-chord angle with an inscribed angle to see their related half-arc structure.
  • Keep a diagram that makes the tangency point and intercepted arc easy to review later.
Learning outcome

What this page helps you do

These takeaways are meant to help you recognize the idea faster, read diagrams more accurately, and use the topic with more confidence in real problems.

1

Tangent-Chord Angle

Recognise tangent-chord angle setups more quickly in theorem and proof questions.

2

Tangent-Chord Angle

Use the half-intercepted-arc rule without confusing it with a different circle-angle configuration.

3

Tangent-Chord Angle

Read mixed tangent-and-chord geometry with greater precision.

06

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An inscribed angle has its vertex on the circle and intercepts an arc.

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