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G.9(C)·Geometry·Circles & PolyhedraSupporting

Secants, Chords, and Circle Angles

Test conjectures about angles and arcs formed by lines that intersect circles.

Students choose circle-angle relationships by locating the angle's vertex, distinguishing central and inscribed angles from angles formed by intersecting chords or exterior secants.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice50 minutes

Learn the concept

At the center or on the circle

A central angle equals the degree measure of its intercepted arc. An inscribed angle has its vertex on the circle and equals half its intercepted arc. Therefore, central and inscribed angles intercepting the same arc have a 2:1 relationship.

Inside the circle

When two chords intersect inside a circle, an angle equals half the sum of the arcs intercepted by that angle and its vertical angle: m∠=12(arc1+arc2)m\angle=\frac12(\text{arc}_1+\text{arc}_2). Use the pair opposite the chosen angle and its vertical partner.

Outside the circle

When two secants meet outside a circle, the exterior angle equals half the far arc minus the near arc: m∠=12(far arc−near arc)m\angle=\frac12(\text{far arc}-\text{near arc}). All arcs around a circle total 360∘360^\circ; a semicircle totals 180∘180^\circ. Locate the vertex before deciding which relationship applies.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 9 · 1 point

Circle with center V. Secant WY and secant ZX intersect inside the circle at point U, which is not the center. Arc WZ, on the left, is labeled 72 degrees, and arc XY, on the right, is labeled 60 degrees.
Secants of circle VOriginal visual from the released STAAR test.

Worked example

From exterior point P, two secants pass through near points A and C and then far points B and D. The far arc BD measures 166° and the near arc AC measures 54°. Find the angle at P and the measure of an inscribed angle intercepting arc BD.

  1. The secants intersect outside the circle, so the angle at P uses half the difference of the far and near arcs.
  2. m∠P=12(166−54)=56∘m\angle P=\frac12(166-54)=56^\circ.
  3. An inscribed angle intercepting arc BD uses only that arc.
  4. Its measure is 12(166)=83∘\frac12(166)=83^\circ.

The exterior angle is 56°; the inscribed angle is 83°.

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Geometry: Conjectures about Circles

UT IPSI · Kid2Kid / Texas Education Agency

Use circle relationships to test conjectures about angles and intercepted arcs.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. Two chords intersect inside a circle. An angle and its vertical angle intercept arcs measuring 92° and 48°. Find the angle.

    Show answer and reasoning

    70°

    The vertex is inside, so use the half-sum: 12(92+48)=70∘\frac12(92+48)=70^\circ.

  2. 2. An inscribed angle measures 37°. Find its intercepted arc and the central angle intercepting that same arc.

    Show answer and reasoning

    The arc and central angle both measure 74°.

    An intercepted arc is twice the inscribed angle. A central angle has the same measure as that arc.

  3. 3. A student says an exterior secant angle equals half the sum of its intercepted arcs. Correct the statement.

    Show answer and reasoning

    Use half the far arc minus the near arc.

    Half the sum is the rule for two chords intersecting inside the circle. Exterior secants use half the positive difference.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 9 · 1 point

Circle with center V. Secant WY and secant ZX intersect inside the circle at point U, which is not the center. Arc WZ, on the left, is labeled 72 degrees, and arc XY, on the right, is labeled 60 degrees.
Secants of circle VOriginal visual from the released STAAR test.

ZX↔\overleftrightarrow{ZX} and WY↔\overleftrightarrow{WY} are secants of circle VV, as shown. Based on this information, which of the following can be proved true?

  1. m∠ZUY=m∠ZVYm\angle ZUY = m\angle ZVY
  2. m∠XUY=12(mWZ^+mXY^)m\angle XUY = \frac{1}{2}(m\widehat{WZ} + m\widehat{XY})
  3. mWX^=180°−mWZ^m\widehat{WX} = 180° - m\widehat{WZ}
  4. m∠WUX=12(mZY^−mWX^)m\angle WUX = \frac{1}{2}(m\widehat{ZY} - m\widehat{WX})
Show STAAR answer and explanation

m∠XUY=12(mWZ^+mXY^)m\angle XUY = \frac{1}{2}(m\widehat{WZ} + m\widehat{XY})

Two chords intersect inside the circle at U, so the angle is half the sum of the intercepted arcs of that angle and its vertical angle. Thus m∠XUY=(72+60)/2=66°m\angle XUY=(72+60)/2=66°.

m∠ZUY=m∠ZVYm\angle ZUY = m\angle ZVY: U is not the center; the intersecting-chord angle is not generally equal to central angle ZVY.

mWX^=180°−mWZ^m\widehat{WX} = 180° - m\widehat{WZ}: This assumes ZX is a diameter; no diameter is given.

m∠WUX=12(mZY^−mWX^)m\angle WUX = \frac{1}{2}(m\widehat{ZY} - m\widehat{WX}): Interior chord angles use half the sum of intercepted arcs, not half their difference.

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Learning objectives

By the end of the lesson, students can:

  • Identify intercepted arcs for central, inscribed, interior, and exterior angles.
  • Use half-sum or half-difference relationships for intersecting lines.
  • Evaluate conjectures using circle-angle theorems and arc totals.

Instructional sequence

  1. 1

    Engage

    5 min

    Draw angles with vertices at the center, on the circle, and outside it. Ask why the same intercepted arc does not always give the same angle.

  2. 2

    Direct instruction

    12 min

    Organize the four angle relationships by vertex location. Trace the intercepted arcs, including the arc of the vertical angle for interior intersections.

  3. 3

    Guided practice

    12 min

    Solve central, inscribed, and intersecting-chord examples. Use full-circle and semicircle totals only when their conditions are shown.

  4. 4

    Conjecture testing

    14 min

    Analyze exterior secant examples and correct claims that confuse arc sums with arc differences. Justify statements with the appropriate theorem.

  5. 5

    Closure

    7 min

    Complete the exit ticket with a vertex classification before writing an equation.

Misconceptions to address

““Every circle angle is half its intercepted arc.””

Teacher move: That rule applies to an inscribed angle. A central angle equals its arc; interior and exterior intersections require two arcs.

““Two secants always mean subtract the arcs.””

Teacher move: Locate their intersection. An intersection inside uses half the sum of the opposite intercepted arcs; an exterior intersection uses half the positive difference.

““Any two arcs shown add to 180°.””

Teacher move: They must together form a semicircle with diameter endpoints. Otherwise use the actual marked relationships or the 360-degree full-circle total.