Grade 10 · Math · Geometry

10th grade Angles formed by chords, secants and tangents worksheet

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Angles formed by chords, secants and tangents

Grade 10 Math Worksheet - Challenge Me

NameDate

Circle the best answer, or write your answer on the line.

Kira maps two paths from Q across a round pond. Secants QCD and QGF meet at Q, outside circle O. What is ∠Q, in degrees?
Luz draws two lines from Q through a clock. Secants QCD and QGF meet at Q, outside circle O. What is ∠Q, in degrees?
Milo cuts two lines from Q through a round pizza. Secants QCD and QGF meet at Q, outside circle O. What is ∠Q, in degrees?
Noah lays two strings from Q across a rug. Secants QCD and QGF meet at Q, outside circle O. What is ∠Q, in degrees?
Pia cuts two crossing lines on a round pizza. Chords CF and DG cross at H, inside circle O. What is ∠CHD, in degrees?
Tala ices two crossing lines on a cookie. Chords CF and DG cross at H, inside circle O. What is ∠CHD, in degrees?
Zee sews two crossing stripes on a rug. Chords CF and DG cross at H, inside circle O. What is ∠CHD, in degrees?
Bo draws two lines from Q through a clock. For secants QCD and QGF, ∠Q = 43°, far arc DF = (3x − 6)° and near arc CG = 34°. What is x?
Esme cuts two lines from Q through a round pizza. For secants QCD and QGF, ∠Q = 31°, far arc DF = (4x + 2)° and near arc CG = 44°. What is x?
Jun lays two strings from Q across a rug. For secants QCD and QGF, ∠Q = 52°, far arc DF = (5x + 4)° and near arc CG = 60°. What is x?

Angles formed by chords, secants and tangents. Every version has 10 questions, and the answer key prints on the second page.

I can find the angle where two chords cross: half the sum of the arcs.
I can find the angle made outside the circle: half the difference of the arcs.
I can find an angle made by a tangent and a chord, and work back to a missing arc.

Standard: G-C.A.2

secant: A line that cuts through the circle at two points.

tangent-chord angle: The angle between a tangent and a chord at the point of tangency.

vertical angles: The two opposite angles where two lines cross; they are equal.

far arc / near arc: From an outside point, the arc farther away / nearer.

half the sum / half the difference: (arc + arc) ÷ 2 inside; (far − near) ÷ 2 outside.

Your child used the outside-angle rule (half the difference) for chords that cross inside the circle.

Your child found the right arcs but did not take half.

Your child made the angle between a tangent and a chord equal to its arc.

Ask: where do the two lines meet, inside or outside the circle?

Ask: is an angle made by two lines bigger or smaller than the arcs it catches?

Ask: where is the corner of this angle, at the center or on the circle?

What is the most common mistake with angles formed by chords, secants and tangents?

Your child used the outside-angle rule (half the difference) for chords that cross inside the circle.

How can I help my child with angles formed by chords, secants and tangents?

Ask: where do the two lines meet, inside or outside the circle?

Are these 10th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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