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9.5: Angles of Chords, Secants, and Tangents

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Learning Objectives

  • Find the measures of angles formed by chords, secants, and tangents.

Review Queue

  1. What is m\angle OML and m\angle OPL? How do you know?
  2. Find m\angle MLP.
  3. Find m\widehat{MNP}.

Know What? The sun’s rays hit the Earth such that the tangent rays determine when daytime and night time are. If the arc that is exposed to sunlight is 178^\circ, what is the angle at which the sun’s rays hit the earth (x^\circ)?

Angle on a Circle

When an angle is on a circle, the vertex is on the edge of the circle. One type of angle on a circle is the inscribed angle, from the previous section. Another type of angle on a circle is one formed by a tangent and a chord.

Investigation 9-6: The Measure of an Angle formed by a Tangent and a Chord

Tools Needed: pencil, paper, ruler, compass, protractor

  1. Draw \bigodot A with chord \overline{BC} and tangent line \overleftrightarrow{ED} with point of tangency C.
  2. Draw in central angle \angle CAB. Find m\angle CAB and m\angle BCE.
  3. Find m\widehat{BC}. How does the measure of this arc relate to m\angle BCE?

Theorem 9-11: The measure of an angle formed by a chord and a tangent that intersect on the circle is half the measure of the intercepted arc.

m\angle DBA=\frac{1}{2} m\widehat{AB}

We now know that there are two types of angles that are half the measure of the intercepted arc; an inscribed angle and an angle formed by a chord and a tangent.

Example 1: Find:

a) m\angle BAD

b) m \widehat{AEB}

Solution: Use Theorem 9-11.

a) m\angle BAD = \frac{1}{2} m \widehat{AB} = \frac{1}{2} \cdot 124^\circ=62^\circ

b) m\widehat{AEB} = 2 \cdot m\angle DAB= 2 \cdot 133^\circ=266^\circ

Example 2: Find a, \ b, and c.

Solution: 50^\circ + 45^\circ + m\angle a & = 180^\circ \qquad \text{straight angle}\\m\angle a & = 85^\circ

m \angle b & = \frac{1}{2} \cdot m \widehat{AC}\\m \widehat{AC} & = 2 \cdot m\angle EAC = 2 \cdot 45^\circ=90^\circ\\m \angle b & = \frac{1}{2} \cdot 90^\circ=45^\circ

85^\circ + 45^\circ + m\angle c & = 180^\circ \qquad \text{Triangle Sum Theorem}\\m\angle c & = 50^\circ

From this example, we see that Theorem 9-8 is true for angles formed by a tangent and chord with the vertex on the circle. If two angles, with their vertices on the circle, intercept the same arc then the angles are congruent.

Angles inside a Circle

An angle is inside a circle when the vertex anywhere inside the circle, but not on the center.

Investigation 9-7: Find the Measure of an Angle inside a Circle

Tools Needed: pencil, paper, compass, ruler, protractor, colored pencils (optional)

  1. Draw \bigodot A with chord \overline{BC} and \overline{DE}. Label the point of intersection P.
  2. Draw central angles \angle DAB and \angle CAE. Use colored pencils, if desired.
  3. Find m\angle DPB, \ m\angle DAB, and m\angle CAE. Find m \widehat{DB} and m \widehat{CE}.
  4. Find \frac{m\widehat{DB} + m\widehat{CE}}{2}.
  5. What do you notice?

Theorem 9-12: The measure of the angle formed by two chords that intersect inside a circle is the average of the measure of the intercepted arcs.

m\angle SVR & = \frac{1}{2} \left ( m\widehat{SR} + m\widehat{TQ} \right )= \frac{m\widehat{SR}+m\widehat{TQ}}{2}=m\angle TVQ\\m\angle SVT& = \frac{1}{2} \left (m\widehat{ST} + m\widehat{RQ} \right )=\frac{m\widehat{ST}+m\widehat{RQ}}{2} = m\angle RVQ

Example 3: Find x.

a)

b)

c)

Solution: Use Theorem 9-12 to write an equation.

a) x=\frac{129^\circ+71^\circ}{2}=\frac{200^\circ}{2}=100^\circ

b) 40^\circ= \frac{52^\circ+x}{2}\!\\80^\circ=52^\circ+x\!\\28^\circ=x

c) x is supplementary to the angle that the average of the given intercepted arcs, y.

y=\frac{19^\circ+107^\circ}{2}=\frac{126^\circ}{2}=63^\circ \qquad  x+63^\circ=180^\circ; \ x=117^\circ

Angles outside a Circle

An angle is outside a circle if the vertex of the angle is outside the circle and the sides are tangents or secants. The possibilities are: an angle formed by two tangents, an angle formed by a tangent and a secant, and an angle formed by two secants.

Investigation 9-8: Find the Measure of an Angle outside a Circle

Tools Needed: pencil, paper, ruler, compass, protractor, colored pencils (optional)

  1. Draw three circles and label the centers A, \ B, and C. In \bigodot A draw two secant rays with the same endpoint. In \bigodot B, draw two tangent rays with the same endpoint. In \bigodot C, draw a tangent ray and a secant ray with the same endpoint. Label the points like the pictures below.
  2. Draw in all the central angles. Using a protractor, measure the central angles and find the measures of each intercepted arc.
  3. Find m\angle EDF, \ m\angle MLN, and m\angle RQS.
  4. Find \frac{m\widehat{EF}-m\widehat{GH}}{2}, \ \frac{m\widehat{MPN}-m\widehat{MN}}{2}, and \frac{m\widehat{RS}-m\widehat{RT}}{2}. What do you notice?

Theorem 9-13: The measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs.

m\angle D & = \frac{m\widehat{EF}-m\widehat{GH}}{2}\\m\angle L & =\frac{m\widehat{MPN}-m\widehat{MN}}{2}\\m\angle Q & =\frac{m\widehat{RS}-m\widehat{RT}}{2}

Example 4: Find the measure of x.

a)

b)

c)

Solution: For all of the above problems we can use Theorem 9-13.

a) x=\frac{125^\circ-27^\circ}{2}=\frac{98^\circ}{2}=49^\circ

b) 40^\circ is not the intercepted arc. The intercepted arc is 120^\circ, \ (360^\circ-200^\circ-40^\circ). x=\frac{200^\circ-120^\circ}{2}=\frac{80^\circ}{2}=40^\circ

c) Find the other intercepted arc, 360^\circ-265^\circ=95^\circ x = \frac{265^\circ-95^\circ}{2}=\frac{170^\circ}{2}=85^\circ

Know What? Revisited From Theorem 9-13, we know x=\frac{182^\circ-178^\circ}{2}=\frac{4^\circ}{2}=2^\circ.

Review Questions

  • Questions 1-3 use the definitions of tangent and secant lines.
  • Questions 4-7 use the definition and theorems learned in this section.
  • Questions 8-25 are similar to Examples 1-4.
  • Questions 26 and 27 are similar to Example 4, but also a challenge.
  • Questions 28 and 29 are fill-in-the-blank proofs of Theorems 9-12 and 9-13.
  1. Draw two secants that intersect:
    1. inside a circle.
    2. on a circle.
    3. outside a circle.
  2. Can two tangent lines intersect inside a circle? Why or why not?
  3. Draw a tangent and a secant that intersect:
    1. on a circle.
    2. outside a circle.

Fill in the blanks.

  1. If the vertex of an angle is on the _______________ of a circle, then its measure is _______________ to the intercepted arc.
  2. If the vertex of an angle is _______________ a circle, then its measure is the average of the __________________ arcs.
  3. If the vertex of an angle is ________ a circle, then its measure is ______________ the intercepted arc.
  4. If the vertex of an angle is ____________ a circle, then its measure is ___________ the difference of the intercepted arcs.

For questions 8-25, find the value of the missing variable(s).

  1. y \ne 60^\circ

Challenge Solve for x.

  1. Fill in the blanks of the proof for Theorem 9-12. Given: Intersecting chords \overline{AC} and \overline{BD}. Prove: m\angle a=\frac{1}{2} \left (m\widehat{DC}+m\widehat{AB}\right )
Statement Reason
1. Intersecting chords \overline{AC} and \overline{BD}.

2. Draw \overline{BC}

Construction
3. m\angle DBC=\frac{1}{2} m\widehat{DC}\!\\m\angle ACB=\frac{1}{2} m\widehat{AB}
4. m\angle a=m\angle DBC+m\angle ACB
5. m\angle a=\frac{1}{2} m\widehat{DC}+\frac{1}{2} m\widehat{AB}
  1. Fill in the blanks of the proof for Theorem 9-13. Given: Secant rays \overrightarrow{AB} and \overrightarrow{AC} Prove: m\angle a = \frac{1}{2} \left (m\widehat{BC}-m\widehat{DE} \right )
Statement Reason
1. Intersecting secants \overrightarrow{AB} and \overrightarrow{AC}.

2. Draw \overline{BE}.

Construction
3. m\angle BEC=\frac{1}{2} m\widehat{BC}\!\\m\angle DBE=\frac{1}{2} m\widehat{DE}
5. m\angle a+m\angle DBE=m\angle BEC
6. Subtraction PoE
7. Substitution
8. m\angle a=\frac{1}{2} \left (m\widehat{BC}-m\widehat{DE} \right )

Review Queue Answers

  1. m \angle OML = m \angle OPL = 90^\circ because a tangent line and a radius drawn to the point of tangency are perpendicular.
  2. 165^\circ + m \angle OML + m \angle OPL + m \angle MLP = 360^\circ\!\\{\;} \qquad \quad \ \ 165^\circ + 90^\circ + 90^\circ + m \angle MLP = 360^\circ\!\\{\;} \qquad \qquad \qquad \qquad \qquad \quad \ \ \ m \angle MLP = 15^\circ
  3. m\widehat{MNP} = 360^\circ - 165^\circ = 195^\circ

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Grades:

8 , 9 , 10

Date Created:

Feb 22, 2012

Last Modified:

Aug 21, 2014
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CK.MAT.ENG.SE.1.Geometry-Basic.9.5

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