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9. Circles
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United States of AmericaFL
Grade 912

9.05 Angles from intersecting chords, secants, and tangents

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Concept summary

A line that intersects two or more points on a curve is called a secant. If that curve is a circle, then it is called the secant of a circle.

A secant segment is a chord that has been extended in one direction.

Minor arc congruency theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Circle C with chords A B and D E. Chords A B and D E are congruent. Minor arcs A B and D E are congruent.
Tangent and intersected chord theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Circle C with points A, B, and E placed clockwise on the circle. Chord A E is drawn. Line D E is tangent to circle C at E.

For the diagram shown, the tangent and intersected chord theorem says thatm \angle AED = \dfrac{1}{2}m \overset{\large\frown}{ABE}

Secant angle theorem

The measure of an angle formed by two secant lines that intersect inside the circle is half the sum of the measures of the intercepted arcs.

A circle and two secant lines intersecting at a point in the circle. The lines divide the circle into 4 arcs. One arc is labeled x degrees, and another arc is labeled y degrees. The two labeled arcs are not adjacent. The two secant lines forms an angle labeled 1. Angle 1 intercepts the arc labeled x degrees.

For the diagram shown, the secant angle theorem says thatm\angle 1 = \dfrac{1}{2}\left(x\degree + y\degree\right)

Outside secant angle theorem

The measure of an angle formed by two tangents, two secants, or a tangent and a secant that intersect outside of a circle is half of the difference of the measures of the intercepted arcs.

In any of the following cases, the same equation results:

Three diagrams are shown. The left diagram labeled Case 1 shows a circle, and two rays that are secant to the circle. The middle diagram labeled Case 2 shows a circle, a ray that is secant to the circle, and another ray that is tangent to the circle. The right diagram labeled Case 3 shows a circle, and two rays that are tangent to the circle. For each case, the rays share a common endpoint that is outside the circle and forms an angle labeled 1. Angle 1 intercepts the circle in two arcs with the smaller arc labeled a degrees and the larger arc labeled b degrees.

m\angle 1 = \dfrac{1}{2}\left( b\degree - a\degree \right)

Worked examples

Example 1

If m\angle FPB = \left(2x+17\right) \degree , m\overset{\large\frown}{AG} = \left(3x+7\right)\degree , and m\overset{\large\frown}{FB} = \left(2x-7\right)\degree , solve for x.

A circle with points A, G, B, and F placed clockwise on the circle. Chords A B and F G are drawn and intersects at a point P in the circle.

Approach

By extending the chords AB and FG we get secants.

We know the measure of two arcs and the measure of the angle between the two secants, so we can use the secant angle theorem to relate these quantities.

Solution

\displaystyle m\angle FPB\displaystyle =\displaystyle \dfrac{1}{2} \left(m\overset{\large\frown}{AG} + m\overset{\large\frown}{FB} \right)Secant angle theorem
\displaystyle 2x+17\displaystyle =\displaystyle \dfrac{1}{2}\left(\left(3x + 7\right) + \left(2x-7\right) \right)Substitute known values
\displaystyle 2\left(2x+17\right)\displaystyle =\displaystyle 3x+7 + 2x-7Multiply both sides by 2
\displaystyle 2\left(2x+17\right)\displaystyle =\displaystyle 5xCombine like terms
\displaystyle 4x+34\displaystyle =\displaystyle 5xDistribute the multiplication by 2
\displaystyle 34\displaystyle =\displaystyle xSubtract 4x from both sides
\displaystyle x\displaystyle =\displaystyle 34Symmetric property of equality

Example 2

Given m\overset{\large\frown}{AC} = 78 \degree,m\overset{\large\frown}{AD}= 170 \degree, and that \overline{AB} is tangent to the circle. Find m\angle ABC.

A circle and with points A, D, and C placed clockwise on the circle. Secant line B D passing through C is drawn. Point C is between B and D. A segment from A to B is drawn. Segment A B is tangent to the circle at A.

Approach

We know the measures of two arcs of the circle, and wish to find the measure of the angle outside of the circle formed by the corresponding secant and tangent. To find this measure, we can use the outside secant angle theorem.

Solution

\displaystyle \angle ABC\displaystyle =\displaystyle \dfrac{1}{2}\left(m\overset{\large\frown}{AD} - m\overset{\large\frown}{AC}\right)Outside secant angle theorem
\displaystyle =\displaystyle \dfrac{1}{2}\left(170\degree-78\degree \right)Substitute known values
\displaystyle =\displaystyle 46\degreeSimplify

The measure of \angle{ABC} is 46 \degree.

Outcomes

MA.912.GR.6.2

Solve mathematical and real-world problems involving the measures of arcs and related angles.

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