Topics
13. Circles
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United States of America
Grades 9-12

13.05 Angles from intersecting chords, secants, and tangents

Lesson
Practice

Introduction

We have previously learned about tangents and chords, and we will now learn about the last type of line found in circles called a secant line. This lesson will explore the relationships of the angles formed when these three types of lines intersect.

Angles from intersecting chords, secants, and tangents

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.

Exploration

Move the points around to explore the relationships between the angles created by the line segments.

  1. When the segments intersect inside the circle, what is the relationship between the angle formed by the segments and the intercepted arcs?
  2. When the segments intersect outside the circle, what is the relationship between the angle formed by the segments and the intercepted arcs?
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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)

Examples

Example 1

Given: \overleftrightarrow{AB} and \overleftrightarrow{CD} are secants to circle O

Prove: m\angle APD=\dfrac{1}{2}\left(m\overset{\large\frown}{AC}-m\overset{\large\frown}{BD}\right)

Worked Solution
Create a strategy

Notice we are trying to prove the first case of the outside secant angle theorem. The proof involves the measures of \overset{\large\frown}{AC} and \overset{\large\frown}{BD}, but we do not have a way to describe these measures yet. What we need to do first is contruct a chord that allows us to describe the measures of \overset{\large\frown}{AC} and \overset{\large\frown}{BD}.

Let's begin by constructing \overline{AD}.

By constructing this chord, we have constructed inscribed angles, \angle ADC and \angle DAB, and a triangle, \triangle ADP. We now have a way to describe the measures of the arcs.

Apply the idea
To prove: m\angle APD=\dfrac{1}{2}\left(m\overset{\large\frown}{AC}-m\overset{\large\frown}{BD}\right)
StatementsReasons
1.\overleftrightarrow{AB} and \overleftrightarrow{CD} are secants to circle OGiven
2.Construct \overline{AD}Any two points define a line
3.m\angle ADC=\dfrac{1}{2}m\overset{\large\frown}{AC}Inscribed angle theorem
4.m\angle DAP=\dfrac{1}{2}m\overset{\large\frown}{BD}Inscribed angle theorem
5.m\angle ADC=m\angle APD+m\angle DAPExterior angle of a triangle theorem
6.\dfrac{1}{2}m\overset{\large\frown}{AC}=m\angle APD+\dfrac{1}{2}m\overset{\large\frown}{BD}Substitution
7.\dfrac{1}{2}m\overset{\large\frown}{AC}-\dfrac{1}{2}m\overset{\large\frown}{BD}=m\angle APDSubtraction property of equality
8.\dfrac{1}{2}\left(m\overset{\large\frown}{AC}-m\overset{\large\frown}{BD}\right)=m\angle APDDistributive property

Example 2

Given m\overset{\large\frown}{AC} = 78 \degree, m\overset{\large\frown}{AD}= 170 \degree, and \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.
Worked Solution
Create a strategy

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.

Apply the idea
\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.

Example 3

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.
Worked Solution
Create a strategy

By extending the chords AB and FG, we get secant lines.

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.

Apply the idea
\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
Reflect and check

If we substitute our answer back into m\angle FPB = \left(2x+17\right) \degree , we find \begin{aligned}m\angle FPB &= 2\left(34\right)+17\\ &= 68+17 \\&=85\end{aligned}In the diagram, m\angle FPB appears to be a right angle, but we have just shown that it is not.

This is another reminder that we should not rely on a diagram for information. We should use the information given in the instructions and only use the diagram as visual inspiration for the direction to take when solving a problem.

Example 4

If m\angle ABC=52\degree in the diagram below, find m\overset{\large\frown}{BDC}.

Worked Solution
Create a strategy

One way to solve this problem is by using the tangent and intersected chord theorem to find m\overset{\large\frown}{BC} first, then subtracting that measure from 360\degree.

Apply the idea

By the tangent and intersected chord theorem, m\angle ABC=\dfrac{1}{2}m\overset{\large\frown}{BC}. Substituting m\angle ABC=52 and solving, we get:

\displaystyle m\angle ABC\displaystyle =\displaystyle \dfrac{1}{2}m\overset{\large\frown}{BC}Original equation
\displaystyle 52\displaystyle =\displaystyle \dfrac{1}{2}m\overset{\large\frown}{BC}Substitute m\angle ABC=52
\displaystyle 104\displaystyle =\displaystyle m\overset{\large\frown}{BC}Multiply both sides by 2

This is the measure of the minor arc, and m\overset{\large\frown}{BDC} is the measure of the corresponding major arc. By the arc addition postulate, m\overset{\large\frown}{BC}+m\overset{\large\frown}{BDC}=360.

\displaystyle m\overset{\large\frown}{BC}+m\overset{\large\frown}{BDC}\displaystyle =\displaystyle 360Arc addition postulate
\displaystyle 104+m\overset{\large\frown}{BDC}\displaystyle =\displaystyle 360Substitute m\overset{\large\frown}{BC}=104
\displaystyle m\overset{\large\frown}{BDC}\displaystyle =\displaystyle 256\degreeSubstract 104 from both sides
Reflect and check

Another way we could have solved this is by using the fact that \overleftrightarrow{AB} has a measure of 180\degree. We were given m\angle ABC=52, so its supplementary angle is 180-52=128. By the tangent and intersecting chords theorem, \begin{aligned}\dfrac{1}{2}m\overset{\large\frown}{BDC}&=128\\m\overset{\large\frown}{BDC}&=256\end{aligned}

Idea summary
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}

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)

In any of the following cases, the outside secant angle theorem says thatm\angle 1 = \dfrac{1}{2}\left( b\degree - a\degree \right)

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.

Outcomes

G.C.A.2

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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