Geometry (B): READ: Angles of Chords, Secants, and Tangents Reading

READ: Angles of Chords, Secants, and Tangents Reading

Angles of Chords, Secants, and Tangents

Learning Objective

find the measures of angles formed by chords, secants, and tangents

Measure of Tangent-Chord Angle

Theorem The measure of an angle formed by a chord and a tangent that intersect on the circle equals half the measure of the intercepted arc.

In other words:

$m\angle{FAB} = \frac{1}{2}m\widehat{ACB}$ and

$m\angle{EAB} = \frac{1}{2}m\widehat{ADB}$

Proof

Draw the radii of the circle to points $A$ and $B$ .

$\triangle{AOB}$ is isosceles, therefore

$m\angle{BAO}= m\angle{ABO} = \frac{1}{2}(180^\circ-m\angle{AOB})=90^\circ-\frac{1}{2}m\angle{AOB}.$

We also know that, $m\angle{BAO}+m\angle{FAB}=90^\circ$ because $FE$ is tangent to the circle.

We obtain $90^\circ -\frac{1}{2}m\angle{AOB}+m\angle{FAB} =90^\circ \Rightarrow m\angle{FAB}=\frac{1}{2}m\angle{AOB}.$

Since $\angle{AOB}$ is a central angle that corresponds to $\widehat{ADB}$ then, $m\angle{FAB} = \frac{1}{2}m \widehat{ADB}$ .

This completes the proof. $\blacklozenge$

Example 1

Find the values of $a, b$ and $c$ .

First we find angle $a: 50^\circ + 45^\circ + \angle{a}=180^\circ \Rightarrow m\angle{a}=85^\circ.$

Using the Measure of the Tangent Chord Theorem we conclude that:

$m\widehat{AB}=2(45^\circ)=90^\circ$

and

$m\widehat{AC}=2(50^\circ)=100^\circ$

Therefore,

$m\angle{b} =\frac{1}{2}10^\circ=50^\circ \\ m\angle{c} =\frac{1}{2}90^\circ=45^\circ$

Angles Inside a Circle

Theorem The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measure of their intercepted arcs. In other words, the measure of the angle is the average (mean) of the measures of the intercepted arcs.

In this figure, $m\angle{a}=\frac{1}{2}(m\widehat{AB}+m\widehat{DC}).$

Proof

Draw a segment to connect points $B$ and $C$ .

$m\angle{DBC} & = \frac{1} {2} m\widehat{DC} && \text{Inscribed angle}\\ m\angle{ACB} &= \frac{1} {2} m\widehat{AB} && \text{Inscribed angle}\\ m\angle{a} &= m\angle{ACB} + m\angle{DBC} && \text{The measure of an exterior angle in a triangle is equal to}\\ &&& \text{the sum of the measures of the remote interior angles.}\\ m\angle{a} &= \frac{1} {2} m\widehat{DC} + \frac{1} {2} m\widehat{AB} && \text{Substitution}\\ m\angle{a} &= \frac{1} {2} \left(m\widehat{DC} + m\widehat{AB}\right) \blacklozenge &&$

Example 2

Find $m\angle{DEC}$ .

$m\angle{AED} &= \frac{1} {2} \left(m\widehat{AD} + m\widehat{BC}\right) = \frac{1} {2} \left(40^\circ + 62^\circ\right) = 51^\circ\\ m\angle{DEC} &= 180^\circ -m\angle{AED}\\ m\angle{DEC} &= 180^\circ - 51^\circ = 129^\circ$

Angles Outside a Circle

Theorem The measure of an angle formed by two secants drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.

In other words: $m\angle{a} = \frac{1} {2} (y^\circ - x^\circ).$

This theorem also applies for an angle formed by two tangents to the circle drawn from a point outside the circle and for an angle formed by a tangent and a secant drawn from a point outside the circle.

Proof

Draw a line to connect points $A$ and $B$ .

$m\angle{DBA} & = \frac{1} {2} x^\circ && \text{Inscribed angle}\\ m\angle{BAC} & = \frac{1} {2} y^\circ && \text{Inscribed angle}\\ m\angle{BAC} & = m\angle{DBA} + m\angle{a} && \text{The measure of an exterior angle in a triangle is equal to}\\ &&& \text{the sum of the measures of the remote interior angles.}\\ \frac{1} {2} y^\circ & = \frac{1} {2} x^\circ + m\angle{a} && \text{Substitution}\\ m\angle{a} & = \frac{1} {2} (y^\circ - x^\circ) \blacklozenge &&$

Example 3

Find the measure of angle $x$ .

$m\angle{x} = \frac{1} {2} \left(220^\circ - 54^\circ\right) = 83^\circ$

Lesson Summary

In this section we learned about finding the measure of angles formed by chords, secants, and tangents. We looked at the relationship between the arc measure and the angles formed by chords, secants, and tangents.

The questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

Find the value of the variable.
Find the measure of the following angles:
1. $m\angle{OAB}$
2. $m\angle{COD}$
3. $m\angle{CBD}$
4. $m\angle{DCO}$
5. $m\angle{AOB}$
6. $m\angle{DOA}$
Find the measure of the following angles:
1. $m\angle{CDE}$
2. $m\angle{BOC}$
3. $m\angle{EBO}$
4. $m\angle{BAC}$
Four points on a circle divide it into four arcs, whose sizes are $44^\circ, 100^\circ, 106^\circ$ , and $110^\circ$ , in consecutive order. The four points determine two intersecting chords. Find the sizes of the angles formed by the intersecting chords.

Review Answers

1. $102.5^\circ$
2. $21^\circ$
3. $100^\circ$
4. $40^\circ$
5. $90^\circ$
6. $60^\circ$
7. $30^\circ$
8. $25^\circ$
9. $100^\circ$
10. $a = 60^\circ, b = 80^\circ, c = 40^\circ$
11. $a = 82^\circ, b = 56^\circ, c = 42^\circ$
12. $45^\circ$
13. $x = 35^\circ, y = 35^\circ$
14. $x = 60^\circ, y = 25^\circ$
15. $20^\circ$
16. $50^\circ$
17. $60^\circ$
18. $45^\circ$
1. $45^\circ$
2. $80^\circ$
3. $40^\circ$
4. $50^\circ$
5. $90^\circ$
6. $110^\circ$
1. $90^\circ$
2. $110^\circ$
3. $55^\circ$
4. $20^\circ$
$75^\circ$ and $105^\circ$

Last modified: Tuesday, 29 June 2010, 10:39 AM

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