Angle of Intersecting Secants Theorem

Chapter 48

Concept Explanation

The Angle of Intersecting Secants Theorem helps in calculating the angle formed when two secants intersect outside a circle. It states that the measure of the angle formed by the secants is half the difference of the measures of the intercepted arcs.

The formula for this theorem is:

$m \angle APB = \frac{1}{2}(m \overset{\frown}{AC} - m \overset{\frown}{BD})$

Where:

$m \angle APB$ is the angle formed by the intersecting secants.
$\overset{\frown}{AC}$ and $\overset{\frown}{BD}$ are the intercepted arcs.

This concept can be used in problems involving intersections outside a circle, such as the ones that follow.

Common Mistakes (Recap):

Incorrect Arc Subtraction: Students often subtract the arcs in the wrong order.
Confusion with Interior Intersections: It’s easy to confuse this theorem with the Angle of Intersecting Chords Theorem.
Missing the Half Factor: Forgetting to divide the difference by 2 is a common mistake.

Helpful Tips:

Label Arcs and Secants: Clearly label the intercepted arcs on your diagram for better visualization.
Check Arc Order: Always ensure you subtract the smaller arc from the larger one.

Hard Questions (with Diagrams):

Q1: Two secants $AB$ and $CD$ intersect outside a circle at point $P$ . The measures of the intercepted arcs are $m \overset{\frown}{AC} = 150^\circ$ and $m \overset{\frown}{BD} = 70^\circ$ . Find $m \angle APB$ .

*** QuickLaTeX cannot compile formula:

\begin{tikzpicture}
\draw (0,0) circle (3cm);
\draw<a href="-3,1">thick</a> -- (4,-2);
\draw<a href="-3,-1">thick</a> -- (3,2);
\node at (-3,1) [left] {A};
\node at (4,-2) [right] {B};
\node at (-3,-1) [left] {C};
\node at (3,2) [right] {D};
\node at (0,-2) [below] {P};
\node at (-1.5,0) [above] {$\overset{\frown}{AC} = 150^\circ$};
\node at (1.5,0) [above] {$\overset{\frown}{BD} = 70^\circ$};
\end{tikzpicture}


*** Error message:
Error: Cannot create svg file

Step-by-Step Solution:

Use the Angle of Intersecting Secants Theorem:

$m \angle APB = \frac{1}{2}(m \overset{\frown}{AC} - m \overset{\frown}{BD})$

Substitute the given values:

$m \angle APB = \frac{1}{2}(150^\circ - 70^\circ) = \frac{1}{2}(80^\circ) = 40^\circ$

Answer:

$m \angle APB = 40^\circ$

Q2: Two secants $PQ$ and $RS$ intersect outside a circle at point $P$ . The measures of the intercepted arcs are $m \overset{\frown}{PR} = 130^\circ$ and $m \overset{\frown}{QS} = 40^\circ$ . Find $m \angle QPR$ .

Step-by-Step Solution:

Apply the theorem:

$m \angle QPR = \frac{1}{2}(m \overset{\frown}{PR} - m \overset{\frown}{QS})$

Substitute the known arc measures:

$m \angle QPR = \frac{1}{2}(130^\circ - 40^\circ) = \frac{1}{2}(90^\circ) = 45^\circ$

Answer:

$m \angle QPR = 45^\circ$

Q3: The measure of arc $\overset{\frown}{AC}$ is 160°, and the measure of arc $\overset{\frown}{BD}$ is 80°. If two secants $AB$ and $CD$ intersect at point $P$ outside the circle, find the measure of $\angle APB$ .

Step-by-Step Solution:

Use the Angle of Intersecting Secants Theorem:

$m \angle APB = \frac{1}{2}(m \overset{\frown}{AC} - m \overset{\frown}{BD})$

Substitute the arc measures:

$m \angle APB = \frac{1}{2}(160^\circ - 80^\circ) = \frac{1}{2}(80^\circ) = 40^\circ$

Answer:

$m \angle APB = 40^\circ$

Q4: If the intercepted arcs are $\overset{\frown}{PR} = 140^\circ$ and $\overset{\frown}{QS} = 60^\circ$ , and the secants intersect outside the circle, what is the measure of the angle formed by the secants?

Step-by-Step Solution:

Apply the formula:

$m \angle QPR = \frac{1}{2}(m \overset{\frown}{PR} - m \overset{\frown}{QS})$

Substitute the known values:

$m \angle QPR = \frac{1}{2}(140^\circ - 60^\circ) = \frac{1}{2}(80^\circ) = 40^\circ$

Answer:

$m \angle QPR = 40^\circ$

Q5: Two secants $XY$ and $AB$ intersect outside a circle at point $P$ . The measure of arc $\overset{\frown}{XA}$ is 170°, and the measure of arc $\overset{\frown}{YB}$ is 90°. Find the measure of $\angle XPY$ .

Step-by-Step Solution:

Use the theorem:

$m \angle XPY = \frac{1}{2}(m \overset{\frown}{XA} - m \overset{\frown}{YB})$

Substitute the given values:

$m \angle XPY = \frac{1}{2}(170^\circ - 90^\circ) = \frac{1}{2}(80^\circ) = 40^\circ$

Answer:

$m \angle XPY = 40^\circ$

In conclusion, the Angle of Intersecting Secants Theorem is vital for solving problems where secants intersect outside a circle. By carefully calculating the difference of the intercepted arcs and dividing by 2, you can solve for the angle formed by the secants. When solving these problems, remember to label the arcs clearly and apply the formula accurately.

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