9.4: Inscribed Angles
Learning Objectives
 Find the measure of inscribed angles and the arcs they intercept.
Review Queue
We are going to use #14 from the homework in the previous section.
 What is the measure of each angle in the triangle? How do you know?
 What do you know about the three arcs?
 What is the measure of each arc?
Know What? The closest you can get to the White House are the walking trails on the far right. You want to get as close as you can (on the trail) to the fence to take a picture (you were not allowed to walk on the grass). Where else can you take a picture from to get the same frame of the White House? Your line of sight in the camera is marked in the picture as the grey lines.
Inscribed Angles
In addition to central angles, we will now learn about inscribed angles in circles.
Inscribed Angle: An angle with its vertex on the circle and sides are chords.
Intercepted Arc: The arc that is inside the inscribed angle and endpoints are on the angle.
The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc.
Investigation 94: Measuring an Inscribed Angle
Tools Needed: pencil, paper, compass, ruler, protractor
 Draw three circles with three different inscribed angles. Try to make all the angles different sizes.
 Using your ruler, draw in the corresponding central angle for each angle and label each set of endpoints.
 Using your protractor measure the six angles and determine if there is a relationship between the central angle, the inscribed angle, and the intercepted arc.
Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
Example 1: Find and .
Solution: From the Inscribed Angle Theorem:
Example 2: Find and .
Solution: The intercepted arc for both angles is . Therefore,
This example leads us to our next theorem.
Theorem 98: Inscribed angles that intercept the same arc are congruent.
and intercept , so .
and intercept , so .
Example 3: Find in .
Solution: is the center, so is a diameter. endpoints are on the diameter, so the central angle is .
Theorem 99: An angle intercepts a semicircle if an only if it is a right angle.
intercepts a semicircle, so .
is a right angle, so is a semicircle.
Anytime a right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter and the diameter is the hypotenuse.
Example 4: Find and .
Solution:
Inscribed Quadrilaterals
Inscribed Polygon: A polygon where every vertex is on a circle.
Investigation 95: Inscribing Quadrilaterals
Tools Needed: pencil, paper, compass, ruler, colored pencils, scissors
 Draw a circle. Mark the center point .
 Place four points on the circle. Connect them to form a quadrilateral. Color in the 4 angles.
 Cut out the quadrilateral. Then cut the diagonal , making two triangles.
 Line up and so that they are next to each other. What do you notice?
By cutting the quadrilateral in half, we are able to show that and form a linear pair when they are placed next to each other, making and supplementary.
Theorem 910: A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.
If is inscribed in , then and .
If and , then is inscribed in .
Example 5: Find the value of the missing variables.
a)
b)
Solution:
a)
b)
Example 6: Algebra Connection Find and in the picture below.
Solution:
Know What? Revisited You can take the picture from anywhere on the semicircular walking path, the frame will be the same.
Review Questions
 Questions 18 use the vocabulary and theorems learned in this section.
 Questions 927 are similar to Examples 15.
 Questions 2833 are similar to Example 6.
 Question 34 is a proof of the Inscribed Angle Theorem.
Fill in the blanks.
 A _______________ polygon has all its vertices on a circle.
 An inscribed angle is ____________ the measure of the intercepted arc.
 A central angle is ________________ the measure of the intercepted arc.
 An angle inscribed in a ________________ is .
 Two inscribed angles that intercept the same arc are _______________.
 The _____________ angles of an inscribed quadrilateral are ________________.
 The sides of an inscribed angle are ___________________.
 Draw inscribed angle in . Then draw central angle . How do the two angles relate?
Quadrilateral is inscribed in . Find:
Quadrilateral is inscribed in . Find:
Find the value of and/or in .
Algebra Connection Solve for .
 Fill in the blanks of the Inscribed Angle Theorem proof. Given: Inscribed and diameter Prove:
Statement  Reason 

1. Inscribed and diameter and 

2.  
3.  All radii are congruent 
4.  Definition of an isosceles triangle 
5. and  
6. and  
7. and  
8.  Arc Addition Postulate 
9.  
10.  Distributive PoE 
11.  
12. 
Review Queue Answers
 , it is an equilateral triangle.
 They are congruent because the chords are congruent.