Lab 4 Part A: Inscribed Angles

This part of the lab is a continuation of the Carpenter Theorem lab.

Arcs and Arc Angles – How Sketchpad Does Arcs

• Draw a circle with center O through R; hide R.  Then construct points A and B on the circle.  These two points divide the circle into two arcs.  Let's see how Sketchpad handles arcs.
• Select the two points A and B (in this order) along with the circle and choose Arc On Circle from the Construct Menu.  The arc will appear as a selected object.  Immediately choose Display>Line Width>Thick so that you can see the arc better.

• Experiment with the arc.  Move A across B.  The Sketchpad arc is always draw counterclockwise from the first selected point to the second one, so when you move a point across, the arc should jump from being the short one to the long one, or vice verse.
• Measure angle AOB and also measure the arc angle.  How are these related?  Does this still hold true when the arc becomes bigger than a semicircle?
• Construct the arc with the points in the opposite order.  The two arcs should together cover the whole circle exactly.  What is the sum of the arc angles?  (You may find it handy to hide the circle itself and color the arcs separately.) Notice that this is a way in Sketchpad to measure angles > 180 degrees.
• Select an arc and choose Construct > Arc Interior > Arc Segment.  Do this for the other arc also.  Compare this with the result of Construct > Arc Interior > Arc Sector. 

Inscribed Angles

• Draw a circle with center O through R and construct points A, B, C are points on the circle.
• Construct point D as the intersection of the line BO with the circle (the other intersection besides B).
• Construct the line through B perpendicular to OB and construct points E and F on this line on opposite sides of B, as in the figure.

• Draw in the segments as in the figure.

Write answers to these questions based on your knowledge of isosceles triangles and your work before (or figure it out again if needed)..

• How are angle DBC and angle DOC related?  Give an explanation. Measure the angles with Sketchpad and confirm your answer.
• What is angle BCD? Give an explanation. Measure the angles with Sketchpad and confirm your answer.
• What is the relationship between angle CBF and angle COB?
• State the comparable relationships in triangle DBA.

Add two arcs and arc segments as shown in this figure.

• An angle ABC with points A, B, C on a circle c is called an inscribed angle in c.  The angle ABC here is an inscribed angle and the unshaded part of the circle is the part of the circle inside the inscribed angle.
• How are angle ABC and angle AOC related?  Explain. (Hint:  Look at each right triangle and add some angles. Or else subtract the shaded part from the whole.)
• What if you drag A so that A and C are both on the same side of line BD.  Is the relationship still true?  Explain why.

Application 1

In this figure, mark the equal angles and find some similar triangles. Write some equal ratios or products resulting from the similar triangles.

Application 2

If a quadrilateral ABCD is inscribed in a circle, the angles BAD and DCB are both inscribed angles with the same end points. What is the relationship between these angles? What is the relationship between the angles at vertices B and D?