Understand and apply theorems about circles
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.
Note: We used the Math Nspired activity Circles – Angles and Arcs as a guide for this lesson.
We started off our lesson on Angles and Arcs just a little differently than before. I always have students explore – to pay attention to what is changing and what is staying the same in a geometric figure. But I am trying to learn from many of you out there, and so I asked students to estimate first.
We say that ∠ABC is inscribed in the circle. What is its measure?
It turns out that just over 50% of the students correctly estimated the measure. I showed the students how others answered, but I didn’t show them the correct answer. Next we played with our dynamic geometry software to find out what students noticed about inscribed angles and central angles that intercept the same arc.
What happens when a right angle is inscribed in a circle?
What is the relationship between the central angle and its intercepted arc?
What is the relationship between the inscribed angle and its intercepted arc?
What is the relationship for inscribed angles and central angles that intercept the same arc?
Students made observations. Before we formalized their observations, I sent another Quick Poll. We were up to just over 75% correct.
After formalizing the observations together, students worked through a few exercises in their groups.
Our next major exploration was to determine the relationship for opposite angles in a cyclic quadrilateral. Students explored by themselves, and then I made someone the Live Presenter.
What did you notice?
B.K. had noticed that the opposite angles had a sum of 180˚.
Can we make sense of why that happens?
What’s true about the opposite angles of a cyclic quadrilateral?
-They’re inscribed angles.
-Their intercepted arcs make the whole circle.
Why is that significant?
The intercepted arcs add to 360˚. The two inscribed angles are half that sum, 180˚.
The opposite angles of a cyclic quadrilateral are supplementary.
And a Quick Poll to assess student understanding:
And the results. 65% have it correct. What happened to those who got 95˚? What happened to those who got 190˚?
And so the journey continues … as we are learning to answer the question being asked, and not just giving the first calculation or even the second calculation that we get when we make sense of the given information.