Name: ___________________________ Period: ________ 10.1 Investigations of Circle theorems Investigation 1: Angles inscribed in a semi-circle Step 1: Construct a large circle, and make a diameter with end points A and B Step 2: Place 1 point somewhere on the arc that connects A and B. Call this point C. Step 3: Construct segments and . Then measure the angle ∠. Step 4: What is the measurement of ∠? Step 5: Repeat steps 2 – 4 using new points D and E. Step 6: Write a summary of your findings that includes your construction, you must include conjecture that demonstrates your findings. Investigation 2: Cyclic quadrilaterals (Quadrilaterals whose vertices all lie on a circle) Step 1: Construct 2 large circles. Place 4 points along the circumference of your circles and connect these points with segments. Please make sure your shape is not a rectangle or similar to any shape of your neighbors. Step 2: Measure each of the angles inside the quadrilaterals. Write the measure of each angle in its appropriate place. Step 3: Carefully examine the relationships of the angles of these quadrilaterals. Step 4: Write a summary of your findings that includes your construction, you must include conjecture that demonstrates your findings.
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Name: ___________________________ Period: ________ 10.1 Investigations of Circle theorems
Investigation 1: Angles inscribed in a semi-circle
Step 1: Construct a large circle, and make a diameter with end points A and B
Step 2: Place 1 point somewhere on the arc that connects A and B. Call this point
C.
Step 3: Construct segments 𝐴𝐶 and 𝐵𝐶. Then measure the angle ∠𝐴𝐶𝐵.
Step 4: What is the measurement of ∠𝐴𝐶𝐵?
Step 5: Repeat steps 2 – 4 using new points D and E.
Step 6: Write a summary of your findings that includes your construction, you must include conjecture
that demonstrates your findings.
Investigation 2: Cyclic quadrilaterals (Quadrilaterals whose vertices all lie on a circle)
Step 1: Construct 2 large circles. Place 4 points along the circumference of your
circles and connect these points with segments. Please make sure your shape is not a
rectangle or similar to any shape of your neighbors.
Step 2: Measure each of the angles inside the quadrilaterals. Write the measure of
each angle in its appropriate place.
Step 3: Carefully examine the relationships of the angles of these quadrilaterals.
Step 4: Write a summary of your findings that includes your construction, you must include conjecture
that demonstrates your findings.
Proof 1
Why is the angle inscribed in a semi-circle always a right angle?
1) O is the center of the circle. Draw a segment 𝑂𝐶.
2) What kind of triangle is ΔAOC? _____________________ label the congruent angles “x”
What kind of triangle is ΔOCB? _____________________ label the congruent angles “y”
3) Since the sum of the angles in a triangle is 180°, express ∠𝐴𝑂𝐶 in terms of x and ∠𝐶𝑂𝐵 in terms of y.