Page 186 - IM_FL_Geometry_Print Sample

Student Task Statement
Here is a cyclic quadrilateral, , centered at point , which is inside the quadrilateral. Prove that the pairs of opposite angles are supplementary.
Student Response
Answers vary. Sample responses:
• Drawing radii from to each point of the quadrilateral creates 4 isosceles triangles. By the Isosceles Triangle Theorem, the base angles in each isosceles triangle are congruent to each other. Label the measures of the base angles of isosceles triangles , , , and
as , , , and respectively. It doesn’t matter whether we measure with radians or degrees, so assume these measures are all in degrees. The central angles go around in a complete circle, so their degree measures sum to 360. Adding up all the angles in all 4 isosceles triangles with the Triangle Angle Sum Theorem, the result is . Subtracting 360 from each side and then dividing each side by 2, we get . The expressions and represent degree measures of one pair of opposite angles, which means, by the equation in the previous sentence, that those opposite angles are supplementary. Similarly, the expressions and represent degree measures of the other pair of opposite angles, and are supplementary for the same reason.
• First, look at opposite angles and . Angle is inscribed to an arc of the circle, and angle is inscribed to the arc that traces out the rest of the circle. By the Inscribed Angle Theorem, , so the angles are supplementary. The other pair of opposite angles are supplementary because of a similar argument.
Activity Synthesis
Select previously identi ed students to share in this order:
• First, discuss the strategy of decomposing the cyclic quadrilateral into isosceles triangles and using arguments involving triangle angle relationships. Here is an image that highlights isosceles triangles:
Unit 7 Lesson 11: Circles Outside of Quadrilaterals 113