In this lesson, students construct a line that is perpendicular to the radius of a circle and that goes through the point where the radius intersects the circle. They prove that this perpendicular line intersects the circle at exactly one point. That is, the line is tangent to the circle. Students also prove the converse: If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.

Students use these findings to show that an angle circumscribed about a circle is supplementary to the central angle defined by the points at which the rays of the circumscribed angle are tangent to the circle. As students determine that a tangent line intersects a circle at exactly one point, then use that property to show additional properties of circumscribed angles, students are making sense of problems and persevering in solving them (MP1).

A particular choice of construction tools in this lesson and throughout this unit is not necessary. Paper folding and straightedge and compass moves are both acceptable methods.

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.