CCSS.Math.Content.HSG-C.A

CCSS.Math.Content.HSG-C.A is a cluster heading. These are the individual standards under it.

• CCSS.Math.Content.HSG-C.A.1

  Prove that all circles are similar.

• CCSS.Math.Content.HSG-C.A.2

  Identify and describe relationships among inscribed angles, radii, and chords.

• CCSS.Math.Content.HSG-C.A.3

  Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

• CCSS.Math.Content.HSG-C.A.4

  (+) Construct a tangent line from a point outside a given circle to the circle.

01

What Students Need to Do

Students identify relationships among arcs, chords, radii, tangents, central angles, and inscribed angles. They use those relationships to calculate measures and prove claims. They also construct inscribed figures, circumscribed figures, and tangent lines.

02

What Mastery Looks Like

Students can find missing arc and angle measures in an unfamiliar circle diagram. They can write a clear proof using definitions and known relationships. They can also construct tangents or inscribed figures and explain why the construction works.

03

Common Misconceptions

Students often confuse central angles with inscribed angles and forget the factor of two. They may assume any line touching a drawn circle is tangent or that every quadrilateral in a circle has four right angles. Some use a theorem without checking that angles intercept the same arc.

04

How to Assess It

Exit ticket: In circle O, tangent PA meets chord AB at A, and the angle between PA and AB is 38°. Find the minor arc AB, angle ACB for C on the opposite arc, and angle OAP, then justify each answer.

1. 01

   Give pairs compasses, rulers, and patty paper to compare two circles by tracing and scaling radii, then record why circles are similar.

2. 02

   Display central and inscribed angles intercepting the same arc, then ask students to explain why their measures have a 2-to-1 relationship.

3. 03

   Run a card sort matching circle diagrams, theorem statements, missing measures, and reasons, with teams correcting any unmatched cards.

4. 04

   Place a bicycle wheel against the floor, mark the contact point, and explain why the spoke there is perpendicular to the floor.

• CCSS.Math.Content.HSG-CO.C.9

  Prove theorems about lines and angles.

• CCSS.Math.Content.HSG-CO.C

  Prove geometric theorems

• CCSS.Math.Content.7.G.B.4

  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumf...