Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Central Angle
- Inscribed Angle
Purpose and Instructional Strategies
In the middle grades, students encountered arc measure, arc length, and central angle by way of circle graphs and random experiments involving spinners, and they learned the formula for the circumference of a circle. In Algebra 1, students rearranged formulas to highlight a quantity of interest and solved linear equations in one variable. In Geometry, students learn a variety of results about angles and arcs in circles and how they relate to one another. In later courses, students will determine the value of trigonometric functions for real numbers by identifying angle measures in the unit circle.
- Instruction includes using precise definitions and language when working with angle measures and arcs involving circles. Students should be able to determine the similarities and differences between each of the various angle measures and arcs and how their relationships interact with one another. (MTR.4.1)
- For example, students should be able to answer questions like “What is the difference between central angle and inscribed angle?”, “Does the diameter have a central angle?” and “What is the difference between the measure of an inscribed angle and the measure of an intercepted arc?”
- Instruction includes student understanding of the Inscribed Angle Theorem and how it relates to angle measures within a circle. Proving this Theorem requires three cases: (1) the center of the circle is on one of the chords, so one of the chords is a diameter; (2) the center of the circle is between the two chords; and (3) the center of the circle is not between the two chords. Below describes the proof of the first case of the Inscribed Angle Theorem.
- For example, given Circle C with the chord AB and the diameter AD, the central angle is BCD and the inscribed angle is DAB, which is the same as CAB. Students should realize that BCD + BCA = 180. Students should also realize that CAB + ABC + BCA = 180. Therefore, after using the Substitution Property of Equality, BCD + BCA = CAB + ABC + BCA which is equivalent to BCD = CAB + ABC. Since two of the sides of triangle ABC are radii, students can classify it as an isosceles triangle, therefore CAB = ABC. Students can use this fact to make the equivalent equation BCD = CAB + CAB which is equivalent to BCD = 2CAB. This equation proves that the central angle, angle BCD, has twice the measure of the inscribed angle, angle CAB, which is the first case of the Inscribed Angle Theorem.
- Instruction includes the connection to coordinate geometry to prove, or justify, that every angle inscribed in a semicircle is a right angle.
- For example, given Circle C below with chords AB and BD, students can use slope criteria to prove that angle ABD is a right angle.
- Instruction includes student understanding of the following relationships between angle measures and arcs in a circle (even though some extend beyond Clarification 1). Students should understand that most relationships between angles in a circle, and relationships between segments in a circle, can be derived from the Inscribed Angle Theorem. (MTR.5.1)
- The measure of a central angle is equal to the measure of its intercepted arc.
- The measure of an inscribed angle is half the measure of its intercepted arc.
- The measure of an inscribed angle is half the measure of the central angle that
intercepts the same arc on the circle.
- In a circle (or congruent circles), any two inscribed angles with the same
intercepted arcs are congruent.
- The measure of an angle formed by two tangents, two secants or a secant and a
tangent from a point outside the circle, is half the difference of the measures of
the intercepted arcs.
- The angle made by two intersecting tangents to a circle is called a circumscribed
angle and it is supplementary to the central angle intercepting the same arc.
- An angle formed by two intersecting chords and whose vertex is inside the circle
equals one-half the sum of its intercepted arcs.
- An angle formed by a chord and a tangent, whose vertex is on the circle, is one-
half its intercepted arc.
- If one chord of a circle is a perpendicular bisector of another chord, then the first
chord is a diameter and bisects the arc intercepted by the chord.
- Instruction includes the understanding and naming of major, minor and semicircle arcs.
- Problem types include determining missing angle measures in both mathematical and
- Instruction includes the connection to arc lengths in circles. (MA.912.GR.6.4)
Common Misconceptions or Errors
- Students may try to apply properties of central angles and their intercepted arcs to inscribed angles and intercepted arcs.
- Students may confuse arc measure and arc length, and may try to measure arcs with linear units rather than degrees.
Instructional Task 1 (MTR.3.1)
- Find the measure of angle E in circle A.
Instructional Task 2 (MTR.2.1, MTR.4.1, MTR.5.1)
- A circle is given below with two intersecting secants,
PA and PC.
- Part A. What is the sum of the measures of angle BCP, angle CPB and angle PBC?
- Part B. What is the sum of the measures of angle PBC and angle ABC?
- Part C. What can you conclude about the relationship between the sum of the measures of the three angles from Part A and the sum of the measures of the two angles from Part B?
- Part D. Using the information from Part C, what can you conclude about the measure of angle CPB? State your conclusion algebraically as an equation where CPB =?.
- Part E. How can you use the information from Part D, to justify the Secant-Secant Angle Theorem which states that
Instructional ItemsInstructional Item 1
- The International Space Station (ISS) passes over the earth 248 miles above the earth’s surface. The angle formed between the two tangents formed from the ISS and the earth measures 140.4 °. What is the measure of the arc of the earth that could have a view of the ISS passing overhead?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.