HSG.C.A.2: Inscribed Angles, Radii, Chords and Tangents
In plain English: HSG.C.A.2 is the Common Core geometry standard that asks students to identify and describe relationships among inscribed angles, radii and chords. Students learn that an inscribed angle is half the central angle on the same arc, that an angle inscribed in a semicircle is a right angle, and that a radius is perpendicular to the tangent at its endpoint. It is usually taught in Geometry.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Common Core State Standards for Mathematics · Domain: Circles (C) · Cluster: Understand and apply theorems about circles Also written as HSG-C.A.2 or G-C.2 · Official standard
Students discover and explain how the angles and segments of a circle depend on each other. The central idea is the inscribed angle theorem: an inscribed angle measures half of its intercepted arc, so it is half of the central angle on the same arc. Two important cases follow, both named in the standard: an angle inscribed in a semicircle is a right angle, and a circumscribed angle (formed by two tangents) is supplementary to the central angle of the same arc, because each radius is perpendicular to its tangent.
Students also work with radii and chords: a radius perpendicular to a chord bisects it, which lets them find chord lengths and distances with the Pythagorean Theorem. Every relationship is justified, not only used: radii are congruent, so the triangles they form are isosceles, and that fact drives the proofs.
Learning Objectives
By the end of this lesson, students will be able to:
Describe the relationship between central, inscribed and circumscribed angles that intercept the same arc, and use it to find angle measures
Explain why an angle inscribed in a semicircle is a right angle, and use this fact to find lengths
Explain why a radius is perpendicular to the tangent at the point of tangency, and use the right angle to find tangent lengths
Use the fact that a radius perpendicular to a chord bisects the chord to find chord lengths and distances from the center
Justify circle relationships with congruent radii, isosceles triangles and angle sums
Prior Knowledge Required
Students should already be comfortable with:
Angle sums in triangles and the exterior angle of a triangle 8.G.A.5
The Pythagorean Theorem for unknown side lengths 8.G.B.7
Base angles of isosceles triangles and triangle proofs HSG.CO.C.10
Vocabulary of circles: radius, diameter, chord, arc, tangent and central angle HSG.CO.A.1
Solving linear equations in one variable HSA.REI.B.3
Give each student a handout with a circle, its center O, and two points A and B on the circle. Students draw three different points C on the larger arc and measure angle ACB each time with a protractor.
Warm-Up Prompt
"Measure angle ACB for three different positions of C, then measure angle AOB. What do you notice? Make a conjecture you think is true for every circle."
Record class results on the board. Measurements vary by a degree or two, but students should notice that the three inscribed angles are about equal and that angle AOB is about twice as large. Write the conjecture in the students' words, then tell them the lesson will prove it and use it.
Direct Instruction20 minutes
Vocabulary. A central angle has its vertex at the center; its measure equals the measure of its intercepted arc. An inscribed angle has its vertex on the circle and sides that are chords. A circumscribed angle has its vertex outside the circle and sides tangent to the circle. Use Diagram 1, where all three angles intercept the same 100° arc, and build the relationships in order:
Inscribed angle theorem, diameter case: if side CB of inscribed angle ACB passes through O, then OA = OC because both are radii, so triangle AOC is isosceles with base angles x. The exterior angle AOB of that triangle equals x + x = 2x, so the inscribed angle is half the central angle.
Other cases: when the center lies inside or outside angle ACB, draw the diameter through C and add or subtract two diameter cases. The result is the same: an inscribed angle measures half of its intercepted arc. Inscribed angles that intercept the same arc are therefore congruent.
Semicircle: if AB is a diameter, the intercepted arc is 180°, so any inscribed angle ACB is 180°/2 = 90°.
Tangent and radius: a tangent line meets the circle at exactly one point T, and radius OT is perpendicular to it. If it were not, the perpendicular from O to the line would be shorter than OT, so a second point of the line would also be at distance OT from O and on the circle.
Circumscribed angle: tangents PA and PB form quadrilateral OAPB with two right angles, so angle APB + angle AOB = 360° - 90° - 90° = 180°.
Then work through the examples, asking before each one: "Which relationship applies, and which segment is a radius?"
Central and inscribed angles
Arc AB measures 110°. Find central angle AOB and inscribed angle ACB, where C is on the other arc.
AB is a diameter of length 26, and C is on the circle with AC = 10. Find BC.
Equation: angle C = 90°, so BC = √(26² - 10²) = √576 = 24
Radius and tangent
Line PT is tangent to a circle with center O at T. The radius is 9 and OP = 15. Find PT.
Equation: OT ⊥ PT, so PT = √(15² - 9²) = √144 = 12
Circumscribed angle
Tangents from P touch the circle at A and B, and central angle AOB = 130°. Find circumscribed angle APB.
Equation: angle APB = 180° - 130° = 50°
Radius perpendicular to a chord
A circle has radius 10, and chord MN has length 16. How far is MN from the center?
Equation: the perpendicular from O bisects MN, so the distance is √(10² - 8²) = 6
Use Diagram 2 to connect the last two examples to pictures: the diameter case of the inscribed angle theorem on the left, and the perpendicular from the center to a chord on the right. Stress that in every example the key step was seeing a radius, and with it an isosceles or right triangle.
Guided Practice10-15 minutes
Pairs solve five short problems on whiteboards and name the relationship before computing: (a) an inscribed angle intercepts an 84° arc (42°); (b) an inscribed angle measures 38°, find the central angle on the same arc (76°); (c) triangle ABC is inscribed with AB a diameter and angle A = 34°, find angle B (56°); (d) a circumscribed angle measures 64°, find the central angle of the same arc (116°); (e) a tangent segment from P has length 12 and the radius is 5, find OP (13). Watch for students who double an inscribed angle to get another inscribed angle, and for students who forget that the right angle in (e) is at the point of tangency, not at P.
Independent Practice15 minutes
Students work alone: (a) two inscribed angles intercept the same arc and measure (3x + 5)° and (5x - 17)°, find x and the angle (x = 11, 38°); (b) a circle has radius 17 and a chord of length 30, find the distance from the center to the chord (8); (c) a point P is 25 units from the center of a circle of radius 7, find the length of a tangent segment from P (24); (d) a right triangle with legs 9 and 12 is inscribed in a circle, find the radius (the hypotenuse 15 is a diameter, so the radius is 7.5). For each, students write one sentence naming the relationship they used.
Closure5 minutes
Exit ticket: "Tangents PA and PB touch circle O at A and B, and minor arc AB measures 140°. Find angle AOB, angle APB, and angle ACB for a point C on the major arc." (Answers: 140°, 40°, 70°.) Ask students to mark every right angle on their sketch before computing.
Differentiation Strategies
For Struggling Students
Give a color-coded reference card: central angle = arc, inscribed angle = half the arc, circumscribed angle = 180° minus the central angle
Have students trace every radius in one color before solving, so isosceles and right triangles stand out
Start with dynamic geometry or patty-paper folding so students measure before they prove
For Advanced Students
Prove the inscribed angle theorem for the case where the center lies outside the angle, by subtracting two diameter cases
Prove the converse: if angle ACB is a right angle, then AB is a diameter of the circle through A, B and C
Prove that the two tangent segments from an outside point are congruent, using the right angles and the shared hypotenuse OP
Assessment Guidance
What to Look For
Check that students name the relationship before they compute: central angle and arc are equal, an inscribed angle is half its arc, and a circumscribed angle is supplementary to the central angle of the same arc. When a student uses a right angle, ask where it comes from (a diameter or a tangent) and at which vertex it is. For explanations, look for the radii: a complete justification points to congruent radii, the isosceles triangles they form, or the right angle between a radius and a tangent.
02
Classroom Activities
3 Activities
1
Prove the Inscribed Angle Theorem
20 minPairs
Pairs confirm the warm-up conjecture with careful drawings, then prove it for the case where one side of the inscribed angle is a diameter. The proof rests on the radii: two radii make an isosceles triangle, and its exterior angle is the central angle.
Procedure
Draw a circle with center O and a diameter CD. Mark a point A on the circle and draw chord CA and radius OA
Measure angle ACD and angle AOD. Sample result: angle ACD = 31° and angle AOD = 62°
Mark OA and OC as congruent radii, label the base angles of triangle AOC as x, and use the exterior angle theorem to show angle AOD = 2x
Repeat with the center inside angle ACB by drawing the diameter through C and adding two cases
Discussion Questions
Which fact about the circle made triangle AOC isosceles?
Why must all inscribed angles that intercept the same arc be congruent?
What happens to the inscribed angle when the intercepted arc is a semicircle?
Modification for Distance Learning
Use a free dynamic geometry tool: students drag C around the circle, record the measure of angle ACB and angle AOB in a shared table, and post a screenshot with their proof.
2
Tangents and Circumscribed Angles
20 minPairs
Students slide a ruler along a paper circle until it touches at exactly one point, measure the angle with the radius, and then use the right angles to connect circumscribed angles to central angles.
Procedure
Draw a circle with a compass, lay a ruler so it touches the circle at one point T, and draw the line. Draw radius OT and measure the angle between them (90°)
Draw a second tangent from an outside point P. Measure central angle AOB and circumscribed angle APB, then add them
Sample card: angle AOB = 124°, so angle APB = 56°. Students check with a protractor
Write the argument: quadrilateral OAPB has angle sum 360° and two right angles, so the other two angles add to 180°
Discussion Questions
A line through T makes an 80° angle with radius OT. Why must it cross the circle again?
What kind of quadrilateral is OAPB when the circumscribed angle is 90°?
Why are the tangent segments PA and PB always the same length?
3
Chords and Diameters Stations
20 minGroups of 3-4
Groups rotate through four stations that use radii and chords to solve practical problems. Each station ends with a one-sentence explanation of the relationship used.
Stations
Station 1, broken plate: on a paper-plate fragment, draw two chords and their perpendicular bisectors. The bisectors meet at the center, because the perpendicular bisector of any chord passes through the center
Station 2, carpenter's square: place the corner of a square (or an index card) on the rim of a paper plate. The two points where the edges cross the rim are the ends of a diameter, because the inscribed angle is 90°. Measure the diameter and check it with a second placement
Station 3, pipe cross-section: a pipe has radius 13 cm and the water surface is 24 cm wide. Find how far the surface is from the center (5 cm)
Station 4, equal chords: in a circle of radius 10, draw two chords of length 12 in different places. Measure their distances from the center (both 8) and explain why congruent chords are equidistant from the center
Challenge Variation
At Station 4, ask groups to prove that congruent chords have congruent central angles, using SSS on the two isosceles triangles formed by the radii.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Central, Inscribed and Circumscribed Angles
All three angles intercept the same 100° arc AB. The central angle AOB equals the arc, the inscribed angle ACB is half of it, and the circumscribed angle APB is 180° - 100° = 80° because each radius meets its tangent at a right angle. Points are computed exactly: A and B are 100° apart on the circle, and P lies on the bisector of angle AOB at distance OA/cos 50° from O.
Diagram 2: Diameters, Radii and Chords
Left: AB is a diameter, so angle ACB intercepts a 180° arc and measures 90°; the three congruent radii split triangle ABC into two isosceles triangles. Right: the radius perpendicular to a chord bisects it. With radius 10 and distance 6, each half of the chord is 8, so the chord is 16. Both panels are drawn to scale.
04
Homework Assignment
~30 min
HSG.C.A.2 Homework: Inscribed Angles, Radii, Chords and Tangents
Directions: Sketch each figure and mark every radius, right angle and congruent segment. Name the relationship you use in each step, and give exact answers unless a problem asks for a decimal.
Part 1: Angles and Arcs (Problems 1-2)
(a) Arc RS measures 128°. Find the central angle ROS and the inscribed angle RTS, where T is on the other arc. (b) An inscribed angle measures 71°. Find the measure of its intercepted arc and of the central angle on the same arc.
Triangle ABC is inscribed in a circle, and AB is a diameter. Angle A measures (2x + 10)° and angle B measures (3x - 5)°. Find x and all three angles of the triangle, and explain why angle C has the measure you found.
Part 2: Tangents (Problems 3-4)
A circular garden bed has a radius of 20 feet. A sprinkler at point P is 29 feet from the center. A straight hose runs from P and just touches the edge of the bed at point T. Explain why angle OTP is a right angle, and find the length of the hose from P to T.
Tangents from P touch circle O at A and B, and angle APB = 48°. Find angle AOB, the measure of the major arc AB, and angle ACB for a point C on the major arc. Explain why angles OAP and OBP are right angles.
Part 3: Chords and Proof (Problems 5-6)
A water pipe has an inner radius of 25 cm. The surface of the water inside is 48 cm wide, and the water is below the center of the pipe. Use a radius perpendicular to the surface to find the distance from the center to the surface and the depth of the water.
Prove that an angle inscribed in a semicircle is a right angle without using the inscribed angle theorem. Let AB be a diameter with center O, and let C be any other point on the circle. Use the fact that OA = OB = OC to find two isosceles triangles, and add their angles.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Angle Relationships
Central, inscribed and circumscribed angles correct, with relationships named
Most angles correct, or relationships not named
Relationships misapplied
Right Angles
Right angles located correctly (diameter or tangent) and used
Right angle used but its source not explained
Right angle missing or misplaced
Lengths
Tangent, chord and depth lengths correct with units
Correct setup with one arithmetic error
Incorrect setup
Proof
Isosceles triangles identified and angle sum argument complete
Isosceles triangles found, argument incomplete
No valid argument
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A central angle measures 72°. What is the measure of an inscribed angle that intercepts the same arc?
Answer: C
The arc measures 72°, and an inscribed angle is half its intercepted arc: 72°/2 = 36°. Choice B doubles the central angle instead of halving it. Choice A treats the inscribed angle as if its vertex were at the center.
Question 2 of 20 · Multiple Choice
An inscribed angle measures 47°. What is the measure of its intercepted arc?
Answer: A
The arc is twice the inscribed angle: 2 · 47° = 94°. Choice C halves the angle, which reverses the relationship. Choice D is 180° - 47°, which confuses the inscribed angle with a circumscribed angle.
Question 3 of 20 · Multiple Choice
Inscribed angles ADB and ACB intercept the same arc AB, and angle ADB = 58°. What is angle ACB?
Answer: D
Both angles are half of the same arc (116°), so they are congruent: 58°. Choice A is the arc, not the angle. Choice C is 180° - 58°, which would apply to opposite angles of an inscribed quadrilateral, not to angles on the same arc.
Question 4 of 20 · Multiple Choice
Triangle ABC is inscribed in a circle with AB as a diameter. Angle A measures 27°. What is angle B?
Answer: B
Angle C is inscribed in a semicircle, so it is 90°. Then angle B = 180° - 90° - 27° = 63°. Choice C forgets the right angle and subtracts only 27° from 180°. Choice D is angle C, not angle B.
Question 5 of 20 · Multiple Choice
AB is a diameter of length 17, and C is a point on the circle with AC = 8. What is BC?
Answer: C
Angle C is a right angle, and AB is the hypotenuse: BC = √(17² - 8²) = √225 = 15. Choice B adds the squares, treating AB as a leg. Choice A subtracts the lengths, and choice D adds them.
Question 6 of 20 · Multiple Choice
Line PT is tangent to circle O at T. The radius is 12 and OP = 37. What is PT?
Answer: A
OT ⊥ PT, so triangle OTP has its right angle at T and hypotenuse OP: PT = √(37² - 12²) = √1225 = 35. Choice D puts the right angle at O. Choice B subtracts the lengths, and choice C adds them.
Question 7 of 20 · Multiple Choice
What is the measure of the angle between a tangent line and the radius drawn to the point of tangency?
Answer: D
A radius is perpendicular to the tangent at the point where it meets the circle, so the angle is 90°. Choice C would mean the tangent line contains the radius, which would make it cross the circle.
Question 8 of 20 · Multiple Choice
Tangents from P touch circle O at A and B. Circumscribed angle APB measures 70°. What is central angle AOB?
Answer: B
Quadrilateral OAPB has right angles at A and B, so angle AOB = 360° - 90° - 90° - 70° = 110°. Choice C doubles the circumscribed angle as if it were inscribed. Choice A assumes the two angles are equal.
Question 9 of 20 · Multiple Choice
A central angle AOB measures 150°, and tangents at A and B meet at P. What is circumscribed angle APB?
Answer: C
The central and circumscribed angles of the same arc are supplementary: 180° - 150° = 30°. Choice A halves the central angle, which gives an inscribed angle, not a circumscribed one. Choice D is the major arc.
Question 10 of 20 · Multiple Choice
A circle has radius 20 and chord AB has length 32. How far is chord AB from the center?
Answer: D
The perpendicular from the center bisects the chord, so each half is 16, and the distance is √(20² - 16²) = √144 = 12. Choice A is half the chord, not the distance from the center. Choice B subtracts the lengths. Choice C adds the squares instead of subtracting them.
Question 11 of 20 · Multiple Choice
A line passes through point T on circle O and makes an 85° angle with radius OT. Which statement is true?
Answer: B
A line through T is tangent exactly when it is perpendicular to radius OT. Any other line through T gets closer to O than the radius on one side, so it meets the circle again. Choice A treats "close to 90°" as good enough, but the relationship is exact.
Question 12 of 20 · Multiple Choice
Points A, B and C lie on a circle, and angle ACB = 90°. What can you conclude about chord AB?
Answer: A
A 90° inscribed angle intercepts an arc of 180°, a semicircle, so A and B are the ends of a diameter. Choice D is impossible because C is on the circle, not at its center. Choice C is impossible because AB is a chord.
Question 13 of 20 · Multiple Choice
In a circle of radius 6, chord AB subtends a central angle AOB of 60°. How long is chord AB?
Answer: C
OA = OB = 6 because they are radii, so triangle AOB is isosceles with a 60° vertex angle. Its base angles are (180° - 60°)/2 = 60°, so it is equilateral and AB = 6. Choice D is the diameter, and choice A is half a radius.
Question 14 of 20 · Multiple Choice
Tangents from P touch a circle of radius 5 at A and B, and angle APB = 90°. What is OP?
Answer: D
Angles OAP and OBP are 90°, and angle AOB = 180° - 90° = 90°, so OAPB is a rectangle with OA = OB = 5, which makes it a square of side 5. OP is its diagonal: 5√2. Choice B doubles the radius. Choice A confuses OP with a side.
Question 15 of 20 · Short Answer
An inscribed angle measures (x + 12)° and the central angle that intercepts the same arc measures (3x - 4)°. Find x and both angles.
The central angle is twice the inscribed angle: 3x - 4 = 2(x + 12), so 3x - 4 = 2x + 24 and x = 28. The inscribed angle is 40° and the central angle is 80°.
Question 16 of 20 · Short Answer
A tangent segment PT is drawn to a circle with center O and radius 9. If OP = 41, find PT and explain where the right angle is.
The right angle is at T, the point of tangency, because the radius OT is perpendicular to the tangent. Then PT = √(41² - 9²) = √(1681 - 81) = √1600 = 40.
Question 17 of 20 · Short Answer
Explain why every angle inscribed in a semicircle is a right angle, using the relationship between inscribed angles and their arcs.
The endpoints of a diameter split the circle into two 180° arcs. An inscribed angle whose sides pass through those endpoints intercepts one of these semicircles, and an inscribed angle is half its intercepted arc, so it measures 180°/2 = 90°, wherever its vertex is on the circle.
Question 18 of 20 · Short Answer
Tangent segments PA and PB touch circle O at A and B. The radius is 15 and PA = 20. Find OP and the perimeter of quadrilateral OAPB.
Angle OAP = 90°, so OP = √(15² + 20²) = 25. Triangles OAP and OBP are right triangles with the same hypotenuse OP and equal legs OA = OB, so PB = PA = 20. The perimeter is 15 + 20 + 20 + 15 = 70.
Question 19 of 20 · Short Answer
A chord is 21 units from the center of a circle with radius 35. How long is the chord?
The perpendicular from the center bisects the chord. Half the chord is √(35² - 21²) = √784 = 28, so the chord is 56 units long.
Question 20 of 20 · Short Answer
A student says: "The inscribed angle is 67°, so the intercepted arc is 33.5°." Find the error and give the correct arc measure.
The student halved the angle, but the inscribed angle is half of the arc, so the arc is twice the angle: 2 · 67° = 134°. A central angle on the same arc would also be 134°.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.C.A.2 mean?
HSG.C.A.2 asks students to identify and describe how inscribed angles, radii and chords of a circle are related. The official text names three relationships to include: central, inscribed and circumscribed angles; right angles inscribed on a diameter; and the radius perpendicular to a tangent.
Is HSG.C.A.2 taught in Geometry or Algebra 2?
It is usually taught in high school Geometry, in the circles unit. It uses the Pythagorean Theorem and triangle angle facts from earlier grades and prepares students for inscribed quadrilaterals (HSG.C.A.3) and tangent constructions (HSG.C.A.4).
What is the difference between a central angle and an inscribed angle?
A central angle has its vertex at the center of the circle, and an inscribed angle has its vertex on the circle. For the same intercepted arc, the central angle equals the arc and the inscribed angle is half of it.
What is a circumscribed angle?
A circumscribed angle is formed by two tangents drawn from a point outside the circle. It is supplementary to the central angle of the arc between the points of tangency, because the quadrilateral formed with the two radii has two right angles: circumscribed angle = 180° - central angle.
Why is an angle inscribed in a semicircle always a right angle?
Because it intercepts an arc of 180°, and an inscribed angle is half its arc. A proof without the theorem also works: the radius to the vertex splits the triangle into two isosceles triangles, and their base angles add up to half of 180°.
Why is a tangent line perpendicular to the radius?
A tangent touches the circle at exactly one point T, so every other point of the line is outside the circle and farther from the center than T. The shortest segment from the center to a line is the perpendicular one, so OT is perpendicular to the tangent.
How do you find the length of a chord from the radius?
Draw the perpendicular from the center to the chord. It bisects the chord, so half the chord, the distance from the center and the radius form a right triangle. For radius 10 and distance 6, half the chord is 8 and the chord is 16.
What mistakes do students make with inscribed angles?
Common errors include:
doubling the arc instead of halving it for an inscribed angle
using the wrong arc (the one the angle does not intercept)
assuming a line is tangent because it looks tangent, without a right angle
putting the right angle at the outside point instead of at the point of tangency
Do inscribed angles that intercept the same arc have to be equal?
Yes. Each one is half of the same arc, so they are congruent, no matter where their vertices sit on the other arc. This is what the warm-up measurements in this lesson show.
How does HSG.C.A.2 connect to other standards?
It is the base for HSG.C.A.3, where the inscribed angle theorem proves that opposite angles of an inscribed quadrilateral are supplementary, and for HSG.C.A.4, where the right angle between a radius and a tangent is used to construct tangent lines. The chord and tangent lengths also give practice with the Pythagorean Theorem (8.G.B.7).
07
Related Standards
6 standards
These standards connect to HSG.C.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.G.A.5Prerequisite
Use informal arguments about angle sums and exterior angles of triangles
