Download TOPIC: INSCRIBED ANGLES (Sect. 9

TOPIC: INSCRIBED ANGLES (Sect. 9-4)
Rachel Whiteside
OBJECTIVES:
Students will recognize and find measures of inscribed angles given measures of
intercepted arcs.
Students will apply the result of Theorem 9-5 and use it to solve problems.
NCTM EXPECTATIONS ADDRESSED:
Establish the validity of geometric conjectures using deduction, proving theorems, and
critiquing arguments made by others.
Analyze properties and determine attributes of circles.
MATERIALS/EQUIPMENT: Guided investigation sheets, patty paper, and proof with blanks on
the board before class starts.
INSTRUCTIONAL ACTIVITIES:
1. Start-up/Motivation: Students will have the partially completed proof below of
Theorem 9-1 on the board they will be asked to complete in their notes as they arrive in
the classroom. Highlighted areas will be left blank on the board.
A
B
E
D
Claim: If two minor arcs of a circle are congruent, then their
corresponding chords are congruent.
̂ ≅ 𝐷𝐶
̂.
Given: In the circle centered at E, 𝐴𝐵
Prove: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐶 .
C
Statements:
Reasons:
̂ ≅ 𝐷𝐶
̂
1. Given
1. 𝐴𝐵
̅̅̅̅
2. All radii of a circle are congruent.
2. ̅̅̅̅
𝐴𝐸 ≅ 𝐶𝐸
̅̅̅̅
̅̅̅̅
𝐵𝐸 ≅ 𝐷𝐸
3. ∠𝐴𝐸𝐵 ≅ ∠𝐶𝐸𝐷
3. Vertical angles are congruent.
4. ∆𝐴𝐸𝐵 ≅ ∆𝐶𝐸𝐷
4. SAS
̅̅̅̅ ≅ 𝐷𝐶
̅̅̅̅
5. CPCTC
5. 𝐴𝐵
Go over completed proof and discuss any problems.
(Ex: Students might only list one set of congruent radii on #2
Students might list the incorrect congruence theorem for #4 – if so trace over the
example so it is obvious SAS is appropriate
Students might just describe #5 instead of using CPCTC – encourage correct
mathematical language)
2. Instruction: Introduce terms inscribed angle and intercepted arc.
 Inscribed Angle – an angle having its vertex lie on a given circle and containing
two chords of the circle.
 Intercepted Arc – An arc having the following properties:
i. The endpoints of the arc lie on the angle.
ii. All points of the arc, except the endpoints, are in the interior of the circle.
iii. Each side of the angle contains an endpoint of the arc.
Given the two definitions, ask two students to draw a circle with an inscribed angle and
ask another student to identify the intercepted arc for each. The intercepted arcs will
be the arcs created by each of the inscribed angles. If students are struggling with
drawing the new terms, walk them through the two definitions so that it is more clear
what they are asking. Emphasize the vertex lying on the outside of the circle and the
two chords forming an angle.
3. Patty Paper Activity: Hand students each the guided investigation worksheet for the
activity with 2 pieces of patty paper for each student. Instructions for activity are on the
worksheet. Observe students as they work through the instructions and especially make
sure they are using the same arc as instructed in step #2. Notice student’s answers to
the final question on the worksheet, make sure that they use the correct term
(congruent). Have students compare their conjecture to Theorem 9-5 – If two
inscribed angles of a circle or congruent circles intercept congruent arcs or the same
arc, then the angles are congruent.
4. Class Instruction: Tell students that when an angle is inscribed in a circle, the measure
of the angle equals ½ the measure of the intercepted arc. There is a proof to this, but
we aren’t going to discuss it in class, if anyone would like to see it, see pages 466 – 467.
Ask if any students know how to locate the center of a circle given only three points on
the circle. Notes for this are on the reverse of the guided investigation for the patty
paper note sheet. Go through this activity on the overhead as students follow along at
their desks. Tell them their segments do not have to be exactly perpendicular, but that
they should get as close as possible. Students can follow along on page 468 for written
instruction if they would like. Relate this to:
Theorem 9-6 – If any inscribed angle of a circle intercepts a semicircle, then the angle
is a right angle.
Point out how this is true in the circle just used with the three points and found center.
5. Summary: Tell students to look over Example 3 on page 469 and tell them there will be
a graded fill in the blank review of this at the beginning of the next class similar to the
one given today. Have students write an exit slip over three things they learned today
and rating the usefulness of the patty paper exercise. Look for answers including either
of the theorems or the terms inscribed angles/arcs.
ASSESSMENT: Exit slips will be collected as students leave the classroom and used to evaluate
whether the students learned the material and how well they understood the concepts. The
opening responses to the proof on the board will be used to evaluate students’ abilities to
prove things in geometry as well as their understanding of the previous day’s material.
VOCABULARY: inscribed angle, intercepted arc.
HOMEWORK: Read Example 3 on page 469. Assign exercises starting on page 470, 17-27
(Should be time for this to be completed in class.) Remind students to study for quiz next class.