Code
G-C.1
G-C.2
G-C.3
G-C.5
G-CO.1
G-SRT.2
G-SRT.3
G-SRT.4
G-SRT.5
Common Core/Essential Standards
Prove that all circles are similar.
Identify and describe relationships among inscribed angles, radii, and chords. Include he 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.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a
quadrilateral inscribed in a circle.
Derive using similarity the fact that length of the arc intercepted by an angle is proportional to the radius,
and define the radian measure of the angle as the constant of proportionality; derive the formula for the area
of a sector.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line distance along a line, and distance around a circular arc
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are
similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other
two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric
figures
Students will be able to…

Know precise definitions of circle, and distance around a circular arc (arc length)

Given two figures, use the definition of similarity (in terms of similarity transformations) to decide if they are similar

Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs
of angles and the proportionality of all corresponding pairs of sides

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures

Prove that all circles are similar

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.)

Construct the inscribed and circumscribed circles of a triangle.

Prove properties of angles for a quadrilateral inscribed in a circle (An angle inscribed in a semicircle = 90 degrees.
Opposite angles of an inscribed quadrilateral are supplementary)

Derive the formula for the area of a sector.
1.
Given:
Circle A has a radius of 5 units.
Line BC is Tangent to Circle A at point B
Prove Radius AB is perpendicular to Tangent BC.
1.
4
The slope of Tangent BC is − .
a.
2.
𝑦2 −𝑦1
𝑥2 −𝑥1
=
3−(−1)
4−7
3
The slope of Radius AB is
a.
3.
4.
𝑚=
𝑚=
𝑦2 −𝑦1
𝑥2 −𝑥1
=
0−3
0−4
4
=
3
=
−3
−4
4
−3
=
=−
4
3
3
4
Slopes of perpendicular lines are opposite reciprocals of one another.
3
The opposite reciprocal of Tangent BC is
3
4
5. The slope of Radius AB is
4
Therefore Radius AB is perpendicular to Tangent BC.
Theory: If a radius intersects a line tangent to a circle, at the point of tangency, then
that intersection is perpendicular.
2.
Given:
Line CD is Tangent to Circle A at point C
Line BD is Tangent to Circle A at point B
Line CD intersects BD at point D
Prove:
Segment CD is the same length as segment BD.
1.
2.
3.
4.
5.
6.
7.
Radius AC is perpendicular to Line CD
Radius AB is perpendicular to Line BD
a. Theorem.
Radius AC is congruent to Radius AB
Draw segment AD
Segment AD is congruent to segment AD
Triangle ACD is congruent to Triangle ABD
a. Hypotenuse-Leg Congruency
Segment CD is congruent to segment BD
a. Congruent Parts of Congruent Triangles are Congruent
Theory: Lines tangent to a circle, that intersect one another at a point, are
equidistant from their point of tangency.
Ex.
1) Find the Length of AD
2) What is the measure of angle ADC
3) Show that AD bisects angle CDB
Ex.
1) What is the measure of Angle ACB?
2) Show how your answer makes triangle ABC an equilateral triangle
3) If Circle A has a radius of 7. What can you expect is the length of BC?
6.1.2 Chords, Arcs, and Central Angles
Definitions:
Chord: a line segment that joins two points of the circle.
Central Angle: an angle of measure less than 180° with vertex at the cener of the
cirlce and sides along radii of the circle
Minor Arc: the arc that lies in the interior of the central angle (Use two points)
Major Arc: the arc that lies in the exterior of the central angle (Use three points)
Points of Notice: The degree measure of a minor arc is equal to the measure of the
corresponding central angle. The degree measure of a major arc is 360 minus the
measure of the corresponding minor arc
1) What is the longest chord in a circle?
2) Suppose that in the diagram above 𝑚∠𝐵𝐴𝐶 = 160°
̂?
a. What is the 𝑚𝐵𝐶
̂
b. What is 𝑚𝐵𝐷𝐶
Given: AB = CD.
̂ = 𝑚𝐷𝐸
̂
Prove: that 𝑚∠𝐵𝐴𝐶 = 𝑚∠𝐷𝐴𝐸 and 𝑚𝐵𝐶
1) AB = AD = AC = AE
a. All radii of a circle are equal
2) ∆𝐵𝐴𝐶 ≅ ∆𝐷𝐴𝐸
a. 𝑆𝑆𝑆 ≅
3) 𝑚∠𝐵𝐴𝐶 = 𝑚∠𝐷𝐴𝐸
a. Congruent Parts of Congruent Triangles are Congruent
̂ = 𝑚𝐷𝐸
̂
4) 𝑚𝐵𝐶
a. Definition of arc measure
Have students prove that a radius can be a perpendicular bisector to a chord.