Module 15: Angles and Segments in Circles

A circle is the set of points in a plane at a given distance ( radius)from a given point (center).You name a circle by its center. The circle on the bicycle wheel withcenter O, is called circle O.Radius: segment from the center to a point on the edge of the circle.

Lessons 15.1 to 15.3

Part 1

Congruent Circles: Circles with the same radius.

Concentric Circles: Coplanar circles with the same center.

Circle Terms 15.1, page 779

Point on Tangency:The point where the tangent touches the circle. Example: point B, and point C.

.R

S

. .Secant:Is a line that intersects the circle twice: 𝑹𝑺

Central Angle: Angle whose vertex is at the center of the circle and the sides are radius of the circle.

15.1, page 779

Inscribed Angle: Angle whose vertex is on the circle and the sides are chords.

15.1, page 779

MAJOR ARC page 780

SEMICIRCLE Page 780

15.1, page 781

15.1, page 782

Theorem:Inscribed Angles intercepting the same arc are congruent.

Since both inscribed ⦟ABC and ⦟ADC intercept the same 𝑨𝑪,then m ⦟ABC = m ⦟ADC

15.1, page 785

Cyclic (inscribed) Quadrilateral:A quadrilateral inscribed in a circle.All vertices are on the circle.

⦟D + ⦟B = 180° and ⦟A + ⦟C = 180°

15.2, page 794

Page 782

15.3, page 806

15.3, page 808

Solution

90°

90°

140°40°

OR 180° - 140° = 40°

𝑨𝑵 ≅ 𝑮𝑵

Tangent Segments

16 cm

7 cm4 cm

By the Tangent Theorem, m ⦟T = 90° and m ⦟ P = 90°.⦟T + ⦟P = 180°, therefore they are supplementary.Since ⦟T and ⦟P are consecutive interior and supplementary, then by the converse of the parallel lines theorem 𝑻𝑵 ǁ 𝑷𝑴,therefore NMPT is a trapezoid .Since NMPT is a trapezoid, then

Area = 𝒉 (𝒃𝟏+𝒃𝟐)

𝟐=

𝟏𝟔 (𝟕+𝟒)

𝟐= 88 𝒄𝒎𝟐

Homework:Online Assignment15.1-15.2, & 15.3

Quiz: Equations of Circles (20 minutes)

Multiple choice

Answer each question from 1-8

Choose one from 9 & 10

Show your work