Math Problem Statement

3) In the given figure, O is the center of the circle and CD = DE = EF = FG. If ∠COE = 80°, then ∠DOG is equal to: 4) In figure, BC is a diameter of the circle and ∠BOA = 70°. Then, ∠ADC is equal to:

Solution

Let's analyze each question in turn.

Question 3

In this problem:

  • OO is the center of the circle.
  • The segments CD=DE=EF=FGCD = DE = EF = FG.
  • We are given COE=80\angle COE = 80^\circ, and we need to find DOG\angle DOG.

Since CD=DE=EF=FGCD = DE = EF = FG, each of these segments subtends an equal angle at the center. This divides the entire angle around the center (360 degrees) into 4 equal parts.

  1. Calculate Each Angle Subtended:
    Since COECOE, EOGEOG, DOGDOG, and CODCOD are equal, Each angle=3604=90\text{Each angle} = \frac{360^\circ}{4} = 90^\circ

  2. Find DOG\angle DOG:
    Since COE=80\angle COE = 80^\circ is less than 90 degrees, it implies that the question is to evaluate the sum of angles involved in the

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Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Angles at the Center of a Circle
Inscribed Angles

Formulas

Angle subtended by equal arcs at the center
Relationship between inscribed and central angles

Theorems

Inscribed Angle Theorem
Central Angle Theorem

Suitable Grade Level

Grades 9-10