Math Problem Statement

∠xと∠yを求めてください

Solution

To solve for angles xx and yy in the given diagram, we will use the properties of circles and angles subtended by arcs.

Step 1: Understanding the diagram

  • 55\angle 55^\circ is an interior angle within the circle.
  • 70\angle 70^\circ is the exterior angle at point AA, formed between the tangent ATAT and the chord ABAB.
  • xx and yy are the angles that need to be determined.

Step 2: Finding angle xx

Angle xx is an angle at the circumference subtended by the same arc as the angle 5555^\circ. According to the circle theorem, angles subtended by the same arc on the circumference are equal.

Therefore, x=55x = 55^\circ

Step 3: Finding angle yy

Angle yy is formed by the tangent ATAT and the chord ABAB. According to the alternate segment theorem, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Therefore, angle yy is equal to angle xx:

y=55y = 55^\circ

However, considering that 70\angle 70^\circ is an exterior angle formed by the tangent and the chord ABAB, yy can also be found as:

y=18070=110y = 180^\circ - 70^\circ = 110^\circ

Thus, angle y=110y = 110^\circ.

Final Answers:

  • x=55x = 55^\circ
  • y=110y = 110^\circ

Would you like further details or have any other questions?


Here are 8 follow-up questions related to this problem:

  1. What is the circle theorem that explains why x=55x = 55^\circ?
  2. How do we apply the alternate segment theorem in this scenario?
  3. What is the sum of angles in any triangle, and how does it relate to this problem?
  4. Can you explain why the exterior angle yy is calculated as 18070180^\circ - 70^\circ?
  5. How would the solution change if the angle at AA was different?
  6. What are other examples where the alternate segment theorem is applicable?
  7. Can you determine the angle subtended at the center by the same chord?
  8. How would the problem change if the tangent was on the opposite side of AA?

Tip:

In geometry, always consider all the theorems that might apply to a situation. Sometimes, different theorems can give the same answer, reinforcing your solution.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Theorems
Alternate Segment Theorem

Formulas

-

Theorems

Circle theorem
Alternate segment theorem

Suitable Grade Level

High School