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Chapter 5: Problem 21

Name the quadrant in which each angle lies. $$85^{\circ}$$

Short Answer

Expert verified
Quadrant I

Step by step solution

01

Understanding Angles and Quadrants

In a coordinate plane, angles are measured from the positive x-axis. The quadrants are as follows: - Quadrant I: 0掳 to 90掳 - Quadrant II: 90掳 to 180掳 - Quadrant III: 180掳 to 270掳 - Quadrant IV: 270掳 to 360掳.
02

Identify the Range

Identify the range in which the given angle falls. Angle is 85掳. Identify which quadrant includes 85掳.
03

Determine the Quadrant

Since the angle 85掳 falls between 0掳 and 90掳, it lies in the Quadrant I.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

angle measurement
To navigate through any exercise involving angles, understanding angle measurement is very important. An angle is formed by two rays (the sides of the angle) sharing a common endpoint (the vertex). In mathematics, angles are usually measured in degrees. Full rotation in a circle is 360 degrees, symbolized as $$360^{\circ}$$.

There are several key types of angles including:
  • Acute Angles: less than $$90^{\circ}$$
  • Right Angles: exactly $$90^{\circ}$$
  • Obtuse Angles: between $$90^{\circ}$$ and $$180^{\circ}$$
  • Straight Angles: exactly $$180^{\circ}$$

This helps in figuring out which quadrant an angle belongs to on the coordinate plane. In exercises like identifying quadrants, the knowledge of angle measurement is the first step.
coordinate plane
The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves. It's fundamental in understanding where angles lie.

The coordinate plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). The point where these axes intersect is called the origin, designated as (0,0).

These are the quadrants:
  • Quadrant I: Located where both x and y are positive (top-right section)
  • Quadrant II: Located where x is negative and y is positive (top-left section)
  • Quadrant III: Located where both x and y are negative (bottom-left section)
  • Quadrant IV: Located where x is positive and y is negative (bottom-right section)

This structure of the coordinate plane is critical for determining the position of angles and other shapes.
angular quadrants
On the coordinate plane, angular quadrants help in determining where an angle is located. When measuring angles from the positive x-axis in a counterclockwise direction, the plane's quadrants play a vital role.

Here is the breakdown of angular quadrants:
  • Quadrant I: Angles between $$0^{\circ}$$ and $$90^{\circ}$$
  • Quadrant II: Angles between $$90^{\circ}$$ and $$180^{\circ}$$
  • Quadrant III: Angles between $$180^{\circ}$$ and $$270^{\circ}$$
  • Quadrant IV: Angles between $$270^{\circ}$$ and $$360^{\circ}$$

For example, an angle of $$85^{\circ}$$ is between $$0^{\circ}$$ and $$90^{\circ}$$. Thus, it is in Quadrant I. This distinction helps in solving various geometric problems and understanding spatial relationships.

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