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Chapter 9: Problem 23

Find the measure of a complement of each angle. $$ 67^{\circ} $$

Short Answer

Expert verified
The measure of the complement is \( 23^\bullet \).

Step by step solution

01

Understand Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees.
02

Set Up the Equation

Let the measure of the unknown complementary angle be represented by \( x \). Using the definition of complementary angles, set up the equation: \( x + 67^{\bullet} = 90^{\bullet} \).
03

Solve for \( x \)

Rearrange the equation to solve for \( x \): \( x = 90^{\bullet} - 67^{\bullet} \).
04

Calculate \( x \)

Subtract 67 from 90: \( x = 23^{\bullet} \).

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Key Concepts

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

angle measurement
Understanding angle measurement is crucial for solving problems related to angles. Angles are measured in degrees, represented by the symbol \( ^{\circ} \). There are different types of angles you should be familiar with:
  • Complementary angles: Two angles whose measures add up to 90 degrees.
  • Supplementary angles: Two angles whose measures add up to 180 degrees.
  • Acute angles: Angles that are less than 90 degrees.
  • Right angles: Exactly 90 degrees.
  • Obtuse angles: More than 90 degrees but less than 180 degrees.
Knowing these types and their properties helps in solving geometric problems. In our example, we've dealt with complementary angles. If one angle is known, the other can be easily found by subtracting the known angle from 90 degrees.
geometry basics
Geometry is the branch of mathematics that deals with shapes, sizes, and properties of figures. Here are some foundational concepts:
  • Points: An exact position in space, usually represented by a dot.
  • Lines: Extend infinitely in both directions, with no endpoints.
  • Line segments: Part of a line that has two endpoints.
  • Angles: Formed by two rays (or line segments) that share a common endpoint (vertex).
  • Planes: Flat, two-dimensional surfaces that extend infinitely.
Understanding these basics helps when working with problems related to angles, shapes, and theorems. For instance, when finding the complement of a 67-degree angle, you're using the properties of complementary angles and line segments.
equation solving
Solving equations is a fundamental skill in mathematics. To find the measure of a complementary angle, follow these steps:
  • Identify the type of problem and relevant properties (complementary angles add up to 90 degrees).
  • Set up the equation based on the property: \( x + 67^{\circ} = 90^{\circ} \).
  • Isolate the variable (\( x \)) by performing arithmetic operations: Subtract 67 from both sides.
  • Calculate the result: \( x = 23^{\circ} \).
Properly setting up and solving equations allows you to find unknown measurements efficiently. Always ensure to check your work for accuracy by plugging the solution back into the original equation wherever applicable.

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