AXIOMS OF EQUALITY COMMON CORE GEOMETRY HOMEWORK REASONING A related axiom to The Whole is the Sum of Its Parts is the following: The Whole is Greater Than Any of Its Parts. Given an example of this using the angle diagram shown below.
The correct answer and explanation is:
The axiom “The Whole is Greater Than Any of Its Parts” refers to the idea that when something is divided into parts, the total or whole is always greater than any of its individual parts. This axiom can be illustrated using an angle diagram where a larger angle is divided into smaller angles.
Example:
Consider an angle ∠ABC\angle ABC, which is split into two smaller angles: ∠ABD\angle ABD and ∠DBC\angle DBC. According to the axiom “The Whole is Greater Than Any of Its Parts,” the measure of the whole angle ∠ABC\angle ABC is greater than either of the smaller angles ∠ABD\angle ABD or ∠DBC\angle DBC, i.e., ∠ABC>∠ABDand∠ABC>∠DBC.\angle ABC > \angle ABD \quad \text{and} \quad \angle ABC > \angle DBC.
Explanation:
In geometry, angles can be divided into smaller parts, and the axiom “The Whole is Greater Than Any of Its Parts” tells us that the total measure of an angle is always greater than any of the parts into which it is divided. This can be clearly observed in the angle diagram where an angle is broken into two or more smaller angles. For example, if ∠ABC=90∘\angle ABC = 90^\circ, and it is divided into two smaller angles, say ∠ABD=40∘\angle ABD = 40^\circ and ∠DBC=50∘\angle DBC = 50^\circ, the sum of the smaller angles is equal to the whole angle: ∠ABD+∠DBC=40∘+50∘=90∘.\angle ABD + \angle DBC = 40^\circ + 50^\circ = 90^\circ.
This verifies that the whole angle ∠ABC\angle ABC is equal to the sum of its parts. However, according to the axiom, ∠ABC\angle ABC is greater than either of the individual smaller angles ∠ABD\angle ABD or ∠DBC\angle DBC. This illustrates that the whole angle is always greater than any one of its parts in terms of size or measure.
Thus, when you divide something into smaller parts, the total remains greater than each individual part. This concept is fundamental in understanding how angles and other geometric objects relate to their components.