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Chapter 1: Problem 40

What is the supplement of the supplement of an angle measuring \(160^{\circ} ?\)

Short Answer

Expert verified
The supplement of the supplement of an angle measuring \(160^{\circ}\) is \(160^{\circ}\).

Step by step solution

01

Understand the concept of supplementary angles

Two angles are supplementary if the sum of their measures is \(180^{\circ}\). For an angle \(A\), its supplement is \(180^{\circ} - A\).
02

Calculate the supplement of the given angle

Given angle is \(160^{\circ}\). The supplement of \(160^{\circ}\) is \(180^{\circ} - 160^{\circ} = 20^{\circ}\).
03

Calculate the supplement of the new angle

The supplement of \(20^{\circ}\) is \(180^{\circ} - 20^{\circ} = 160^{\circ}\).

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Key Concepts

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

angle measurement
In geometry, an angle is formed by two rays (or lines) that share a common endpoint, called the vertex. The measure of an angle is determined by the amount of rotation from one ray to the other around their common vertex. Angles are typically measured in degrees (掳). A full rotation around a point is 360掳. A right angle is exactly 90掳, and a straight angle is exactly 180掳. Understanding angle measurement is fundamental for solving problems related to angles.
Using tools such as protractors can help physically measure an angle, while calculations assist in theoretical problems.
supplement of an angle
When two angles add up to 180掳, they are called supplementary angles. An easy way to find the supplement of an angle is by subtracting it from 180掳. For example, if you have an angle measuring 70掳, its supplement would be calculated as follows:
\[180^{\text{掳}} - 70^{\text{掳}} = 110^{\text{掳}}\].
This concept is useful in various geometric problems, including finding unknown angles in polygons or determining interior and exterior angles. In our exercise, the supplement of an angle measuring 160掳 was found using:
\[180^{\text{掳}} - 160^{\text{掳}} = 20^{\text{掳}}\].
geometry calculations
Geometry involves various calculations to solve problems related to shapes and angles. For example, when calculating the supplement of an angle, you use basic arithmetic operations.
In our exercise, we tackled the supplement of the supplement of an angle.
We had an initial angle of 160掳. First, we found its supplement:
\[180^{\text{掳}} - 160^{\text{掳}} = 20^{\text{掳}}\].
Then, we found the supplement of the new angle 20掳:
\[180^{\text{掳}} - 20^{\text{掳}} = 160^{\text{掳}}\].
This involves simple subtraction, which is a fundamental calculations used in many aspects of geometry. Understanding such calculations is crucial for more complex geometric problem-solving.

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Most popular questions from this chapter

In Exercises 1 and \(2,\) supply reasons for the statements in each proof. Prove Theorem 1.4. Giver: \(D\) is interior to \(\angle A B C\) \(S\) is interior to \(\angle P Q R\) \(m \angle A B C=m \angle P Q R\) \(m \angle D B C=m \angle S Q R\) Prove: \(m \angle A B D=m \angle P Q S\) (IMAGE CAN'T COPY) $$\begin{array}{ll} \text { Statements } & \text { Reasons } \\ \text { 1. } D \text { is interior to } \angle A B C & 1 \text { . } \\ \text { 2. } S \text { is interior to } \angle P Q R & \text { 2. } \\ \text { 3. } m \angle A B C= \text { }m \angle P Q R \text { and }\\\ m \angle D B C= m \angle S Q R & { 3. } \\ \text { 4. } m \angle A B C-m \angle D B C= \\ m \angle P Q R-m \angle S Q R & { 4. } \\ \text { 5. } m \angle A B D=m \angle A B C-m \angle D B C\\\ \text { and } m \angle P Q S=m \angle P Q R-m \angle S Q R & { 5. } \\ \text { 6. }m \angle A B D=m \angle P Q S & \text { 6. } \\ \end{array}$$

For problems 13-26, explain the reasoning in one or two complete sentences. 19\. Can two vertical angles both be obtuse?

State the hypothesis and conclusion for each statement. If a triangle is isosceles, then the triangle has two congruent sides.

In the Declaration of Independence, the statement is made that "All men are created equal." Is this statement a postulate or a theorem?

Find the complement of each angle in Exercises \(27-30\). $$36^{\circ} 40^{\prime}$$ (GRAPH CANT COPY)
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