Question
Write four mathematical equations for two complementary and two supplementary angles. make your answers long
Answers
Answered by
GPT-4o mini
Sure! Let's start by defining complementary and supplementary angles:
- **Complementary Angles**: Two angles are complementary if the sum of their measures is \(90^\circ\).
- **Supplementary Angles**: Two angles are supplementary if the sum of their measures is \(180^\circ\).
### Complementary Angles
1. **Equation 1**:
Let angle \(A\) be defined as \(A = x\), and angle \(B\) be defined as \(B = 90^\circ - x\), where \(x\) is a variable representing a certain degree measure. The equation representing their relationship will be:
\[
A + B = 90^\circ \implies x + (90^\circ - x) = 90^\circ
\]
Here, the equation holds true for any value of \(x\) such that \(0 < x < 90\).
2. **Equation 2**:
Suppose angle \(C\) is defined as \(C = 2y\) and angle \(D\) is defined as \(D = 90^\circ - 2y\). The complementary relationship can thus be expressed as:
\[
C + D = 90^\circ \implies 2y + (90^\circ - 2y) = 90^\circ
\]
This equation illustrates how angle \(C\) is twice the measure of angle \(D\) while both angles still sum to \(90^\circ\).
### Supplementary Angles
3. **Equation 3**:
Define angle \(E\) as \(E = 3z\) and angle \(F\) as \(F = 180^\circ - 3z\). The supplementary nature of these angles can be described by the equation:
\[
E + F = 180^\circ \implies 3z + (180^\circ - 3z) = 180^\circ
\]
In this scenario, angle \(E\) is three times angle \(F\), maintaining the relationship that their sum equals \(180^\circ\).
4. **Equation 4**:
Let angle \(G\) be given by \(G = 45^\circ + m\) and angle \(H\) by \(H = 135^\circ - m\), where \(m\) is a variable. The supplementary relationship can be expressed in the following manner:
\[
G + H = 180^\circ \implies (45^\circ + m) + (135^\circ - m) = 180^\circ
\]
This equation emphasizes the balance achieved through \(m\), as the sum of angles \(G\) and \(H\) indeed returns to \(180^\circ\).
These equations effectively illustrate the relationships between complementary and supplementary angles, providing mathematical expressions for the concepts.
- **Complementary Angles**: Two angles are complementary if the sum of their measures is \(90^\circ\).
- **Supplementary Angles**: Two angles are supplementary if the sum of their measures is \(180^\circ\).
### Complementary Angles
1. **Equation 1**:
Let angle \(A\) be defined as \(A = x\), and angle \(B\) be defined as \(B = 90^\circ - x\), where \(x\) is a variable representing a certain degree measure. The equation representing their relationship will be:
\[
A + B = 90^\circ \implies x + (90^\circ - x) = 90^\circ
\]
Here, the equation holds true for any value of \(x\) such that \(0 < x < 90\).
2. **Equation 2**:
Suppose angle \(C\) is defined as \(C = 2y\) and angle \(D\) is defined as \(D = 90^\circ - 2y\). The complementary relationship can thus be expressed as:
\[
C + D = 90^\circ \implies 2y + (90^\circ - 2y) = 90^\circ
\]
This equation illustrates how angle \(C\) is twice the measure of angle \(D\) while both angles still sum to \(90^\circ\).
### Supplementary Angles
3. **Equation 3**:
Define angle \(E\) as \(E = 3z\) and angle \(F\) as \(F = 180^\circ - 3z\). The supplementary nature of these angles can be described by the equation:
\[
E + F = 180^\circ \implies 3z + (180^\circ - 3z) = 180^\circ
\]
In this scenario, angle \(E\) is three times angle \(F\), maintaining the relationship that their sum equals \(180^\circ\).
4. **Equation 4**:
Let angle \(G\) be given by \(G = 45^\circ + m\) and angle \(H\) by \(H = 135^\circ - m\), where \(m\) is a variable. The supplementary relationship can be expressed in the following manner:
\[
G + H = 180^\circ \implies (45^\circ + m) + (135^\circ - m) = 180^\circ
\]
This equation emphasizes the balance achieved through \(m\), as the sum of angles \(G\) and \(H\) indeed returns to \(180^\circ\).
These equations effectively illustrate the relationships between complementary and supplementary angles, providing mathematical expressions for the concepts.