If `angleA` and `angleB` are supplementary angles. If `4angleA=5angleB`, then find `angleA` and `angleB`.

Answer

To solve the problem step by step, we will follow the properties of supplementary angles and the given relationship between angle A and angle B.

### Step 1: Understand the property of supplementary angles
Supplementary angles are two angles whose sum is equal to 180 degrees. Therefore, we can write:
\[ \text{angle A} + \text{angle B} = 180^\circ \]

### Step 2: Set up the equation from the given relationship
We are also given that:
\[ 4 \cdot \text{angle A} = 5 \cdot \text{angle B} \]
From this, we can express angle A in terms of angle B:
\[ \text{angle A} = \frac{5}{4} \cdot \text{angle B} \]

### Step 3: Substitute angle A in the supplementary angle equation
Now, we will substitute the expression for angle A into the supplementary angle equation:
\[ \frac{5}{4} \cdot \text{angle B} + \text{angle B} = 180^\circ \]

### Step 4: Combine like terms
To combine the terms on the left side, we can express angle B as a fraction:
\[ \frac{5}{4} \cdot \text{angle B} + \frac{4}{4} \cdot \text{angle B} = 180^\circ \]
This simplifies to:
\[ \frac{9}{4} \cdot \text{angle B} = 180^\circ \]

### Step 5: Solve for angle B
To isolate angle B, we multiply both sides by the reciprocal of \(\frac{9}{4}\):
\[ \text{angle B} = 180^\circ \cdot \frac{4}{9} \]
Calculating this gives:
\[ \text{angle B} = 80^\circ \]

### Step 6: Find angle A
Now that we have angle B, we can find angle A using the supplementary angle property:
\[ \text{angle A} = 180^\circ - \text{angle B} \]
Substituting the value of angle B:
\[ \text{angle A} = 180^\circ - 80^\circ = 100^\circ \]

### Final Answer
Thus, the measures of the angles are:
- \( \text{angle A} = 100^\circ \)
- \( \text{angle B} = 80^\circ \)

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