Question

Two angles of a quadrilateral measure 300° and 28°. The other two angles are in a ratio of 7:9. What are the measures of those two angles?

° and
°

Answer 1

To find the measures of the two unknown angles in the quadrilateral, we first need to use the fact that the sum of the angles in any quadrilateral is \(360^\circ\).

Given:
- The first two angles measure \(300^\circ\) and \(28^\circ\).
- Let the two unknown angles be \(x\) and \(y\), with the given ratio \(x:y = 7:9\).

1. First, calculate the sum of the known angles:
\[
300^\circ + 28^\circ = 328^\circ
\]

2. Calculate the sum of the unknown angles:
\[
x + y = 360^\circ - 328^\circ = 32^\circ
\]

3. Since \(x\) and \(y\) are in the ratio \(7:9\), we can express them in terms of a variable \(k\):
\[
x = 7k \quad \text{and} \quad y = 9k
\]

4. Substitute these expressions into the equation for the sum of the angles:
\[
7k + 9k = 32^\circ
\]
\[
16k = 32^\circ
\]
\[
k = 2^\circ
\]

5. Now, substitute \(k = 2^\circ\) back into the expressions for \(x\) and \(y\):
\[
x = 7k = 7 \times 2^\circ = 14^\circ
\]
\[
y = 9k = 9 \times 2^\circ = 18^\circ
\]

Thus, the measures of the two unknown angles are:
\[
\text{First angle: } 14^\circ
\]
\[
\text{Second angle: } 18^\circ
\]

Final answer:
**14° and 18°**