Sure, let's break down the problem step-by-step.
1. Understanding Complementary Angles:
- Complementary angles are two angles that sum up to 90 degrees. So, let's denote the two complementary angles by [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Therefore, we have the equation:
[tex]\[ A + B = 90^\circ \][/tex]
2. Angles in a Quadrilateral:
- The sum of all the angles in a quadrilateral is 360 degrees. Hence, if the quadrilateral has four angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], [tex]\( C \)[/tex], and [tex]\( D \)[/tex], then:
[tex]\[ A + B + C + D = 360^\circ \][/tex]
3. Angles in the Ratio 4:5:
- According to the problem, the other two angles in the quadrilateral are in the ratio 4:5. Let these angles be represented as [tex]\( 4x \)[/tex] and [tex]\( 5x \)[/tex], respectively.
4. Forming the Equation:
- Substitute the expressions for the complementary angles and the ratio-based angles into the sum of angles for a quadrilateral:
[tex]\[ A + B + 4x + 5x = 360^\circ \][/tex]
5. Using the Complementary Angle Relationship:
- Since we know [tex]\( A + B = 90^\circ \)[/tex], substitute this into the quadrilateral angle sum equation:
[tex]\[ 90^\circ + 4x + 5x = 360^\circ \][/tex]
[tex]\[ 90^\circ + 9x = 360^\circ \][/tex]
6. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], subtract 90 from 360:
[tex]\[ 9x = 270^\circ \][/tex]
- Now, divide both sides by 9:
[tex]\[ x = 30^\circ \][/tex]
7. Calculate the Measures:
- Now that we have [tex]\( x \)[/tex], we can find the two angles in the ratio 4:5:
- The first angle is [tex]\( 4x \)[/tex]:
[tex]\[ 4 \times 30^\circ = 120^\circ \][/tex]
- The second angle is [tex]\( 5x \)[/tex]:
[tex]\[ 5 \times 30^\circ = 150^\circ \][/tex]
Thus, the measures of the two angles in the quadrilateral, given in the ratio of 4:5, are [tex]\( 120^\circ \)[/tex] and [tex]\( 150^\circ \)[/tex].
