Solve for the angles and indicate the angle.

[tex]\begin{array}{l}
\text { 4- solve, the angles } \
\text { and indicate the angle } \
\underbrace{3 x}_{7 x+6} \frac{4 x^0 / 5 x-18}{5 x+22^0}\end{array}[/tex]

The angles of the quadrilateral are approximately 62.82°, 83.76°, 86.71°, and 126.71°. This is determined by using the equation representing the sum of the interior angles, solving for x, and substituting back to find each angle. Verification shows the sum equals 360°, confirming the calculations are correct.
;

Answered by Anonymous | 2025-07-04

Set up an equation using the fact that the sum of interior angles in a quadrilateral is 36 0 ∘ : 3 x + 4 x + ( 5 x − 18 ) + ( 5 x + 22 ) = 360 .
Solve the equation for x : x = 17 356 ​ ≈ 20.94 .
Substitute the value of x into the expressions for each angle to find their values.
The angles are approximately 62.8 2 ∘ , 83.7 6 ∘ , 86.7 1 ∘ , and 126.7 1 ∘ . 62.8 2 ∘ , 83.7 6 ∘ , 86.7 1 ∘ , 126.7 1 ∘ ​

Explanation

Problem Analysis The figure shows a quadrilateral with angles labeled as 3 x , 4 x , 5 x − 18 , and 5 x + 22 . The sum of two of the angles, 3 x and 4 x , is given as 7 x + 6 . Our objective is to determine the values of all angles in the quadrilateral.

Set up the equation The sum of the interior angles of a quadrilateral is 360^\\\circ . Therefore, we can set up the equation: 3 x + 4 x + ( 5 x − 18 ) + ( 5 x + 22 ) = 360

Solve for x Combine the terms on the left side of the equation: 17 x + 4 = 360 Subtract 4 from both sides: 17 x = 356 Divide both sides by 17: x = f r a c 356 17 ≈ 20.94

Calculate the angles Now, substitute the value of x back into the expressions for each angle:


Angle 1: 3x = 3 \times \frac{356}{17} = \frac{1068}{17} \approx 62.82^\\circ
Angle 2: 4x = 4 \times \frac{356}{17} = \frac{1424}{17} \approx 83.76^\\circ
Angle 3: 5x - 18 = 5 \times \frac{356}{17} - 18 = \frac{1780}{17} - \frac{306}{17} = \frac{1474}{17} \approx 86.71^\\circ
Angle 4: 5x + 22 = 5 \times \frac{356}{17} + 22 = \frac{1780}{17} + \frac{374}{17} = \frac{2154}{17} \approx 126.71^\\circ

Verification Verify that the sum of the angles is 360^\\circ : 17 1068 ​ + 17 1424 ​ + 17 1474 ​ + 17 2154 ​ = 17 6120 ​ = 360 Also, we are given that 3 x + 4 x = 7 x + 6 . Let's check this: 7 x + 6 = 7 × 17 356 ​ + 6 = 17 2492 ​ + 17 102 ​ = 17 2594 ​ ≈ 152.59 And 3 x + 4 x = 17 1068 ​ + 17 1424 ​ = 17 2492 ​ ≈ 146.59 There seems to be an inconsistency in the problem statement, as 3 x + 4 x is not exactly equal to 7 x + 6 when x = 17 356 ​ . However, using x = 17 356 ​ , the angles are approximately 62.82^\\circ , 83.76^\\circ , 86.71^\\circ , and 126.71^\\circ .

Final Answer The angles of the quadrilateral are approximately 62.82^\\circ , 83.76^\\circ , 86.71^\\circ , and 126.71^\\circ .


Examples
Understanding angles in quadrilaterals is crucial in architecture and construction. For example, when designing a building, architects must ensure that the angles of the walls and corners are precise to maintain structural integrity and aesthetic appeal. Knowing that the sum of angles in a quadrilateral is 360 degrees helps in planning layouts and ensuring that all angles fit together correctly, preventing misalignments and structural weaknesses.

Answered by GinnyAnswer | 2025-07-04