Question

∠a and ∠b are complementary angles. find m∠a and m∠b.

1. m∠a = x°, m∠b=(2x - 75)°
2. m∠a=(4x + 34)°, m∠b=(x + 36)°
3. m∠a=(4x - 18)°, m∠b=(6x - 18)°
4. m∠a=(2x + 10)°, m∠b=(-x + 55)°

Explanation:

Step1: Recall complementary - angle property

Complementary angles add up to 90°. So, for each pair, we set up the equation \(m\angle A + m\angle B=90^{\circ}\).

Step2: Solve for \(x\) in problem 25

Given \(m\angle A = x^{\circ}\) and \(m\angle B=(2x - 75)^{\circ}\), we have the equation \(x+(2x - 75)=90\).
Combining like - terms: \(3x-75 = 90\).
Adding 75 to both sides: \(3x=90 + 75=165\).
Dividing both sides by 3: \(x = 55\).
Then \(m\angle A=x = 55^{\circ}\) and \(m\angle B=2x-75=2\times55 - 75=110 - 75 = 35^{\circ}\).

Step3: Solve for \(x\) in problem 26

Given \(m\angle A=(4x + 34)^{\circ}\) and \(m\angle B=(x + 36)^{\circ}\), we set up the equation \((4x + 34)+(x + 36)=90\).
Combining like - terms: \(5x+70 = 90\).
Subtracting 70 from both sides: \(5x=90 - 70 = 20\).
Dividing both sides by 5: \(x = 4\).
Then \(m\angle A=4x + 34=4\times4+34=16 + 34 = 50^{\circ}\) and \(m\angle B=x + 36=4 + 36 = 40^{\circ}\).

Step4: Solve for \(x\) in problem 27

Given \(m\angle A=(4x - 18)^{\circ}\) and \(m\angle B=(6x - 18)^{\circ}\), we set up the equation \((4x - 18)+(6x - 18)=90\).
Combining like - terms: \(10x-36 = 90\).
Adding 36 to both sides: \(10x=90 + 36=126\).
Dividing both sides by 10: \(x = 12.6\).
Then \(m\angle A=4x - 18=4\times12.6-18=50.4 - 18 = 32.4^{\circ}\) and \(m\angle B=6x - 18=6\times12.6-18=75.6 - 18 = 57.6^{\circ}\).

Step5: Solve for \(x\) in problem 28

Given \(m\angle A=(2x + 10)^{\circ}\) and \(m\angle B=(-x + 55)^{\circ}\), we set up the equation \((2x + 10)+(-x + 55)=90\).
Combining like - terms: \(x+65 = 90\).
Subtracting 65 from both sides: \(x = 25\).
Then \(m\angle A=2x + 10=2\times25+10=50 + 10 = 60^{\circ}\) and \(m\angle B=-x + 55=-25 + 55 = 30^{\circ}\).

Answer:

1. \(m\angle A = 55^{\circ}\), \(m\angle B = 35^{\circ}\)
2. \(m\angle A = 50^{\circ}\), \(m\angle B = 40^{\circ}\)
3. \(m\angle A = 32.4^{\circ}\), \(m\angle B = 57.6^{\circ}\)
4. \(m\angle A = 60^{\circ}\), \(m\angle B = 30^{\circ}\)