Question

b 24. $mangle a = 2x$, $mangle b=x - 15$ if $angle c$ and $angle d$ are complementary, find the value of $y$, $mangle c$, and $mangle d$. 26. $mangle c = 3y + 5$, $mangle d=2y$ 27. $mangle c=y - 8$, $mangle d = 3y+2$

Explanation:

Step1: Recall the property of complementary angles

Complementary angles add up to 90 degrees. So, \(m\angle C + m\angle D=90\).

Step2: Set up the equation for problem 26

Given \(m\angle C = 3y + 5\) and \(m\angle D=2y\), we have \((3y + 5)+2y=90\).
Combining like - terms: \(3y+2y+5 = 90\), which simplifies to \(5y+5 = 90\).
Subtract 5 from both sides: \(5y=90 - 5=85\).
Divide both sides by 5: \(y=\frac{85}{5}=17\).
Then \(m\angle C=3y + 5=3\times17+5=51 + 5=56\) degrees and \(m\angle D=2y=2\times17 = 34\) degrees.

Step3: Set up the equation for problem 27

Given \(m\angle C=y - 8\) and \(m\angle D=3y + 2\), we have \((y - 8)+(3y + 2)=90\).
Combining like - terms: \(y+3y-8 + 2=90\), which simplifies to \(4y-6 = 90\).
Add 6 to both sides: \(4y=90 + 6=96\).
Divide both sides by 4: \(y=\frac{96}{4}=24\).
Then \(m\angle C=y - 8=24-8 = 16\) degrees and \(m\angle D=3y + 2=3\times24+2=72 + 2=74\) degrees.

Answer:

For problem 26: \(y = 17\), \(m\angle C = 56^{\circ}\), \(m\angle D = 34^{\circ}\)
For problem 27: \(y = 24\), \(m\angle C = 16^{\circ}\), \(m\angle D = 74^{\circ}\)