Question

see examples 4 and 5

1. if (mangle qpr = 24) and (mangle qps = 59), what is (mangle rps)?
2. if (mangle qpr = 4x - 8), (mangle rps = 3x + 15), (mangle spt = x + 13), and (mangle qpt = 76), what is the value of (x)?
3. if (mangle rpt = 46), (mangle qps = 61), and (mangle qpr = 28), what is (mangle spt)?

in the figure, (mangle dbc=mangle cad), (mangle dba = mangle cab), and (ec = ed). if (mangle cba = 5x + 2), (ec = 3), and (eb = 8), find each value. see examples 3 and 5

Explanation:

26.

Step1: Identify angle - relationship

We know that $\angle QPS=\angle QPR+\angle RPS$.

Step2: Solve for $\angle RPS$

$m\angle RPS=m\angle QPS - m\angle QPR$. Substitute $m\angle QPR = 24$ and $m\angle QPS=59$ into the formula. So $m\angle RPS=59 - 24$.
$m\angle RPS = 35$.

Step1: Use angle - addition property

Since $\angle QPT=\angle QPR+\angle RPS+\angle SPT$, we substitute the given angle - measures: $(4x - 8)+(3x + 15)+(x + 13)=76$.

Step2: Combine like - terms

$(4x+3x + x)+(-8 + 15+13)=76$, which simplifies to $8x+20 = 76$.

Step3: Solve for $x$

Subtract 20 from both sides: $8x=76 - 20=56$. Then divide both sides by 8: $x=\frac{56}{8}=7$.

Step1: First, find $\angle RPS$

We know that $\angle QPS=\angle QPR+\angle RPS$. So $m\angle RPS=m\angle QPS - m\angle QPR$. Substitute $m\angle QPR = 28$ and $m\angle QPS = 61$ into the formula, we get $m\angle RPS=61 - 28=33$.

Step2: Then, find $\angle SPT$

Since $\angle RPT=\angle RPS+\angle SPT$, then $m\angle SPT=m\angle RPT - m\angle RPS$. Substitute $m\angle RPT = 46$ and $m\angle RPS = 33$ into the formula. So $m\angle SPT=46 - 33$.
$m\angle SPT = 13$.

Answer:

$35$

27.