Question

m < pqh =

∠eqf ≅ ∠pqh
m∠eqh = 4(2x + 1), m∠pqh = 9x + 29,
m∠hqf = 2(5x - 1)
find m∠pqh.

Explanation:

Step1: Set up an equation based on angle - addition

Since $\angle EQH+\angle HQF=\angle EQF$ and $\angle EQF\cong\angle PQH$, we have $4(2x + 1)+2(5x - 1)=9x + 29$.
First, expand the left - hand side:
$4(2x + 1)=8x+4$ and $2(5x - 1)=10x - 2$.
So the left - hand side becomes $(8x + 4)+(10x - 2)=18x+2$.
The equation is $18x+2 = 9x + 29$.

Step2: Solve the equation for x

Subtract $9x$ from both sides:
$18x-9x + 2=9x-9x + 29$, which simplifies to $9x+2 = 29$.
Then subtract 2 from both sides:
$9x+2 - 2=29 - 2$, getting $9x=27$.
Divide both sides by 9:
$x=\frac{27}{9}=3$.

Step3: Find the measure of $\angle PQH$

Substitute $x = 3$ into the expression for $m\angle PQH=9x + 29$.
$m\angle PQH=9\times3+29$.
$m\angle PQH=27 + 29=56$.

Answer:

$56$