Question

given ∠fgh and ∠hgd are a linear pair, m∠fgh=(x² + 19)° and m∠hgd=(x + 71)°, find the measure of both angles.

Explanation:

Step1: Use linear - pair property

Since $\angle FGH$ and $\angle HGD$ are a linear pair, $m\angle FGH + m\angle HGD=180^{\circ}$. So, $(x^{2}+19)+(x + 71)=180$.

Step2: Simplify the equation

Combine like - terms: $x^{2}+x+19 + 71=180$, which gives $x^{2}+x+90 = 180$. Then, rewrite it in standard quadratic form: $x^{2}+x - 90=0$.

Step3: Factor the quadratic equation

Factor $x^{2}+x - 90$ as $(x + 10)(x - 9)=0$.

Step4: Solve for x

Set each factor equal to zero: $x+10 = 0$ gives $x=-10$; $x - 9=0$ gives $x = 9$.

Step5: Find the measure of the angles for valid x - values

When $x = 9$:
$m\angle FGH=x^{2}+19=9^{2}+19=81 + 19=100^{\circ}$
$m\angle HGD=x + 71=9+71 = 80^{\circ}$
When $x=-10$:
$m\angle FGH=x^{2}+19=(-10)^{2}+19=100 + 19=119^{\circ}$
$m\angle HGD=x + 71=-10+71 = 61^{\circ}$

Answer:

When $x = 9$, $m\angle FGH = 100^{\circ}$ and $m\angle HGD = 80^{\circ}$; when $x=-10$, $m\angle FGH = 119^{\circ}$ and $m\angle HGD = 61^{\circ}$