Question

in the triangle below, $angle i$ is a right - angle. suppose that $mangle h=(5x + 7)^{circ}$ and $mangle j=(2x + 41)^{circ}$. (a) write an equation to find $x$. make sure you use an \=\ sign in your answer. (b) find the degree measure of each angle.

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. Since $\angle I = 90^{\circ}$, we have $m\angle H+m\angle J + m\angle I=180^{\circ}$. Substituting the given angle measures, we get $(5x + 7)+(2x + 41)+90=180$.
Simplifying the left - hand side gives $5x+7 + 2x+41+90=7x + 138$. So the equation is $7x+138 = 180$.

Step2: Solve the equation for x

Subtract 138 from both sides of the equation $7x+138 = 180$.
$7x=180 - 138$, so $7x = 42$.
Divide both sides by 7: $x=\frac{42}{7}=6$.

Step3: Find the measure of $\angle H$

Substitute $x = 6$ into the expression for $m\angle H$.
$m\angle H=(5x + 7)^{\circ}=(5\times6 + 7)^{\circ}=(30 + 7)^{\circ}=37^{\circ}$.

Step4: Find the measure of $\angle J$

Substitute $x = 6$ into the expression for $m\angle J$.
$m\angle J=(2x + 41)^{\circ}=(2\times6+41)^{\circ}=(12 + 41)^{\circ}=53^{\circ}$.
And $m\angle I = 90^{\circ}$ (given).

Answer:

(a) Equation: $7x + 138=180$
(b)
$m\angle H = 37^{\circ}$
$m\angle I = 90^{\circ}$
$m\angle J = 53^{\circ}$