Question

suppose that △hij is isosceles with base ij. suppose also that m∠h=(5x + 54)° and m∠i=(4x + 24)°. find the degree - measure of each angle in the triangle.

Explanation:

Step1: Use isosceles - triangle property

In an isosceles triangle $\triangle HIJ$ with base $\overline{IJ}$, $\angle H$ and $\angle J$ are congruent. So $m\angle H=m\angle J=(5x + 54)^{\circ}$ and $m\angle I=(4x + 24)^{\circ}$.

Step2: Apply angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So $(5x + 54)+(4x + 24)+(5x + 54)=180$.
Combining like - terms: $5x+4x + 5x+54 + 24+54 = 180$, which simplifies to $14x+132 = 180$.
Subtract 132 from both sides: $14x=180 - 132=48$.
Then $x=\frac{48}{14}=\frac{24}{7}$.

Step3: Find $m\angle H$

Substitute $x = \frac{24}{7}$ into the expression for $m\angle H$: $m\angle H=5x + 54=5\times\frac{24}{7}+54=\frac{120}{7}+54=\frac{120 + 378}{7}=\frac{498}{7}\approx71.14^{\circ}$.

Step4: Find $m\angle I$

Substitute $x=\frac{24}{7}$ into the expression for $m\angle I$: $m\angle I=4x + 24=4\times\frac{24}{7}+24=\frac{96}{7}+24=\frac{96+168}{7}=\frac{264}{7}\approx37.71^{\circ}$.

Step5: Find $m\angle J$

Since $m\angle J=m\angle H$, $m\angle J=\frac{498}{7}\approx71.14^{\circ}$.

Answer:

$m\angle H=\frac{498}{7}^{\circ}$
$m\angle I=\frac{264}{7}^{\circ}$
$m\angle J=\frac{498}{7}^{\circ}$