Question

$overleftrightarrow{al}paralleloverleftrightarrow{dh}$, where $overleftrightarrow{ph}$ is a transversal. given $mangle dhj=(4x - 9)^{circ}$ and $mangle ajh = 10x^{circ}$, complete the statement. the measure of $angle dhj$ is and the measure of $angle ajh$ is. they are because they are angles.

Explanation:

Step1: Identify angle - relationship

Since $\overleftrightarrow{AL}\parallel\overleftrightarrow{DH}$ and $\overleftrightarrow{PH}$ is a transversal, $\angle DHJ$ and $\angle AJH$ are same - side interior angles. Same - side interior angles are supplementary, so $m\angle DHJ + m\angle AJH=180^{\circ}$.

Step2: Set up the equation

We know that $m\angle DHJ=(4x - 9)^{\circ}$ and $m\angle AJH = 10x^{\circ}$. Substitute these into the equation: $(4x - 9)+10x=180$.

Step3: Combine like - terms

Combine the $x$ terms: $4x+10x-9 = 180$, which simplifies to $14x-9 = 180$.

Step4: Solve for $x$

Add 9 to both sides of the equation: $14x=180 + 9=189$. Then divide both sides by 14: $x=\frac{189}{14}=13.5$.

Step5: Find the measure of $\angle DHJ$

Substitute $x = 13.5$ into the expression for $m\angle DHJ$: $m\angle DHJ=4x-9=4\times13.5-9=54 - 9=45^{\circ}$.

Step6: Find the measure of $\angle AJH$

Substitute $x = 13.5$ into the expression for $m\angle AJH$: $m\angle AJH=10x=10\times13.5 = 135^{\circ}$.

Answer:

The measure of $\angle DHJ$ is $45^{\circ}$ and the measure of $\angle AJH$ is $135^{\circ}$. They are supplementary because they are same - side interior angles.