Question

al||dh, where ph is a transversal. given m∠dhj=(4x - 9)° and m∠ajh = 10x°, complete the statement. the measure of ∠dhj is and the measure of ∠ajh is. they are because they are angles.

Explanation:

Step1: Identify angle - relationship

Since $\overrightarrow{AL}\parallel\overrightarrow{DH}$ and $\overrightarrow{PH}$ 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

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

Step3: Solve the equation for $x$

Combine like terms:
$$4x+10x-9 = 180$$
$$14x-9=180$$
Add 9 to both sides:
$$14x=180 + 9$$
$$14x=189$$
Divide both sides by 14:
$$x=\frac{189}{14}=13.5$$

Step4: Find the measure of $\angle DHJ$

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

Step5: 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.