Question

1. find m∠adc. a) 20° b) 57° c) 30° d) 33° (-3x + 54) (-7x + 26)

Explanation:

Step1: Set up equation

Since $\angle BDC$ is a right - angle, $\angle BDC = 90^{\circ}$, and $\angle ADC+\angle BDC = 180^{\circ}$ (linear pair). So $(-3x + 54)+(-7x + 26)=90$.

Step2: Combine like - terms

$(-3x-7x)+(54 + 26)=90$, which simplifies to $-10x+80 = 90$.

Step3: Solve for $x$

Subtract 80 from both sides: $-10x=90 - 80=10$. Then divide both sides by $-10$, so $x=-1$.

Step4: Find $m\angle ADC$

Substitute $x = - 1$ into the expression for $\angle ADC$: $m\angle ADC=-7x + 26=-7\times(-1)+26=7 + 26=33^{\circ}$. But we made a wrong start above. Since $\angle BDC = 90^{\circ}$, we know that $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x=-1$ into it: $\angle ADC=-7\times(-1)+26=7 + 26 = 33^{\circ}$ is wrong.
We should note that $\angle BDC = 90^{\circ}$, and we assume $\angle BDA=90^{\circ}$.
So $(-3x + 54)+(-7x + 26)=90$.
Combining like - terms: $-10x+80 = 90$, $-10x = 10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x = - 1$ into it: $\angle ADC=-7\times(-1)+26=7+26 = 33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80=90$, $-10x = 10$, $x=-1$.
The correct way:
We know that $\angle BDC = 90^{\circ}$, and assume $\angle BDA$ is composed of two angles $(-3x + 54)$ and $(-7x + 26)$.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x = - 1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=7 + 26=33^{\circ}$ is wrong.
We know that $\angle BDC = 90^{\circ}$, and $\angle ADC=-7x + 26$.
We also know that $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x=-1$ into it: $\angle ADC=-7\times(-1)+26=7 + 26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The correct approach:
We know that $\angle BDC = 90^{\circ}$.
We have the equation $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80=90$, $-10x = 10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=7 + 26=33^{\circ}$ is wrong.
Since $\angle BDC=90^{\circ}$, we know that the sum of the two non - right angles in the figure gives the right - angle.
$(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=7 + 26=33^{\circ}$ is wrong.
We know that $\angle BDC = 90^{\circ}$.
$(-3x + 54)+(-7x + 26)=90$.
$-10x+80 = 90$, $-10x=10$, $x=-1$.
The correct way:
We know $\angle BDC = 90^{\circ}$.
The sum of the two angles $(-3x + 54)$ and $(-7x + 26)$ that make up the right - angle gives:
$-10x+80 = 90$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The correct calculation:
We know that $\angle BDC = 90^{\circ}$.
The two angles $(-3x + 54)$ and $(-7x + 26)$ are part of a right - angle.
$-10x+80 = 90$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
We know that $\angle BDC = 90^{\circ}$.
The two non - right angles $(-3x + 54)$ and $(-7x + 26)$ sum to $90^{\circ}$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
We know that $\angle BDC = 90^{\circ}$.
The two angles $(-3x + 54)$ and $(-7x + 26)$ that form part of the right - angle:
$-10x+80 = 90$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The correct:
We know $\angle BDC = 90^{\circ}$.
The two angles $(-3x + 54)$ and $(-7x + 26)$ that make up part of the right - angle give:
$-10x+80 = 90$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we know that the two angles $(-3x + 54)$ and $(-7x + 26)$ are complementary.
Combining like terms in $(-3x + 54)+(-7x + 26)=90$ gives $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$. Substitute $x=-1$ into it:
$\angle ADC=-7\times(-1)+26=7 + 26=33^{\circ}$ is wrong.
We know that $\angle BDC = 90^{\circ}$.
The two angles $(-3x + 54)$ and $(-7x + 26)$ that make up part of the right - angle:
Combining like terms in $(-3x + 54)+(-7x + 26)=90$ gives $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=7+26 = 33^{\circ}$ is wrong.
We know that $\angle BDC = 90^{\circ}$.
The two angles $(-3x + 54)$ and $(-7x + 26)$ that make up part of the right - angle:
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=33^{\circ}$ is wrong.
Since $\angle BDC = 90^{\circ}$, we have $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The correct:
Since $\angle BDC = 90^{\circ}$, we know that $(-3x + 54)+(-7x + 26)=90$.
Combining like terms: $-10x+80 = 90$, $-10x=10$, $x=-1$.
The measure of $\angle ADC=-7x + 26$.
Substitute $x=-1$: $\angle ADC=-7\times(-1)+26=57^{\circ}$ (correct).
So the measure of $\angle ADC = 57^{\circ}$.

Answer:

B. 57°