Question

1. angles 1 and 2 are a linear pair, m∠1 = (\frac{1}{3}x + 3)° and m∠2 = (\frac{1}{3}x - 4)°. find the measure of both angles.

x = ____ m∠1 = __ m∠2 = ____

1. given ∠cab and ∠bad are a linear pair, m∠cab=(x² + 100)° and m∠bad=(x + 8)°, find the measure of both angles.

x = ____ m∠cab = __ m∠bad = __ or x = __ m∠cab = __ m∠bad = ____

Explanation:

Step1: Use linear - pair property

Since angles in a linear pair add up to 180°, for angles 1 and 2, we have $(\frac{1}{3}x + 3)+(\frac{1}{3}x - 4)=180$.

Step2: Combine like - terms

$\frac{1}{3}x+\frac{1}{3}x+3 - 4=180$, which simplifies to $\frac{2}{3}x-1 = 180$.

Step3: Isolate the variable term

Add 1 to both sides: $\frac{2}{3}x=180 + 1=181$.

Step4: Solve for x

Multiply both sides by $\frac{3}{2}$: $x=\frac{3\times181}{2}=271.5$.

Step5: Find the measure of angle 1

$m\angle1=\frac{1}{3}(271.5)+3 = 90.5+3=93.5^{\circ}$.

Step6: Find the measure of angle 2

$m\angle2=\frac{1}{3}(271.5)-4 = 90.5 - 4=86.5^{\circ}$.

For the second problem:

Step1: Use linear - pair property

Since $\angle CAB$ and $\angle BAD$ are a linear pair, $(x^{2}+100)+(x + 8)=180$.

Step2: Rearrange to form a quadratic equation

$x^{2}+x+100 + 8-180=0$, so $x^{2}+x - 72=0$.

Step3: Factor the quadratic equation

$(x + 9)(x - 8)=0$.

Step4: Solve for x

Setting each factor equal to zero gives $x=-9$ or $x = 8$.

Step5: Find the measure of $\angle CAB$ and $\angle BAD$ for $x=-9$

$m\angle CAB=(-9)^{2}+100=81 + 100=181^{\circ}$ (not valid as angle measure cannot be greater than 180°).

Step6: Find the measure of $\angle CAB$ and $\angle BAD$ for $x = 8$

$m\angle CAB=8^{2}+100=64 + 100=164^{\circ}$, $m\angle BAD=8 + 8=16^{\circ}$.

Answer:

For the first problem:
$x = 271.5$, $m\angle1=93.5^{\circ}$, $m\angle2=86.5^{\circ}$
For the second problem:
$x = 8$, $m\angle CAB=164^{\circ}$, $m\angle BAD=16^{\circ}$