Question

1. ∠1 and ∠2 form a linear pair. if (mangle1=(5x + 9)^{circ}) and (mangle2=(3x + 11)^{circ}), find the measure of each angle.
2. ∠1 and ∠2 are vertical angles. if (mangle1=(17x + 1)^{circ}) and (mangle2=(20x - 14)^{circ}), find (mangle2).
3. ∠k and ∠l are complementary angles. if (mangle k=(3x + 3)^{circ}) and (mangle l=(10x - 4)^{circ}), find the measure of each angle.

Explanation:

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Step1: Use linear - pair property

Since $\angle1$ and $\angle2$ form a linear pair, $m\angle1 + m\angle2=180^{\circ}$. So, $(5x + 9)+(3x + 11)=180$.

Step2: Simplify the left - hand side

Combine like terms: $5x+3x + 9 + 11=180$, which gives $8x+20 = 180$.

Step3: Solve for $x$

Subtract 20 from both sides: $8x=180 - 20=160$. Then divide both sides by 8, so $x = 20$.

Step4: Find the measure of each angle

$m\angle1=(5x + 9)^{\circ}=(5\times20 + 9)^{\circ}=109^{\circ}$.
$m\angle2=(3x + 11)^{\circ}=(3\times20+11)^{\circ}=71^{\circ}$.

Step1: Use vertical - angle property

Since $\angle1$ and $\angle2$ are vertical angles, $m\angle1=m\angle2$. So, $17x + 1=20x-14$.

Step2: Solve for $x$

Subtract $17x$ from both sides: $1 = 20x-17x-14$. Then add 14 to both sides: $1 + 14=3x$, so $3x = 15$ and $x = 5$.

Step3: Find $m\angle2$

Substitute $x = 5$ into the expression for $m\angle2$: $m\angle2=(20x - 14)^{\circ}=(20\times5-14)^{\circ}=86^{\circ}$.

Step1: Use complementary - angle property

Since $\angle K$ and $\angle L$ are complementary angles, $m\angle K+m\angle L = 90^{\circ}$. So, $(3x + 3)+(10x-4)=90$.

Step2: Simplify the left - hand side

Combine like terms: $3x+10x+3 - 4=90$, which gives $13x-1 = 90$.

Step3: Solve for $x$

Add 1 to both sides: $13x=90 + 1=91$. Then divide both sides by 13, so $x = 7$.

Step4: Find the measure of each angle

$m\angle K=(3x + 3)^{\circ}=(3\times7 + 3)^{\circ}=24^{\circ}$.
$m\angle L=(10x - 4)^{\circ}=(10\times7-4)^{\circ}=66^{\circ}$.

Answer:

$m\angle1 = 109^{\circ}$, $m\angle2 = 71^{\circ}$

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