Question

1. select all that apply from the figure on the left.

$\angle 2+\angle 3 = 180^{\circ}$

$\angle 2$ & $\angle 3$ are adjacent $\angle$s

$\angle 2 = 40^{\circ}$

$\angle 2$ & $\angle 3$ are not a linear pair

$\angle 3$ & $\angle 4$ are vertical $\angle$s

$\angle 4 = 140^{\circ}$

$\angle 3$ & $\angle 4$ are complementary

$\angle 3$ is congruent to $\angle 4$

$\angle 2$ & $\angle 4$ are a linear pair

$\angle 3 = 40^{\circ}$

Explanation:

Step1: Check $\angle2+\angle3 = 180^{\circ}$

Since $\angle2$ and $\angle3$ form a linear pair (they are adjacent and their non - common sides form a straight line), by the linear pair postulate, $\angle2+\angle3 = 180^{\circ}$.

Step2: Check if $\angle2$ and $\angle3$ are adjacent

$\angle2$ and $\angle3$ have a common vertex and a common side. So, $\angle2$ and $\angle3$ are adjacent angles.

Step3: Find $\angle2$

$\angle2$ and the $40^{\circ}$ angle are vertical angles. Vertical angles are congruent. So, $\angle2=40^{\circ}$.

Step4: Check if $\angle2$ and $\angle3$ are a linear pair

They are adjacent and their non - common sides form a straight line. So, they are a linear pair. The statement “$\angle2$ & $\angle3$ are NOT a Linear Pair” is false.

Step5: Check if $\angle3$ and $\angle4$ are vertical angles

Vertical angles are opposite angles formed by the intersection of two lines. $\angle3$ and $\angle4$ are not opposite angles. So, they are not vertical angles.

Step6: Find $\angle4$

Since $\angle3+\angle4 = 180^{\circ}$ (linear pair) and $\angle3 = 140^{\circ}$ (because $\angle2 = 40^{\circ}$ and $\angle2+\angle3=180^{\circ}$), then $\angle4 = 40^{\circ}$.

Step7: Check if $\angle3$ and $\angle4$ are complementary

Complementary angles sum to $90^{\circ}$. $\angle3 = 140^{\circ}$ and $\angle4 = 40^{\circ}$, $\angle3+\angle4=180^{\circ}$. So, they are not complementary.

Step8: Check if $\angle3$ is congruent to $\angle4$

$\angle3 = 140^{\circ}$ and $\angle4 = 40^{\circ}$. So, they are not congruent.

Step9: Check if $\angle2$ and $\angle4$ are a linear pair

They are not adjacent (they do not have a common side). So, they are not a linear pair.

Step10: Find $\angle3$

Since $\angle2+\angle3 = 180^{\circ}$ and $\angle2 = 40^{\circ}$, then $\angle3=180^{\circ}-\angle2=140^{\circ}$.

Answer:

$\angle2+\angle3 = 180^{\circ}$, $\angle2$ & $\angle3$ are adjacent $\angle$'s, $\angle2 = 40^{\circ}$