Question

math ii
assessment 1: angles relationships and solving equations (show your work)
determine the measure of each unknown:

Explanation:

Step1: Identify angle - relationships for question 1

Since the angles are corresponding angles (assuming parallel lines), they are equal. So the unknown angle is $34^{\circ}$.

Step2: Identify angle - relationships for question 2

The sum of the angles in a triangle formed by the transversals is $180^{\circ}$. First, find the angle adjacent to $117^{\circ}$, which is $180 - 117=63^{\circ}$. Then, using the angle - sum property of a triangle, $x = 180-(63 + 56)=61^{\circ}$.

Step3: Identify angle - relationships for question 3

The given $20^{\circ}$ angle and the unknown $x$ are alternate - interior angles (assuming parallel lines), so $x = 20^{\circ}$.

Step4: Identify angle - relationships for question 4

The angle adjacent to $108^{\circ}$ is $180 - 108 = 72^{\circ}$. Since the lines are parallel, the unknown angle $x$ and the $72^{\circ}$ angle are corresponding angles, so $x = 72^{\circ}$.

Step5: Identify angle - relationships for question 5

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So $120=2x + 6+x$. Combining like terms gives $120 = 3x+6$. Subtract 6 from both sides: $114 = 3x$. Divide both sides by 3: $x = 38^{\circ}$.

Step6: Identify angle - relationships for question 6

The exterior angle of the triangle is $126^{\circ}$. So $126=x+(2x + 6)$. Combine like terms: $126 = 3x+6$. Subtract 6 from both sides: $120 = 3x$. Divide by 3: $x = 40^{\circ}$.

Step7: Identify angle - relationships for question 7

The angle adjacent to $123^{\circ}$ is $180 - 123=57^{\circ}$. Using the angle - sum property of a triangle in $\triangle ABC$, if we assume the triangle formed by the parallel lines and the transversals, we can find the required angle. However, if we consider the angle relationship with the parallel lines, the angle we want is equal to the $57^{\circ}$ angle (corresponding or alternate - interior depending on the parallel - line setup).

Answer:

1. $34^{\circ}$
2. $61^{\circ}$
3. $20^{\circ}$
4. $72^{\circ}$
5. $38^{\circ}$
6. $40^{\circ}$
7. $57^{\circ}$