Verified answer

1. Yes, <2 and <26 are vertical angles. Vertical angles are formed when two lines intersect, and they are opposite angles that share the same vertex. In this case, <2 and <26 are formed when line l intersects with line m, and they are opposite angles that share the vertex at point A.

2. Yes, <3 and <27 are vertical angles. Vertical angles are formed when two lines intersect, and they are opposite angles that share the same vertex. In this case, <3 and <27 are formed when line l intersects with line m, and they are opposite angles that share the vertex at point B.

3. To determine the measure of <5, we need to know the relationship between <5 and <8.

If <5 and <8 are vertical angles, then they are congruent and have the same measure. In this case, if m<8 = 40, then m<5 = 40 as well.

4. The union of <3 and <7 will form vertical angles.

5. To find the measure of angle <4, we need to use the fact that angles 4 and 8 are vertical angles, which means they are opposite to each other and have the same measure.

The vertical angles theorem states that if two lines intersect, then the vertical angles formed are congruent (i.e., they have the same measure).

So, if m<8 = 52, then we know that m<4 (the measure of angle 4) is also 52, since angles 4 and 8 are vertical angles.

Therefore, m<4 = 52.

6. To find the measure of angle <2, we need to use the fact that angles 2 and 5 are corresponding angles, which means they are in the same position relative to the transversal and the two parallel lines.

The corresponding angles theorem states that if two parallel lines are intersected by a transversal, then the corresponding angles formed are congruent (i.e., they have the same measure).

So, if m<5 = 55, then we know that m<2 (the measure of angle 2) is also 55, since angles 2 and 5 are corresponding angles.

Therefore, m<2 = 55.

7. No, angles 4, 5, and 7 are not adjacent angles or are not part of the same figure, then the equation would not be true.

8. Vertical angles are formed by two intersecting lines. When two lines intersect, they form four angles, and the opposite angles are called vertical angles. Vertical angles are always congruent, which means they have the same measure.

In the given scenario, we know that <8 and <2 are not vertical angles because they are adjacent angles, which means they share a common vertex and a common side. To find the two angles that form vertical angles with <8 and <2, we need to look for the angles that are opposite to them.

When two lines intersect, the vertical angles are formed by the pairs of opposite angles. So, the two angles that form vertical angles with <8 and <2 are the angles that are opposite to them. Let's call these angles <1 and <7.

Therefore, the union of <8 and <2 and the union of <1 and <7 will form vertical angles. We can represent this as:

<8 ∪ <2 and <1 ∪ <7 are vertical angles.

9. The sum of adjacent angles is always equal to 180 degrees. Therefore, we can write:

m<4 + m<6 = 180

Substituting m<4 = 30, we get:

30 + m<6 = 180

Solving for m<6, we get:

m<6 = 180 - 30

m<6 = 150

Therefore, <4 and <6 are adjacent angles and share a common vertex and a common side, and m<4 = 30, then m<6 is equal to 150 degrees.

10. If m<2 = 5x - 10 and m<6 = 2x + 20, we can use the fact that the sum of the measures of angles in a triangle is 180 degrees to find the value of x and the measure of each angle.

In a triangle, the sum of the measures of the three angles is always 180 degrees. Therefore, we can write:

m<2 + m<4 + m<6 = 180

Substituting m<2 = 5x - 10 and m<6 = 2x + 20, we get:

5x - 10 + m<4 + 2x + 20 = 180

Simplifying the equation, we get:

7x + 10 + m<4 = 180

Subtracting 10 from both sides, we get:

7x + m<4 = 170

We also know that m<2 and m<4 are vertical angles, which means they are congruent. Therefore, we can write:

m<2 = m<4

Substituting m<2 = 5x - 10, we get:

5x - 10 = m<4

Substituting this into the equation 7x + m<4 = 170, we get:

7x + (5x - 10) = 170

Simplifying the equation, we get:

12x - 10 = 170

Adding 10 to both sides, we get:

12x = 180

Dividing both sides by 12, we get:

x = 15

Therefore, the value of x is 15.

To find the measure of each angle, we can substitute x = 15 into the given expressions for m<2 and m<6:

m<2 = 5x - 10 = 5(15) - 10 = 65 degrees

m<6 = 2x + 20 = 2(15) + 20 = 50 degrees

We also know that m<2 and m<4 are congruent, so m<4 must also be 65 degrees.

Therefore, the measure of angle <2 is 65 degrees, the measure of angle <4 is 65 degrees, and the measure of angle <6 is 50 degrees.