1.8 Geometry angle pairs

An obtuse angle is an angle that measures more than 90° but less than 180°.

It's larger than a right angle but smaller than a straight angle.

For example, if two line segments meet and form an angle that's past a right angle, it's obtuse. You can also think of it as a car seat that's reclined past the 90° upright position. 

Non-adjacent complementary angles are two angles that are not adjacent and add up to 90°. They do not share a common vertex or arm.

1. 1. Supplementary angles add up to 180°.

3. 2. Angle ∠CBF is the angle between segments BC and BF.

4. Angle ∠JBF is the angle between segments BJ and BF.

3. Notice that line BJ and line BC appear to be part of the same straight line when extended. This suggests that angles ∠JBF and ∠CBF are on the same straight line and thus form a linear pair.

4. A linear pair of angles is always supplementary because they lie on a straight line, which measures 180°. In this case, angles ∠JBF and ∠CBF are adjacent angles that share the ray BF and form a straight line.

$$\left(step\ 1\right)\ 2x\ +4x\ +\ 108=180$$

$$\left(step\ 2\right)\ 6x\ +\ 108\ =\ 180$$

$$\left(step\ 3\right)6x\ =\ 72$$

x = 12

Look at these steps above first before you click on the picture

Since angles $$\angle\ 2y\ +\ 6$$ and $$\angle\ 8y\ -102$$ are across from one another they are vertical angles. Therefore, they are equal to one another.

so , $$2y\ +6\ =\ 8y\ -102$$

solve for y , so $$y\ =\ 18\$$

Since $$\angle3x$$ and $$\angle\ 8y\ -102$$ are adjacent to one another, they form a line which means they add up to 180 degrees. so substitute in 18 for y so $$3x+8\left(18\right)-102=180$$
solve for x and

x = 46

Solve the equation:

so ,

$$180=2x-76$$

$$256=\ 2x$$

x = 128,

Therefore the supplement is

180 - 128 = 52

Since $$\overrightarrow{BH\ }$$ bisects $$\angle EBC$$ , then $$m\angle\ EBH\ =\ m\ \angle\ HBC$$ so you can set them equal to one another because a ray that divides an angle into two Congruent angles is called an angle bisector in notes 1.8

$$8x-10\ =\ 6x\ +12$$

$$2x\ =\ 22$$

x = 11

, therefore,

$$6\ \left(11\right)\ +\ 12\ =\ 78$$