Question

Need help with the statements and proofs pls help I’m in need

Answer 1

Explanation:

AB is a straight line. all angles around one point (here O) on one side of the line are 180°. after all, a line can be seen as the diameter of a circle, one side deals with a half-circle and therefore 180° (half of 360° of a full circle).

so, AOC and COB are supplementary angles (together they have 180°).

AOC + COB = 180° | dividing both sides by 2

AOC/2 + COB/2 = 180/2 = 90°

AOD = DOC = AOC/2

COE = EOB = COB/2

we can therefore put any of these bisected angles in place of the half main angles in the main equation :

DOC + COE = 90°

therefore, OD and OE enclose a right angle (90°).

Answer 2

See below for proof.

Explanation:

`\textsf{If $\overrightarrow{OD}$ bisects $\angle AOC$ then}:`

`\angle AOD=\angle DOC`

`\textsf{If $\overrightarrow{OE}$ bisects $\angle COB$ then}:`

`\angle COE=\angle EOB`

`\textsf{Angles on a straight line sum to $180^(\circ)$}:`

`\implies \angle AOD + \angle DOC + \angle COE + \angle EOB = 180^(\circ)`

`\implies 2 \angle DOC + 2\angle COE = 180^(\circ)`

`\implies 2 \left(\angle DOC + \angle COE\right) = 180^(\circ)`

`\implies \angle DOC + \angle COE = 90^(\circ)`

`\textsf{Perpendicular lines are lines that are at $90^(\circ)$ to each other}.`

`\textsf{Therefore, $\overrightarrow{OD} \perp \overrightarrow{OE}$}.`