Question

find the measure of the angle without using a protractor. (1.4 and 1.6)

1. \\( \angle a o b \\)
2. \\( \angle d o e \\)
3. \\( \angle e o f \\)
4. \\( \angle b o f \\)

Explanation:

Step1: Analyze $\angle AOB$

We know that $\angle AOC = 90^{\circ}$ (right - angle). Given $\angle BOC=62^{\circ}$. Using the formula $\angle AOB=\angle AOC-\angle BOC$.
$$\angle AOB = 90^{\circ}-62^{\circ}$$

Step2: Calculate $\angle AOB$

$$\angle AOB=28^{\circ}$$

Step3: Analyze $\angle DOE$

$\angle AOC = \angle COD=\angle DOF=\angle FOA = 90^{\circ}$ (right - angles). $\angle AOB$ and $\angle EOD$ are vertical angles. Since vertical angles are equal.
$$\angle DOE=\angle AOB$$

Step4: Calculate $\angle DOE$

$$\angle DOE = 28^{\circ}$$

Step5: Analyze $\angle EOF$

$\angle DOF = 90^{\circ}$ (right - angle). We know $\angle DOE = 28^{\circ}$. Using the formula $\angle EOF=\angle DOF-\angle DOE$
$$\angle EOF=90^{\circ}-28^{\circ}$$

Step6: Calculate $\angle EOF$

$$\angle EOF = 62^{\circ}$$

Step7: Analyze $\angle BOF$

$\angle AOF=90^{\circ}$ (right - angle). We know $\angle AOB = 28^{\circ}$. Using the formula $\angle BOF=\angle AOF+\angle AOB$
$$\angle BOF=90^{\circ}+28^{\circ}$$

Step8: Calculate $\angle BOF$

$$\angle BOF = 118^{\circ}$$

Answer:

1. $28^{\circ}$
2. $28^{\circ}$
3. $62^{\circ}$
4. $118^{\circ}$