In the given figure, if angle SEC = 120°, angle DCE = 25°, then find angle BAC. In the given figure, if angle DAB = 60°, angle ABD = 50°, then find angle ACB.

Steps to Solve

1. Finding angle BCE To find angle BAC, we first need to find angle BCE using the fact that angles on a straight line add up to 180°.

Since angle SEC = 120° and angle DCE = 25°, we can write: $$ \angle BCE = 180° - \angle SEC - \angle DCE = 180° - 120° - 25° $$ Calculating this gives: $$ \angle BCE = 35° $$

2. Using the Inscribed Angle Theorem The next step is to use the Inscribed Angle Theorem. According to this theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Here, angle BAC intercepts arc BEC.

So, we have: $$ \angle BAC = \frac{1}{2} \angle BCE = \frac{1}{2} \times 35° $$ Calculating this gives: $$ \angle BAC = 17.5° $$

3. Finding angle ACB Now, to find angle ACB, we can use the sum of angles in triangle ABD: $$ \angle ADB = 180° - \angle DAB - \angle ABD $$ Plugging in the values: $$ \angle ADB = 180° - 60° - 50° $$ This simplifies to: $$ \angle ADB = 70° $$

4. Applying the Inscribed Angle Theorem for angle ACB Using the Inscribed Angle Theorem again, angle ACB intercepts arc AB: $$ \angle ACB = \frac{1}{2} \angle ADB = \frac{1}{2} \times 70° $$ Calculating this gives: $$ \angle ACB = 35° $$