Question

(d) problem 20: (first taught in lesson 31) if ℓ || m, and (overrightarrow{de}) bisects (angle cdf), find y.

Explanation:

Step1: Find $\angle CDF$

Since $\ell\parallel m$, and $\angle C$ and the angle adjacent to $\angle CDF$ are same - side interior angles. The adjacent angle to $\angle CDF$ and $\angle C$ are supplementary. Given $\angle C = 130^{\circ}$, the adjacent angle to $\angle CDF=180 - 130=50^{\circ}$, so $\angle CDF = 130^{\circ}$.

Step2: Use the angle - bisector property

Since $\overrightarrow{DE}$ bisects $\angle CDF$, then $\angle CDE=\frac{1}{2}\angle CDF$. Substituting $\angle CDF = 130^{\circ}$, we get $\angle CDE = 65^{\circ}$.

Step3: Find the value of $y$

$\ell\parallel m$, and $\angle CDE$ and $y$ are alternate - interior angles. Alternate - interior angles are congruent when two parallel lines are cut by a transversal. So $y = 65^{\circ}$.

Answer:

$65$