Question

a. explain why the measure of angle ecf is 19 degrees. if you get stuck, consider translating line $ell$ by moving b to c.
b. what is the measure of angle bcd? explain.

Explanation:

Step1: Recall translation property

Translation is a rigid - motion. Rigid - motions preserve angle measures. When line $\ell$ is translated by moving $B$ to $C$, the angles formed by the original line and the new line (related to the translation) are congruent. So, if the angle formed in the process of translation is $\angle ECF = 19^{\circ}$, it is because translation preserves angle - measure.

Step2: Identify linear - pair relationship

$\angle ECF$ and $\angle BCD$ form a linear pair. The sum of the measures of angles in a linear pair is $180^{\circ}$. Let $m\angle BCD=x$. Then $m\angle ECF + x=180^{\circ}$.

Step3: Solve for $\angle BCD$

Since $m\angle ECF = 19^{\circ}$, we substitute into the equation $19^{\circ}+x = 180^{\circ}$. Solving for $x$, we get $x=180^{\circ}-19^{\circ}=161^{\circ}$.

Answer:

a. Translation is a rigid - motion that preserves angle measures. So the measure of $\angle ECF$ is the same as the angle related to the translation of line $\ell$, which is $19^{\circ}$.
b. The measure of $\angle BCD$ is $161^{\circ}$ because $\angle ECF$ and $\angle BCD$ form a linear pair and the sum of the measures of angles in a linear pair is $180^{\circ}$, so $m\angle BCD=180^{\circ}-19^{\circ}=161^{\circ}$.