Question

construct a line through p parallel to ℓ (hint: copy either ∠pca or ∠pcb so that one of the sides of the angle is parallel to ℓ.) what theorem justifies your construction? to construct a line through p and parallel to ℓ, start by drawing a line m through c and p. put the point of the compass at c, and draw an arc to the right of m that intersects ℓ at a point x and m at a point y. place the compass point at p and draw an arc intersecting m at a point q above and to the right of p. place the compass point at x, and open the compass to the distance between and x. without changing the setting, place the compass point at q and draw an arc intersecting the arc that passes through q below and to the right of q. label the point r where the two arcs intersect. draw a line n through the angle thus constructed, is a copy of ∠ycx (which is equivalent to ∠pcb) and therefore is to it. this means those angles are angles that are therefore, lines ℓ and n are parallel by the video textbook get more help ▼ clear all check answer

Explanation:

Step1: Recall parallel - line construction principle

When constructing a line parallel to a given line through a point, we use the property of equal corresponding angles or alternate - interior angles.

Step2: Analyze the construction steps

We first draw a transversal (line m) through the given point P and the given line $\ell$. Then, by copying an angle (either a corresponding or alternate - interior angle) formed by the transversal and the given line $\ell$, we can construct a parallel line. In this case, we are copying an angle (either $\angle PCA$ or $\angle PCB$). When we construct an equal angle, the new line n and the given line $\ell$ are parallel because if two lines are cut by a transversal and the corresponding angles (or alternate - interior angles) are equal, then the two lines are parallel.

Step3: Identify the justifying theorem

The theorem that justifies this construction is the corresponding - angles postulate or the alternate - interior angles theorem. If two lines are cut by a transversal and the corresponding angles are equal, the lines are parallel. Similarly, if the alternate - interior angles are equal, the lines are parallel.

Answer:

The construction is justified by the corresponding - angles postulate or the alternate - interior angles theorem.