Chapter 3.2, Problem 3WE

To determine

To find: all the angles congruent to $\angle 2$ if $a\parallel b$ and $c\parallel d$ .

Answer to Problem 3WE

  $\angle 4,\angle 5,\angle 7,\angle 10,\angle 12,\angle 13,\angle 15$

Explanation of Solution

Given:

The diagram:

Concept Used:

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent

Theorem 2-3: Vertically opposite angles are congruent.

Consider the given figure:

In this figure, by theorem 2-3,

  $\angle 2\cong \angle 5$ (Vertically opposite angles) ….. (1)

Now, lines $c\parallel d$ and a is a transversal, so by postulate 10,

  $\angle 2\cong \angle 10$ (Corresponding angles) ….. (2)

By theorem 2-3,

  $\angle 10\cong \angle 13$ (Vertically opposite angles)

Now, by using the transitive property on $\angle 2\cong \angle 10$ and $\angle 10\cong \angle 13$ .

Thus,

  $\angle 2\cong \angle 13$ ….. (3)

Now, lines $a\parallel b$ and c is a transversal, so by postulate 10,

  $\angle 2\cong \angle 4$ (Corresponding angles) ….. (4)

By theorem 2-3,

  $\angle 4\cong \angle 7$ (Vertically opposite angles)

Now, by using the transitive property on $\angle 2\cong \angle 4$ and $\angle 4\cong \angle 7$ .

Thus,

  $\angle 2\cong \angle 7$ ….. (5)

Now, lines $c\parallel d$ and a is a transversal, so by postulate 10,

  $\angle 4\cong \angle 12$ (Corresponding angles)

Now, by using the transitive property on $\angle 2\cong \angle 4$ and $\angle 4\cong \angle 12$ .

Thus,

  $\angle 2\cong \angle 12$ ….. (6)

By theorem 2-3,

  $\angle 12\cong \angle 15$ (Vertically opposite angles)

Now, by using the transitive property on $\angle 2\cong \angle 12$ and $\angle 12\cong \angle 15$ .

Thus,

  $\angle 2\cong \angle 15$ ….. (7)

Thus, the angles congruent to $\angle 1$ are $\angle 4,\angle 5,\angle 7,\angle 10,\angle 12,\angle 13,\angle 15$ .