Question

Line l is parallel to line m as shown in the above figure. Transversals t and u intersect at point A on l and intersect m at points C and B, respectively. Point X lies on line m, the measure of $\angle ACX$ is ${130}^{\circ }$, and the measure of $\angle BAC$ is ${80}^{\circ }$. Find the number of angles formed by rays l, m, t, and u which have measure ${50}^{\circ }$.

• A. $4$
• B. $6$
• C. $8$
• D. $10$
• E. $12$

Answer 1

The correct option is C $8$

From the figure,

Given,

$\angle ACX={130}^{\circ }$ and $\angle BAC={80}^{\circ }$

In a triangle, exterior angle is equal to sum of interior opposite angles.

$\angle ACX=\angle ABC+\angle BAC={130}^{\circ }$

$\angle ABC={50}^{\circ }$

$\angle ABC=\angle WBY={50}^{\circ }$ ------- opposite angles

$\angle BCA+\angle ABC+\angle BAC={180}^{\circ }$ ------sum of interior angles of a triangle

$\therefore \angle BCA={50}^{\circ }$

$\angle BCA=\angle XCZ={50}^{\circ }$ ------- opposite angles

$\angle SAC=\angle ACX={130}^{\circ }$ --------angles formed by a traversal cutting two parallel lines / alternate angles

But, $\angle SAC=\angle SAB+\angle BAC$

$\therefore \angle SAB={50}^{\circ }$

$\angle SAB=\angle VAU={50}^{\circ }$ -------vertically opposite angles

$\angle SAC+\angle SAT=\angle TAC={180}^{\circ }$ -------straight line

$\therefore \angle SAT={50}^{\circ }$

$\angle SAT=\angle UAC={50}^{\circ }$ -------vertically opposite angles

$\therefore$ Total of $8$ ${50}^{\circ }$ angles are formed.

Option C.