Question 12

A transversal line creates various angle relationships when it intersects two other lines.

Step 1 — About Angle 6

Angle 6 and angle 5 form a linear pair. Their sum is 180 degrees. Angle 6 and angle 8 are vertically opposite angles. So, their measures are equal. Angle 6 is an interior angle. If lines $l$ and $m$ are parallel, then: Angle 6 and angle 2 are corresponding angles. Their measures are equal. Angle 6 and angle 3 are alternate interior angles. Their measures are equal. Angle 6 and angle 4 are consecutive interior angles. Their sum is 180 degrees.

Step 2 — About Angle 5

Angle 5 and angle 6 form a linear pair. Their sum is 180 degrees. Angle 5 and angle 7 are vertically opposite angles. So, their measures are equal. Angle 5 is an an interior angle. If lines $l$ and $m$ are parallel, then: Angle 5 and angle 4 are corresponding angles. Their measures are equal. Angle 5 and angle 1 are alternate interior angles. Their measures are equal. Angle 5 and angle 3 are consecutive interior angles. Their sum is 180 degrees.

Step 3 — About Angle 4

Angle 4 and angle 1 form a linear pair. Their sum is 180 degrees. Angle 4 and angle 2 are vertically opposite angles. So, their measures are equal. Angle 4 is an interior angle. If lines $l$ and $m$ are parallel, then: Angle 4 and angle 5 are corresponding angles. Their measures are equal. Angle 4 and angle 8 are alternate interior angles. Their measures are equal. Angle 4 and angle 6 are consecutive interior angles. Their sum is 180 degrees.

Step 4 — About Angle 3

Angle 3 and angle 2 form a linear pair. Their sum is 180 degrees. Angle 3 and angle 1 are vertically opposite angles. So, their measures are equal. Angle 3 is an interior angle. If lines $l$ and $m$ are parallel, then: Angle 3 and angle 7 are corresponding angles. Their measures are equal. Angle 3 and angle 6 are alternate interior angles. Their measures are equal. Angle 3 and angle 5 are consecutive interior angles. Their sum is 180 degrees.

Step 5 — About Angle 2

Angle 2 and angle 1 form a linear pair. Their sum is 180 degrees. Angle 2 and angle 4 are vertically opposite angles. So, their measures are equal. Angle 2 is an exterior angle. If lines $l$ and $m$ are parallel, then: Angle 2 and angle 6 are corresponding angles. Their measures are equal. Angle 2 and angle 7 are alternate exterior angles. Their measures are equal. Angle 2 and angle 5 are supplementary. Their sum is 180 degrees.

Answer

(i) Angle 6 is an interior angle. It forms a linear pair with angle 5. It is vertically opposite to angle 8. If lines $l$ and $m$ are parallel, angle 6 is equal to angle 2 (corresponding) and angle 3 (alternate interior). Also, if parallel, angle 6 and angle 4 sum to 180 degrees (consecutive interior). (ii) Angle 5 is an interior angle. It forms a linear pair with angle 6. It is vertically opposite to angle 7. If lines $l$ and $m$ are parallel, angle 5 is equal to angle 4 (corresponding) and angle 1 (alternate interior). Also, if parallel, angle 5 and angle 3 sum to 180 degrees (consecutive interior). (iii) Angle 4 is an interior angle. It forms a linear pair with angle 1. It is vertically opposite to angle 2. If lines $l$ and $m$ are parallel, angle 4 is equal to angle 5 (corresponding) and angle 8 (alternate interior). Also, if parallel, angle 4 and angle 6 sum to 180 degrees (consecutive interior). (iv) Angle 3 is an interior angle. It forms a linear pair with angle 2. It is vertically opposite to angle 1. If lines $l$ and $m$ are parallel, angle 3 is equal to angle 7 (corresponding) and angle 6 (alternate interior). Also, if parallel, angle 3 and angle 5 sum to 180 degrees (consecutive interior). (v) Angle 2 is an exterior angle. It forms a linear pair with angle 1. It is vertically opposite to angle 4. If lines $l$ and $m$ are parallel, angle 2 is equal to angle 6 (corresponding) and angle 7 (alternate exterior). Also, if parallel, angle 2 and angle 5 sum to 180 degrees.