Adjacent Angles and Non-Adjacent Angles

Angles and Lines

• Ray BD is the common arm and point B is the common vertex of two angles.
• Ray BA and ray BC are non-common arms of two angles.
• The sum of two adjacent angles is equal to the angle formed by the two non-common arms.

Adjacent Angles

• ∠ABC and ∠ABD are not adjacent angles because their non-common arms BD and BC lie on the same side of the common arm BA.
• When two angles are adjacent, their sum is equal to the angle formed by the two non-common arms.

Linear Pair of Angles

• When the non-common arms of two angles form a line, the angles are called a linear pair of angles.
• Example: ∠ABD and ∠DBC are a linear pair of angles in Fig. 6.3.

Vertically Opposite Angles

• When two lines intersect, they form vertically opposite angles.
• Example: ∠AOD and ∠BOC are vertically opposite angles in Fig. 6.4.

Intersecting and Non-Intersecting Lines

• Two lines can be drawn in two different ways: intersecting or non-intersecting (parallel).
• Example: Fig. 6.5 shows two different ways of drawing lines PQ and RS.

Properties of Lines

• A line extends indefinitely in both directions.
• Lines can be intersecting or parallel.

Example Problems

• If two lines are parallel, corresponding angles are equal.
• Example: In Fig. 6.19, if PQ || RS, ∠MXQ = 135° and ∠MYR = 40°, then ∠XMY = 85°.
• If a transversal intersects two lines and the bisectors of a pair of corresponding angles are parallel, then the two lines are parallel.
• Example: In Fig. 6.21, if ray BE is the bisector of ∠ABQ and ray CG is the bisector of ∠BCS, and BE || CG, then PQ || RS.