Refresher

Pairs of Angles
There are some special relationships between "pairs" of angles.

bullet Adjacent Angles are two angles that share a common vertex, a common side, and no common interior points. (They share a vertex and side, but do not overlap.)

adjangles

∠1 and ∠2 are adjacent angles.

ABC and ∠1 are NOT adjacent angles.
(∠ABC overlaps ∠1.)

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bullet A Linear Pair is two adjacent angles whose non-common sides form opposite rays.
linearpair
∠1 and ∠2 form a linear pair.

The line through points A, B and C is a straight line.

∠1 and ∠2 are supplementary.

theorem
If two angles form a linear pair, the angles are supplementary.
A linear pair forms a straight angle which contains 180º, so you have 2 angles whose measures add to 180, which means they are supplementary.
theorem
If two congruent angles form a linear pair, the angles are right angles.
If two congruent angles add to 180º, each angle contains 90º, forming right angles.

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bullet Vertical Angles are two angles whose sides form two pairs of opposite rays (straight lines).
vertical
Vertical angles are located across from one another in the corners of the "X" formed by the two straight lines.

∠1 and ∠2 are vertical angles.
∠3 and ∠4 are vertical angles.

Vertical angles are not adjacent.
∠1 and ∠3 are not vertical angles (they are a linear pair).

Vertical angles are always equal in measure.

theorem
Vertical angles are congruent.
Vertical angles, such as ∠1 and ∠2, form linear pairs with the same angle, ∠4, giving m∠1 + m∠4 = 180 and m∠2 + m∠4 = 180. With substitution, m∠1 = m∠2 and they are congruent.

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bullet Complementary Angles are two angles the sum of whose measures is 90º.
comp
Complementary angles can be placed so they form perpendicular lines, or they may be two separate angles.
∠1 and ∠2 are complementary.
P and ∠Q are complementary.

perp

theorem
Complements of the same angle, or congruent angles, are congruent.
If m∠a is complementary to the m∠b, and m∠c is complementary to m∠b, then m∠a = m∠c. Consider m∠a = 60º, m∠b = 30º and m∠c = 60º.
theorem
The acute angles of a right triangle are complementary.
The sum of the angles in a triangle add to 180º. After subtracting 90º for the right angle, there are 90º left for the two acute angles, making them complementary.

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bullet Supplementary Angles are two angles the sum of whose measures is 180º.
supp
Supplementary angles can be placed so they form a linear pair (straight line), or they may be two separate angles.
∠1 and ∠2 are supplementary.
P and ∠Q are supplementary.

The line through points A, B and C is a straight line.

theorem
Supplements of the same angle, or congruent angles, are congruent.
If m∠a is supplementary to the m∠b, and m∠c is supplementary to m∠b, then m∠a = m∠c.
Consider m∠a = 60º, m∠b = 120º and m∠c = 60º.

 


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