Complementary, Supplementary, and Vertical Angles - Sixth Grade Mathematics
When pairs of angles interact at intersections or within geometric shapes, their angle sums and spatial positions establish foundational algebraic relationships.
Complementary Angles
Two angles are called complementary angles if the sum of their degree measures equals exactly 90 degrees. Each angle is said to be the complement of the other.
Complementary Angles (Sum = 90°):
Adjacent Complements Non-Adjacent Complements
| |
| / Ray B | /
| / | / 35°
| / +-------
|/ 35° Angle 1 = 35°
+----------- Ray C
55° /
Ray A / 55°
*-------
Angle 1 + Angle 2 = 35° + 55° = 90° Angle 2 = 55°
(Form a right angle) 35° + 55° = 90°
Complementary angles do not need to share a side or vertex; they only need to sum to 90 degrees. To find the complement of an angle measuring x degrees: Complement = 90 - x. For example, the complement of 62 degrees is 90 - 62 = 28 degrees.
Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measures equals exactly 180 degrees. Each angle is said to be the supplement of the other.
Supplementary Angles (Sum = 180°):
Linear Pair (Adjacent & Supplementary):
Ray C
\
\
\
120° \ 60°
<----------*----------*---------->
Ray A Vertex B Ray D
Angle ABC + Angle CBD = 120° + 60° = 180° (Forms a straight line)
When two adjacent angles share a common ray and their non-common rays form a straight line, they are called a linear pair. All linear pairs are supplementary. To find the supplement of an angle measuring y degrees: Supplement = 180 - y. For example, the supplement of 115 degrees is 180 - 115 = 65 degrees.
Memory Trick: C and S
A helpful way to remember which is 90 degrees and which is 180 degrees: Letter C comes before S in the alphabet, just as 90 comes before 180 on the number line: C = Complementary = Corner (90 degrees). S = Supplementary = Straight line (180 degrees).
Vertical Angles
Vertical angles are the opposite angles formed when two straight lines intersect. Vertical angles are across from each other at the intersection vertex and share no common rays.
Vertical Angles at an Intersection:
Line 1
\ / Line 2
\ 1 /
\ /
4 \ / 2
X
/ \
/ \
/ 3 \
/ \
Angle 1 and Angle 3 are Vertical Angles: m(Angle 1) = m(Angle 3)
Angle 2 and Angle 4 are Vertical Angles: m(Angle 2) = m(Angle 4)
The Vertical Angles Theorem states that vertical angles are ALWAYS congruent (equal in measure). Proof: Angle 1 and Angle 2 form a straight line, so m(Angle 1) + m(Angle 2) = 180 degrees. Angle 3 and Angle 2 form a straight line, so m(Angle 3) + m(Angle 2) = 180 degrees. Subtracting m(Angle 2) from both equations proves that m(Angle 1) = m(Angle 3).
Solving Algebraic Equations with Angle Relationships
Geometric relationships allow us to set up and solve algebraic equations for unknown variables.
Problem: Angles with measures (3x + 10)° and (2x + 15)° are complementary. Find x.
Step 1: Write equation based on definition of complementary:
(3x + 10) + (2x + 15) = 90
Step 2: Combine like terms:
5x + 25 = 90
Step 3: Solve the two-step equation:
5x = 90 - 25
5x = 65
x = 13
Step 4: Check angle values:
Angle 1 = 3(13) + 10 = 39 + 10 = 49°
Angle 2 = 2(13) + 15 = 26 + 15 = 41°
49° + 41° = 90°. (Confirmed!)
Exercises
Exercise 1. Find the complement and supplement of each angle, or state if it is not possible: (a) 34 degrees; (b) 75 degrees; (c) 110 degrees; (d) 90 degrees.
Exercise 2. Two angles form a linear pair. If one angle measures 128 degrees, what is the measure of the second angle?
Exercise 3. Two straight lines intersect. One of the angles formed measures 47 degrees. Find the measures of all three remaining angles around the vertex.
Exercise 4. Angle A and Angle B are supplementary. Angle A is twice as large as Angle B. Write an equation with variable b for the measure of Angle B, and find the measures of both angles.
Exercise 5. Two vertical angles have algebraic expressions (4x - 12) degrees and (2x + 28) degrees. Set up an equation to find the value of x, and determine the degree measure of each angle.
Solutions
Solution to Exercise 1. For (a), the complement of 34 degrees is 90 - 34 = 56 degrees, and the supplement is 180 - 34 = 146 degrees. For (b), the complement of 75 degrees is 90 - 75 = 15 degrees, and the supplement is 180 - 75 = 105 degrees. For (c), 110 degrees is greater than 90 degrees, so it has no real positive complement; its supplement is 180 - 110 = 70 degrees. For (d), the complement of 90 degrees is 90 - 90 = 0 degrees (or none), and its supplement is 180 - 90 = 90 degrees.
Solution to Exercise 2. Because angles in a linear pair are supplementary, their sum is 180 degrees. The second angle measures 180 - 128 = 52 degrees.
Solution to Exercise 3. The angle vertically opposite the 47-degree angle is also 47 degrees because vertical angles are congruent. The two angles adjacent to it form linear pairs with it on straight lines, so each measures 180 - 47 = 133 degrees. Therefore, the four angles around the intersection are 47 degrees, 133 degrees, 47 degrees, and 133 degrees.
Solution to Exercise 4. Let b represent the measure of Angle B. Then Angle A is 2b. Because they are supplementary: 2b + b = 180, so 3b = 180. Dividing by 3 gives b = 60 degrees. Therefore, Angle B measures 60 degrees, and Angle A measures 2(60) = 120 degrees. Check: 120 + 60 = 180 degrees.
Solution to Exercise 5. Because vertical angles are congruent, set their expressions equal to each other: 4x - 12 = 2x + 28. Subtract 2x from both sides: 2x - 12 = 28. Add 12 to both sides: 2x = 40. Divide by 2: x = 20. Substitute x = 20 back into either expression: 4(20) - 12 = 80 - 12 = 68 degrees, and 2(20) + 28 = 40 + 28 = 68 degrees. Each angle measures 68 degrees.