Complementary and Supplementary Angles

1.Complementary Angles

Two angles are complementary if the sum of their measures is 90 degrees.

$$\angle A+\angle B=90^\circ$$

Each angle is called the complement of the other. They often form a right angle.

Example

One angle measures 35 degrees. Find its complementary angle.

Show solution

Step 1: Use the complementary formula: $$A + B = 90$$

Step 2: Substitute the known value: $$35 + B = 90$$

Step 3: Subtract 35 from 90 to find the missing measure: $$B = 55$$ degrees

Practice problems

Problem 1
Angle A and Angle B are complementary. If Angle A measures 42 degrees, what is the measure of Angle B?

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Answer: 48 degrees

Step 1: Set up the benchmark equation for complementary angles: $$A + B = 90$$
Step 2: Substitute 42 into the formula: $$42 + B = 90$$
Step 3: Subtract to solve for the missing variable: $$B = 90 - 42 = 48$$ degrees
Problem 2
An angle is exactly twice the size of its complement. Find the measure of the smaller angle.

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Answer: 30 degrees

Step 1: Define the two complementary angles algebraically: x and $$2x$$
Step 2: Combine them according to the definition: $$x + 2x = 90$$-> $$3x = 90$$
Step 3: Divide by 3 to isolate the value of x: $$x = 30$$ degrees
Problem 3
Find the complement of an angle that measures 15 degrees.

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Answer: 75 degrees

Step 1: Apply the complementary angles subtraction rule: $$B = 90 - 15$$
Step 2: Perform the arithmetic step: $$90 - 15 = 75$$
Step 3: Confirm the final target measurement: 75 degrees

Remember

If you know one angle, you can always find its complement by subtracting it from 90.

2.Supplementary Angles

Two angles are supplementary if the sum of their measures is 180 degrees.

$$\angle A+\angle B=180^\circ$$

Each angle is called the supplement of the other. They often form a linear pair (straight line).

Example

One angle measures 110 degrees. Find its supplementary angle.

Show solution

Step 1: Use the supplementary formula: $$A + B = 180$$

Step 2: Substitute the known value: $$110 + B = 180$$

Step 3: Subtract 110 from 180 to find the missing measure: $$B = 70$$ degrees

Practice problems

Problem 4
Two adjacent angles form a linear pair along a straight line. If one angle measures 75 degrees, find the missing supplementary angle value.

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Answer: 105 degrees

Step 1: Recall that a linear pair is supplementary: $$A + B = 180$$
Step 2: Substitute the known value into the equation: $$75 + B = 180$$
Step 3: Subtract 75 from 180 to calculate the value: $$B = 105$$ degrees
Problem 5
Angle X and Angle Y are supplementary angles. If Angle X measures 135 degrees, what is the value of Angle Y?

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Answer: 45 degrees

Step 1: Use the standard supplementary equation: $$X + Y = 180$$
Step 2: Isolate the target variable: $$Y = 180 - 135$$
Step 3: Calculate the difference: $$Y = 45$$ degrees
Problem 6
An engineer measures two angles that form a straight line. If the first angle is 60 degrees, find the measure of the second angle.

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Answer: 120 degrees

Step 1: Identify that a straight line line means the angles must sum to 180: $$A + B = 180$$
Step 2: Set up the subtraction: $$B = 180 - 60$$
Step 3: Compute the remaining value: $$B = 120$$ degrees

Remember

If you know one angle, you can always find its supplement by subtracting it from 180.

Complimentary Angles

Geometry - Angles

$$\angle A + \angle B = 90^\circ$$

Two angles are complementary when their measures add up to 90 degrees (forming a right angle).

Example

If $$\angle A = 35^\circ$$, find its complement ∠B.

Show solution

Step 1: Use formula $$\angle A + \angle B = 90^\circ$$.
Step 2: Substitute ∠A: $$35^\circ + \angle B = 90^\circ$$.
Step 3: Solve: $$\angle B = 90^\circ - 35^\circ = 55^\circ$$.

=> Conclusion: The complement is 55°.

Practice problems

Problem 7
Find the complement of an angle measuring 40°.
Show solution
Answer: 50°
- Step 1: Use complementary formula: Angle + Complement = 90°.
- Step 2: Complement = $$90^\circ - 40^\circ = 50^\circ$$.
=> Conclusion: 50°.
=> Correct Option: A (50°).
Problem 8
Two complementary angles are in the ratio $$2:3$$. Find the larger angle.
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Answer: 54°
- Step 1: Let the angles be $$2x$$ and $$3x$$.
- Step 2: $$2x + 3x = 90^\circ \Rightarrow 5x = 90^\circ \Rightarrow x = 18^\circ$$.
- Step 3: Larger angle $$= 3x = 3 \cdot 18^\circ = 54^\circ$$.
=> Conclusion: 54°.
=> Correct Option: A (54°).
Problem 9
Angle A is $$(2x + 10)$$° and Angle B is $$(3x + 5)$$°. If A and B are complementary, find x.
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Answer: $$x = 15$$
- Step 1: $$(2x + 10) + (3x + 5) = 90^\circ$$.
- Step 2: $$5x + 15 = 90^\circ \Rightarrow 5x = 75^\circ \Rightarrow x = 15$$.
=> Conclusion: $$x = 15$$.
=> Correct Option: B ($$x = 15$$)

Remember

The angles do not need to be adjacent (next to each other) to be complementary.
Each angle is called the complement of the other.

Supplementary Angles

Geometry - Angles

$$\angle A + \angle B = 180^\circ$$

Two angles are supplementary when their measures add up to 180 degrees (forming a straight line).

Example

If $$\angle A = 110^\circ$$, find its supplement ∠B.

Show solution

Step 1: Use formula $$\angle A + \angle B = 180^\circ$$.
Step 2: Substitute ∠A: $$110^\circ + \angle B = 180^\circ$$.
Step 3: Solve: $$\angle B = 180^\circ - 110^\circ = 70^\circ$$.

=> Conclusion: The supplement is 70°.

Practice problems

Problem 10
Find the supplement of an angle measuring 65°.
Show solution
Answer: 115°
- Step 1: Use supplementary formula: Angle + Supplement = 180°.
- Step 2: Supplement = $$180^\circ - 65^\circ = 115^\circ$$.
=> Conclusion: 115°.
=> Correct Option: B (115°).
Problem 11
An angle is 40° more than its supplement. Find the angle.
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Answer: 110°
- Step 1: Let the supplement be x. The angle is $$x + 40^\circ$$.
- Step 2: $$x + (x + 40^\circ) = 180^\circ \Rightarrow 2x + 40^\circ = 180^\circ \Rightarrow 2x = 140^\circ \Rightarrow x = 70^\circ$$.
- Step 3: The angle $$= 70^\circ + 40^\circ = 110^\circ$$.
=> Conclusion: 110°.
=> Correct Option: A (110°).
Problem 12
Two supplementary angles are $$(4x - 20)$$° and $$(2x + 80)$$°. Find the measure of the smaller angle.
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Answer: 60°
- Step 1: $$(4x - 20) + (2x + 80) = 180^\circ \Rightarrow 6x + 60 = 180^\circ \Rightarrow 6x = 120^\circ \Rightarrow x = 20$$.
- Step 2: Angle 1 $$= 4(20) - 20 = 60^\circ$$.
- Step 3: Angle 2 $$= 2(20) + 80 = 120^\circ$$.
=> Conclusion: Smaller angle is 60°.
=> Correct Option: B (60°).

Remember

The angles do not need to be together to be supplementary.
Each angle is called the supplement of the other.

Vertical and Adjacent Angles in Intersecting Lines

Geometry - Intersecting Lines

Vertical: $$\angle 1 = \angle 3$$, $$\angle 2 = \angle 4$$

Adjacent (Linear Pair): $$\angle 1 + \angle 2 = 180^\circ$$

When two lines intersect, vertical (opposite) angles are equal, and adjacent angles on a straight line are supplementary (linear pair).

Example

Two lines intersect. If one angle is 50°, find all other 3 angles.

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Step 1: Vertical angle to 50° is equal = 50°.
Step 2: Adjacent angle forms a linear pair: $$180^\circ - 50^\circ = 130^\circ$$.
Step 3: Vertical angle to 130° is equal = 130°.

=> Conclusion: The four angles are 50°, 130°, 50°, 130°.

Practice problems

Problem 13
Two intersecting lines form an angle of 72°. Find the vertical angle.
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Answer: 72°
- Step 1: Vertical angles are equal.
=> Conclusion: 72°.
=> Correct Option: B (72°).
Problem 14
Two lines intersect. One angle is $$(3x + 10)$$° and its vertical angle is $$(5x - 30)$$°. Find x.
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Answer: $$x = 20$$
- Step 1: Set vertical angles equal: $$3x + 10 = 5x - 30$$.
- Step 2: Solve for x: $$2x = 40 \Rightarrow x = 20$$.
=> Conclusion: $$x = 20$$.
=> Correct Option: A ($$x = 20$$)
Problem 15
Two intersecting lines form adjacent angles $$(2x + 15)$$° and $$(3x + 25)$$°. Find both angle measures.
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Answer: 71° and 109°
- Step 1: Adjacent angles on a straight line are supplementary: $$(2x + 15) + (3x + 25) = 180^\circ$$.
- Step 2: $$5x + 40 = 180^\circ \Rightarrow 5x = 140^\circ \Rightarrow x = 28$$.
- Step 3: Angle 1 $$= 2(28) + 15 = 71^\circ$$; Angle 2 $$= 3(28) + 25 = 109^\circ$$.
=> Conclusion: 71° and 109°.
=> Correct Option: B (71° and 109°).

Remember

Vertical angles are across from each other and ALWAYS equal.
Adjacent angles share a common ray and form a straight line (sum = 180°).