Angle Relationships

Complementary, supplementary, and vertical angles

Angle Relationships

Basic Angle Types

Acute angle: $0° < \theta < 90°$

Right angle: $\theta = 90°$

Obtuse angle: $90° < \theta < 180°$

Straight angle: $\theta = 180°$

Complementary Angles

Two angles are complementary if their sum is $90°$ .

Example: $35°$ and $55°$ are complementary because $35° + 55° = 90°$

Supplementary Angles

Two angles are supplementary if their sum is $180°$ .

Example: $120°$ and $60°$ are supplementary because $120° + 60° = 180°$

Vertical Angles

When two lines intersect, vertical angles are opposite each other.

Key Property: Vertical angles are always congruent (equal).

Linear Pair

Two adjacent angles that form a straight line.

Property: Linear pairs are supplementary (sum to $180°$ ).

Angle Addition Postulate

If point $B$ is in the interior of $\angle AOC$ , then: $m\angle AOB + m\angle BOC = m\angle AOC$

📚 Practice Problems

1Problem 1easy

❓ Question:

Two angles are complementary. One angle measures 35°. What is the measure of the other angle?

💡 Show Solution

Step 1: Recall the definition of complementary angles: Complementary angles are two angles whose measures add up to 90°

Step 2: Set up the equation: Let x = measure of the unknown angle 35° + x = 90°

Step 3: Solve for x: x = 90° - 35° x = 55°

Step 4: Verify: 35° + 55° = 90° ✓

Answer: The other angle measures 55°

2Problem 2easy

❓ Question:

Find the complement of a $42°$ angle.

💡 Show Solution

Complementary angles sum to $90°$ .

$90° - 42° = 48°$

Answer: $48°$

3Problem 3easy

❓ Question:

Two angles are supplementary. One angle is 3 times the measure of the other. Find both angles.

💡 Show Solution

Step 1: Recall supplementary angles: Supplementary angles add up to 180°

Step 2: Define variables: Let x = measure of the smaller angle Then 3x = measure of the larger angle

Step 3: Set up the equation: x + 3x = 180°

Step 4: Solve: 4x = 180° x = 45°

Step 5: Find both angles: Smaller angle: x = 45° Larger angle: 3x = 3(45°) = 135°

Step 6: Verify: 45° + 135° = 180° ✓ 135° = 3(45°) ✓

Answer: The angles are 45° and 135°

4Problem 4medium

❓ Question:

Two angles are supplementary. One angle measures $3x$ and the other measures $2x + 15$ . Find the value of $x$ .

💡 Show Solution

Supplementary angles sum to $180°$ .

$3x + (2x + 15) = 180$

$5x + 15 = 180$

$5x = 165$

$x = 33$

Answer: $x = 33°$

5Problem 5medium

❓ Question:

Two angles are vertical angles. If one angle measures (2x + 10)° and the other measures (3x - 20)°, find the value of x and the measure of each angle.

💡 Show Solution

Step 1: Recall vertical angles: Vertical angles are congruent (equal in measure)

Step 2: Set up the equation: 2x + 10 = 3x - 20

Step 3: Solve for x: 10 + 20 = 3x - 2x 30 = x x = 30

Step 4: Find the angle measures: First angle: 2x + 10 = 2(30) + 10 = 60 + 10 = 70° Second angle: 3x - 20 = 3(30) - 20 = 90 - 20 = 70°

Step 5: Verify: Both angles equal 70° ✓ (vertical angles are congruent)

Answer: x = 30, both angles measure 70°

6Problem 6medium

❓ Question:

Angles A and B are complementary. Angles B and C are supplementary. If angle A measures 28°, find the measures of angles B and C.

💡 Show Solution

Step 1: Use the complementary relationship: A + B = 90° 28° + B = 90° B = 90° - 28° B = 62°

Step 2: Use the supplementary relationship: B + C = 180° 62° + C = 180° C = 180° - 62° C = 118°

Step 3: Verify both relationships: A + B = 28° + 62° = 90° ✓ (complementary) B + C = 62° + 118° = 180° ✓ (supplementary)

Step 4: Visual understanding:

A and B are complementary (make a right angle together)
B and C are supplementary (make a straight line together)
B is shared between both relationships

Answer: Angle B = 62°, Angle C = 118°

7Problem 7hard

❓ Question:

Two lines intersect. One angle measures $(4x - 10)°$ and its vertical angle measures $(3x + 15)°$ . Find $x$ and the measure of both angles.

💡 Show Solution

Vertical angles are congruent, so:

$4x - 10 = 3x + 15$

$x = 25$

Angle measure: $4(25) - 10 = 100 - 10 = 90°$

Check: $3(25) + 15 = 75 + 15 = 90°$ ✓

Answer: $x = 25$ , both angles measure $90°$

8Problem 8hard

❓ Question:

Four angles are formed when two lines intersect. The angles can be labeled as angles 1, 2, 3, and 4 going clockwise. If angle 1 = (4x + 15)° and angle 3 = (6x - 25)°, find the measures of all four angles.

💡 Show Solution

Step 1: Identify the angle relationships: When two lines intersect, they form two pairs of vertical angles Angles 1 and 3 are vertical angles (opposite each other) Angles 2 and 4 are vertical angles (opposite each other)

Step 2: Use vertical angles property: Angle 1 = Angle 3 4x + 15 = 6x - 25

Step 3: Solve for x: 15 + 25 = 6x - 4x 40 = 2x x = 20

Step 4: Find angles 1 and 3: Angle 1 = 4x + 15 = 4(20) + 15 = 80 + 15 = 95° Angle 3 = 6x - 25 = 6(20) - 25 = 120 - 25 = 95°

Step 5: Find angles 2 and 4 using supplementary angles: Adjacent angles at intersection are supplementary Angle 1 + Angle 2 = 180° 95° + Angle 2 = 180° Angle 2 = 85°

Angle 4 = Angle 2 = 85° (vertical angles)

Step 6: Verify all relationships: Angles 1 and 3: 95° = 95° ✓ (vertical) Angles 2 and 4: 85° = 85° ✓ (vertical) Angles 1 and 2: 95° + 85° = 180° ✓ (supplementary) All four angles: 95° + 85° + 95° + 85° = 360° ✓

Answer: Angle 1 = 95°, Angle 2 = 85°, Angle 3 = 95°, Angle 4 = 85°

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