Learn about angles and their pairs through these examples with detailed step-by-step explanations.
Example 1: Types of Angles
Identify and describe different types of angles based on their measure.
Concept: Angles are formed by two rays sharing a common endpoint called the vertex. Angles are measured in degrees (°).
Type 1: **Acute Angle**
An angle whose measure is greater than 0° and less than 90°.
Example: An angle of 45°.
Type 2: **Right Angle**
An angle whose measure is exactly 90°. It is often marked with a square symbol at the vertex.
Example: The corner of a square.
Type 3: **Obtuse Angle**
An angle whose measure is greater than 90° and less than 180°.
Example: An angle of 130°.
Type 4: **Straight Angle**
An angle whose measure is exactly 180°. It forms a straight line.
Example: A straight line.
Type 5: **Reflex Angle**
An angle whose measure is greater than 180° and less than 360°.
Example: The larger angle formed by clock hands at 3:00.
Example 2: Adjacent Angles
Explain what adjacent angles are and provide an example.
Definition: Two angles are called adjacent angles if they have:
A common vertex.
A common arm.
Their non-common arms are on opposite sides of the common arm.
Example: Consider angles Angle ABC and Angle CBD.
Diagram showing angles ABC and CBD sharing vertex B and arm BC.
Angle ABC and Angle CBD are adjacent angles.
Note: Adjacent angles can be added to form a larger angle. For example, Angle ABD = Angle ABC + Angle CBD.
Example 3: Complementary Angles
Explain complementary angles and find the complement of a given angle.
Definition: Two angles are called complementary angles if the sum of their measures is 90°. The angles do not need to be adjacent to be complementary.
Example: Find the complement of an angle measuring 40°.
Step 1: Let the complement of the 40° angle be x.
Step 2: The sum of complementary angles is 90°. So, we have the equation: 40° + x = 90°.
Step 3: Solve for x: x = 90° - 40° = 50°.
Result: The complement of 40° is 50°.
Diagram:
Diagram showing two angles, 40° and 50°, adding up to a right angle.
40° + 50° = 90°
Example 4: Supplementary Angles
Explain supplementary angles and find the supplement of a given angle.
Definition: Two angles are called supplementary angles if the sum of their measures is 180°. The angles do not need to be adjacent to be supplementary.
Example: Find the supplement of an angle measuring 110°.
Step 1: Let the supplement of the 110° angle be y.
Step 2: The sum of supplementary angles is 180°. So, we have the equation: 110° + y = 180°.
Step 3: Solve for y: y = 180° - 110° = 70°.
Result: The supplement of 110° is 70°.
Diagram:
Diagram showing two angles, 110° and 70°, adding up to a straight angle.
110° + 70° = 180°
Example 5: Linear Pair of Angles
Explain what a linear pair of angles is and their property.
Definition: A linear pair of angles is a pair of adjacent angles whose non-common arms form a straight line.
Property: Angles in a linear pair are always supplementary. That is, the sum of the measures of angles in a linear pair is 180°.
Example: Consider angles Angle PQR and Angle RQS where PQS is a straight line.
Diagram showing angles PQR and RQS forming a straight line PQS.
Angle PQR and Angle RQS form a linear pair.
Relationship: Angle PQR + Angle RQS = 180°.
Example 6: Vertically Opposite Angles
Explain what vertically opposite angles are and their property.
Definition: Vertically opposite angles are formed when two lines intersect. They are the angles opposite to each other at the point of intersection.
Property: Vertically opposite angles are always equal in measure.
Example: Consider two lines AB and CD intersecting at point O.
Diagram showing lines AB and CD intersecting at O, forming angles AOC, COB, BOD, DOA.
Angle AOC and Angle BOD are vertically opposite angles.
Angle COB and Angle DOA are vertically opposite angles.
Relationship:
Angle AOC = Angle BOD
Angle COB = Angle DOA
Practice Mode
Enter a problem related to angle types or pairs of angles.
Note: This basic solver can identify angle types, find complements/supplements, or find a missing angle in a linear pair or vertically opposite pair (e.g., "type of 75", "complement of 30", "supplement of 120", "find x if linear pair angles are x and 50", "find x if vertically opposite to 70 is x").
Related Concepts
Explore these related mathematical concepts to deepen your understanding of angles.
Point
A location in space, represented by a dot.
Line
A straight path that extends infinitely in both directions.
Ray
A part of a line that has one endpoint and extends infinitely in one direction.
Line Segment
A part of a line that has two endpoints.
Vertex
The common endpoint where two rays or line segments meet to form an angle.