The Secrets of Triangles

The Secrets of Triangles

Discover the properties of triangles. Learn about their angles, sides, and relationships.

Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This rule helps determine if a given set of side lengths can form a "valid" triangle.

• Example: For a triangle with side lengths of 5, 7, and 10, the theorem holds true: 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5.

Multiple Choice

What does the Triangle Inequality Theorem state?

• The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

• The sum of the lengths of any two sides of a triangle must be equal to the length of the third side.

• The sum of the lengths of any two sides of a triangle must be less than the length of the third side.

• The sum of the lengths of any two sides of a triangle is irrelevant.

Triangle Inequality Theorem (Answer)

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem is important in determining if a set of side lengths can form a triangle. If the sum of the two shorter sides is not greater than the longest side, a triangle can't be formed.

Triangle Angles

In a triangle, the sum of all interior angles is always 180 degrees. The exterior angle is equal to the sum of the two opposite interior angles. Exterior angles are always greater than the other interior angles.

Multiple Choice

What is the relationship between exterior angles and interior angles in a triangle?

• Exterior angles are always equal to the corresponding interior angles.

• Exterior angles are always smaller than the corresponding interior angles.

• Exterior angles are always greater than the corresponding interior angles.

• Exterior angles have no relationship with the corresponding interior angles.

Exterior angles are greater...

Explained: In a triangle, the sum of all three exterior angles is always 360 degrees. This means that each exterior angle is greater than its corresponding interior angle by 180 degrees. Example: if an interior angle is 60 degrees, the corresponding exterior angle will be 240 degrees.

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Special Segments of Triangles

• Altitude: The perpendicular height of a triangle

• Median: A segment that bisects the side of a triangle

• Perpendicular Bisector: A perpendicular segment that bisects the side of a triangle

• Midsegment: A segment that bisects TWO aides of a triangle

• Angle Bisector: A segment that bisects an angle

Multiple Choice

Which line segment divides an angle into two congruent angles?

• Altitude

• Median

• Angle Bisector

• Perpendicular Bisector

Angle Bisector

TExplanation: An angle bisector is a line segment that divides an angle into two congruent angles. It is like its specific task in that it splits the angle evenly. An angle bisector is different from an altitude, median, or perpendicular bisector.

Types of Triangles

• Equilateral Triangle: All sides are equal in length
• Isosceles Triangle: Two sides are equal in length
• Scalene Triangle: No sides are equal in length
• Right Triangle: One angle is 90 degrees
• Obtuse Triangle: One angle is greater than 90 degrees
• Acute Triangle: All angles are less than 90 degrees

Multiple Choice

Which type of triangle has all sides equal in length?

• Equilateral Triangle

• Isosceles Triangle

• Scalene Triangle

• Right Triangle

Equilateral Triangle

Explanation: An equilateral triangle is a type of triangle that has all sides equal in length. It is also known as an equiangular triangle because all three angles are equal.

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PYTHAGOREAN THEOREM 101

• Pythagorean Theorem: A fundamental concept in geometry

• Relationship: Relates the lengths of the sides in a right triangle

• Formula: a² + b² = c². C is the hypotenuse

Multiple Choice

What is the formula for the Pythagorean Theorem?

• a² + b² = c²

• a² + b² = √c

• a + b = c

• a + b = √c

Pythagorean Theorem:

a² + b² = c² is the formula for the Pythagorean Theorem. It shows that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.