Geometry: Triangles Properties and Theorems

Triangles: Properties and Applications

Angle Properties

• Angle Sum Property: The sum of interior angles of a triangle is always 180 degrees.
• Exterior Angle Property: The exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it.

Pythagoras Theorem

• A² + B² = C², where A and B are the lengths of the legs of a right-angled triangle, and C is the length of the hypotenuse.
• Used to find the length of the hypotenuse or the length of one side of a right-angled triangle.

Types of Triangles

• Equilateral Triangle: A triangle with all sides of equal length.
• Isosceles Triangle: A triangle with two sides of equal length.
• Scalene Triangle: A triangle with all sides of different lengths.

Altitude and Median

• Altitude: A line passing through the vertex of a triangle and perpendicular to the opposite side.
• Median: A line passing through the midpoint of a side of a triangle and dividing the opposite side into two equal parts.
• Used to find the length of sides, altitude, and median of a triangle.

Real-Life Applications

• Pythagoras Theorem is used in construction, physics, and engineering to calculate distances and lengths.
• Triangles are used in architecture to design buildings and bridges.

Problem-Solving Strategies

• Use the Angle Sum Property to find the interior angle of a triangle when one interior angle and an exterior angle are given.
• Use the Exterior Angle Property to find the exterior angle of a triangle when two interior angles are given.
• Use Pythagoras Theorem to find the length of the hypotenuse or one side of a right-angled triangle.
• Use Altitude and Median to find the length of sides, altitude, and median of a triangle.

Higher-Order Thinking Skills

• Analyze the properties of triangles to solve complex problems.
• Apply the properties of triangles to real-life scenarios.
• Develop problem-solving strategies using triangles.
• Compare and contrast different types of triangles.

Word Problems

• A building is 15 meters tall, and the length of its shadow is 20 meters. Find the distance from the building to the point where the shadow ends.
• A carpenter needs to construct a roof with a triangular shape. If the length of the roof is 10 meters, and the width is 8 meters, find the height of the roof.
• A surveyor needs to measure the distance between two points. If the distance between the two points is 12 meters, and the angle between them is 60 degrees, find the length of the third side of the triangle.

Test your understanding of triangles, including angle properties and Pythagoras theorem. Learn about the sum of interior angles, exterior angle property, and the famous A² + B² = C² formula.