In this chapter, you will learn
- —Classify triangles by sides and angles
- —Understand and apply the angle sum property
- —Use the exterior angle property in problem-solving
- —Apply the triangle inequality property
- —Understand median, altitude, and their properties
Types of Triangles
Triangles can be classified based on their sides and angles.
Classification by Sides:
- Equilateral Triangle: All three sides equal, all angles 60°
- Isosceles Triangle: Two sides equal, two angles equal
- Scalene Triangle: All sides and angles different
Classification by Angles:
- Acute Triangle: All angles less than 90°
- Right Triangle: One angle exactly 90° (Right angle)
- Obtuse Triangle: One angle greater than 90°
Key Facts: A triangle has 3 sides, 3 angles, and 3 vertices. The sum of angles is always 180°.
Exam Tip
Remember the differences between classifications. A triangle can be both equilateral AND acute.
Common Mistake
Students confuse isosceles with equilateral. Isosceles has only 2 equal sides, equilateral has all 3.
Angle Sum Property of Triangles
Angle Sum Property: The sum of all three angles of a triangle is always 180°.
∠A + ∠B + ∠C = 180°
Applications:
- If two angles are known, find the third: ∠C = 180° - ∠A - ∠B
- In an equilateral triangle, each angle = 180° ÷ 3 = 60°
- In an isosceles triangle, the two equal angles are at the base
Example: In triangle ABC, if ∠A = 50° and ∠B = 70°, then ∠C = 180° - 50° - 70° = 60°
Exam Tip
The angle sum property is fundamental. Use it to find missing angles in almost every triangle problem.
Common Mistake
Students forget that ALL angles sum to 180°. Not just two angles.
Exterior Angle Property
Exterior Angle Property: An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
Exterior angle = Sum of two remote interior angles
Example: If triangle ABC has an exterior angle at C (formed by extending BC to D), then:
- ∠ACD = ∠A + ∠B
- Also: ∠ACD + ∠ACB = 180° (linear pair)
Relationship: An exterior angle is always greater than either of the remote interior angles.
Exam Tip
Exterior angle property is often used in multi-step problems. Identify the exterior angle carefully.
Common Mistake
Students mix up which angles are remote interior angles. Remote means non-adjacent angles.
Triangle Inequality Property
Triangle Inequality Property: The sum of any two sides of a triangle must be greater than the third side.
For a valid triangle with sides a, b, c:
- a + b > c
- b + c > a
- c + a > b
Application: To check if three lengths can form a triangle:
- Lengths: 3, 4, 5 → 3+4=7>5, 4+5=9>3, 5+3=8>4 ✓ Valid triangle
- Lengths: 1, 2, 3 → 1+2=3≮3 ✗ Cannot form triangle
Also Note: Difference of any two sides must be less than the third side: |a - b| < c
Exam Tip
Use triangle inequality to quickly check if three given lengths form a valid triangle.
Common Mistake
Students check only one condition instead of all three conditions.
Median and Altitude of Triangles
Median: A line segment from a vertex to the midpoint of the opposite side.
- A triangle has 3 medians (one from each vertex)
- All medians meet at a point called the centroid
- The centroid divides each median in 2:1 ratio
Altitude: A line segment from a vertex perpendicular to the opposite side (or extended opposite side).
- A triangle has 3 altitudes (one from each vertex)
- All altitudes meet at a point called the orthocenter
- In a right triangle, the orthocenter is at the right angle vertex
Key Differences:
- Median goes to the midpoint; Altitude is perpendicular
- Medians and altitudes may or may not be the same line
- In an equilateral triangle, median = altitude for each vertex
Exam Tip
Remember: Median to midpoint, Altitude perpendicular. Don't confuse these definitions.
Common Mistake
Students think median and altitude are always the same. They're only equal in equilateral triangles.
Chapter Summary
Properties of Triangles form the foundation of geometry. Key points:
- Triangle Types: Equilateral, Isosceles, Scalene (by sides); Acute, Right, Obtuse (by angles)
- Angle Sum: All three angles sum to 180°
- Exterior Angle: Equals sum of two remote interior angles
- Triangle Inequality: Sum of any two sides > third side
- Median: From vertex to midpoint of opposite side; meets at centroid
- Altitude: From vertex perpendicular to opposite side; meets at orthocenter
Exam Focus: Finding missing angles, verifying triangle validity, identifying triangle types, properties of median and altitude.