Triangle Basics

TipLearning Objectives
  • Classify triangles by angles (acute, right, obtuse) and by sides (scalene, isosceles, equilateral).
  • Use the Triangle Angle Sum Theorem.
  • Identify key terminology: vertices, interior angles, sides.
  • Understand properties of special triangles.

Key Ideas

  • A triangle has 3 sides, 3 angles, and the sum of its angles is: \[ 180^\circ \]
  • Classifications:

By Angles

  • Acute: all angles < 90°
  • Right: one angle = 90°
  • Obtuse: one angle > 90°

By Sides

  • Scalene: all sides different
  • Isosceles: two sides equal → base angles equal
  • Equilateral: all sides equal → all angles 60°

Diagram showing triangle classifications by angles (acute, right, obtuse) and by sides (scalene, isosceles with equal-side ticks, and equilateral with all sides marked equal).

Common Problem Types

Using Angle Sum

Find missing angle using
\[A + B + C = 180^\circ.\]

Classifying by Angles

Given angle measures → determine type.

Classifying by Sides

Given all side lengths → determine type.

Isosceles Base-Angle Properties

Equal sides → equal base angles.

Checking for Valid Triangles

Triangle Inequality:
\[ a + b > c, \; b + c > a, \; a + c > b. \]

Strategies

  • Draw and label triangles before solving.
  • Check whether the triangle is isosceles — base angles often match.
  • Use the angle sum as the first step in finding missing angles.

Worked Examples

Example 1 — Angle Sum

Angles are 50° and 65°. Find the third angle. \[ 180 - (50 + 65) = 65^\circ \]

Example 2 — Classifying by Sides

Side lengths: 5, 5, 8 → isosceles.

WarningCommon Mistakes
  • Forgetting angle sum = 180°.
  • Assuming sides are equal without checking.
  • Mixing isosceles with equilateral classifications.
  • Violating the triangle inequality.

Practice Problems

  1. Triangle has angles 40°, 70°, and x°. Find x.
  2. Sides: 6, 7, 10 → classify by sides.
  3. Sides: 7, 7, 7 → classify fully.
  4. Angles: 30°, 30°, 120° → classify by angles.
  1. \(180 - 110 = 70^\circ\)
  2. Scalene
  3. Equilateral → 60°, 60°, 60°
  4. One angle > 90° → obtuse

Summary

  • All triangle angles sum to 180°.
  • Classify by sides and angles separately.
  • Isosceles → equal base angles; equilateral → 60° each.
  • Always check triangle inequality.
  • For isosceles, mark the equal sides immediately.
  • Equilateral is always acute.