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°

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. The angles of a triangle add to \(180^\circ\).

First add the known angles:

\[ 40^\circ + 70^\circ = 110^\circ \]

Then subtract from \(180^\circ\):

\[ x = 180^\circ - 110^\circ = \boxed{70^\circ} \]

2. The side lengths are 6, 7, and 10.

All three sides have different lengths, so the triangle is:

\[ \boxed{\text{Scalene}} \]

3. The side lengths are 7, 7, and 7.

All three sides are equal, so the triangle is equilateral. An equilateral triangle also has three equal angles.

Since the angles sum to \(180^\circ\):

\[ 180^\circ \div 3 = 60^\circ \]

So the triangle has angles \(60^\circ\), \(60^\circ\), and \(60^\circ\) and is:

\[ \boxed{\text{Equilateral and acute}} \]

4. The angles are \(30^\circ\), \(30^\circ\), and \(120^\circ\).

Because one angle is greater than \(90^\circ\), the triangle is:

\[ \boxed{\text{Obtuse}} \]

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.