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MathChapter 17: Triangles

Notes

Angles of a Triangle Notes

Use triangle angle sum, exterior angles, and both directions of the isosceles theorem in numeric and algebraic problems.

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Triangle-angle problems become manageable when every marked fact is translated into one precise theorem. Never estimate a measure from appearance.

The Angle Sum Theorem

Interior-angle sum
\[m\angle A+m\angle B+m\angle C=180^\circ\]

Every nondegenerate triangle has three positive interior angles whose measures total exactly 180 degrees.

Triangle angle sumTriangle ABC has angles A equals 47 degrees, B equals 68 degrees, and C equals 65 degrees; the three measures total 180 degrees.
Triangle angle sumTriangle ABC has angles A equals 47 degrees, B equals 68 degrees, and C equals 65 degrees; the three measures total 180 degrees.47°68°65°ABC47° + 68° + 65° = 180°

Unknown-angle workflow

  1. Identify

    List the three interior angles, including any algebraic expressions.

  2. Equation

    Set their sum equal to (180^\circ).

  3. Solve

    Find the variable, then substitute if the question asks for an angle.

  4. Validate

    Confirm every angle lies strictly between (0^\circ) and (180^\circ).

Worked example

Algebraic angle sum

A triangle has angles (2x+8), (3x-4), and (x+20) degrees. Find the largest angle.

  1. Sum

    ((2x+8)+(3x-4)+(x+20)=180).

  2. Solve

    (6x+24=180), so (x=26).

  3. Evaluate

    The angles are (60^\circ,74^\circ,46^\circ).

The largest angle is (74^\circ).

Exterior angles

Exterior Angle TheoremTriangle ABC has side AC extended past C. The exterior angle at C equals the sum of remote interior angles A and B, while interior angle C is adjacent to it.
Exterior Angle TheoremTriangle ABC has side AC extended past C. The exterior angle at C equals the sum of remote interior angles A and B, while interior angle C is adjacent to it.113°remote: 42° + 71° = 113°42°71°67°ABC

Figure not drawn to scale.

Two valid routes

Remote-angle route

Use (E=A+B) when the remote angles are given.

Linear-pair route

Use (E+C=180^\circ) when the adjacent interior angle is given.

Mini check

Remote, not adjacent

Remote interior angles are (38^\circ) and (77^\circ). What is the exterior angle?

  1. (115^\circ)
  2. (65^\circ)
Show answer and explanation

Answer: (115^\circ).

Add the two remote angles: (38+77=115).

Isosceles triangles: theorem and converse

Isosceles triangle theoremIsosceles triangle ABC has congruent legs AB and BC marked with ticks; the opposite base angles A and C are marked congruent.
Isosceles triangle theoremIsosceles triangle ABC has congruent legs AB and BC marked with ticks; the opposite base angles A and C are marked congruent.base anglevertex anglebase angleABBCACABC
Triangle-angle theorem summary
GivenConcludeReason
Three triangle anglesSum is (180^\circ)Angle Sum Theorem
One exterior angleEquals two remote interior anglesExterior Angle Theorem
Two congruent sidesOpposite angles are congruentIsosceles Triangle Theorem
Two congruent anglesOpposite sides are congruentConverse of Isosceles Triangle Theorem

Vertex-bisector corollary

Isosceles vertex-bisector corollaryIn isosceles triangle ABC, segment BD bisects the vertex angle, bisects base AC, and is perpendicular to AC; matching marks show every stated relationship.
Isosceles vertex-bisector corollaryIn isosceles triangle ABC, segment BD bisects the vertex angle, bisects base AC, and is perpendicular to AC; matching marks show every stated relationship.leglegDangle bisector + perpendicular bisectorABC
Key takeaways

Key takeaways

  • Triangle interiors total (180^\circ).
  • An exterior angle uses the two remote interiors.
  • Equal sides and equal opposite angles are two directions of the isosceles theorem.
  • The vertex-bisector corollary requires an isosceles triangle and a bisected vertex angle.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.