Triangle Fundamentals

Definition & Terminology

• Triangle (also called trilateral) - A polygon composed of three line segments joined end-to-end to form a closed, two-dimensional shape.

  • Each joining point is a vertex; plural = vertices.

  • Not every set of three lines forms a triangle; the segments must intersect only at their endpoints (no overlapping in the middle) and must not be collinear.

• Fundamental parts -

  • Side (line segment): One of the three segments forming the perimeter of the triangle.

  • Vertex / Vertices: The point where two sides meet, forming an angle. There are three vertices in a triangle.

  • Interior angle – The angle formed by two adjacent sides inside the triangle.

  • Exterior angle – An angle formed by extending one side of the triangle beyond its vertex; it forms a linear pair with its adjacent interior angle. This means the exterior angle and its adjacent interior angle sum to $180^{\circ}$ because they lie on a straight line.

• Special names in right & isosceles triangles -

  • Hypotenuse: The longest side of a right triangle, always located directly opposite the $90^{\circ}$ (right) angle.

  • Legs: The two sides that form the right angle in a right triangle.

  • Base (in an isosceles triangle): The side that is unequal in length to the two congruent sides (legs) of an isosceles triangle. The angles opposite the congruent sides are called base angles and are also congruent.

Fundamental Angle Facts

• Interior-angle sum theorem: The sum of the measures of the three interior angles of any triangle is always $180^{\circ}$. So, for a triangle with angles $\angle A, \angle B, \angle C$, we have $\angle A + \angle B + \angle C = 180^{\circ}$.

  • Proof sketch (presented in class):

    1. Draw a line through one vertex (e.g., $\angle B$) that is parallel to the opposite side (e.g., $\overline{AC}$).

    2. By using the properties of alternate-interior angles (formed by a transversal intersecting parallel lines), the angles formed by this new parallel line with the triangle's sides can be shown to be equal to two of the interior angles of the triangle.

    3. Together with the remaining third interior angle, these three angles form a linear pair on the straight parallel line, meaning their combined sum is $180^{\circ}$. This demonstrates that the sum of the original three interior angles is also $180^{\circ}$.

• Exterior-angle theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles (the interior angles that are not adjacent to the exterior angle).
  For an exterior angle $\angle 1$ adjacent to interior $\angle C$,
  $\angle 1 = \angle A + \angle B$
  (where $\angle A$ and $\angle B$ are the two remote interior angles).

• Angle vocabulary -

  • Complementary: Two angles are complementary if their sum is exactly $90^{\circ}$.

  • Supplementary: Two angles are supplementary if their sum is exactly $180^{\circ}$.

  • Explementary / full-turn: Two angles are explementary if their sum is exactly $360^{\circ}$ (labeled “explanatory/exam” in lecture).

Classification by Sides

• Scalene - A triangle in which all three side lengths are different from each other (e.g., sides measuring $5, 6, 7$ units). Consequently, all three interior angles are also different.

• Isosceles - A triangle that has exactly two equal sides (called legs) and one base. The angles opposite the two equal sides (base angles) are also congruent.

• Equilateral - A triangle in which all three sides are congruent (equal in length).

  • An equilateral triangle is automatically equiangular, meaning all three interior angles are also congruent, with each measuring exactly $60^{\circ}$ constantly.

Classification by Angles

• Acute - A triangle in which all three interior angles are acute, meaning each angle measures less than $90^{\circ}$.

• Right - A triangle that has exactly one interior angle measuring $90^{\circ}$. This angle is known as the right angle.

• Obtuse - A triangle that has exactly one interior angle that is obtuse, meaning this angle measures greater than $90^{\circ}$ but less than $180^{\circ}$.

• Equiangular - A triangle in which all three interior angles are congruent (equal in measure).

  • An equiangular triangle always has each interior angle measuring $60^{\circ}$ and is therefore always an acute triangle.

Connections Between Classifications

• A single triangle can (and should) be described by both side & angle type; this provides a more complete classification. For example:

  • “Acute-scalene”: A triangle where all angles are acute, and all sides are unequal in length.

  • “Obtuse-isosceles”: A triangle with one obtuse angle and two equal sides.

• Every equilateral triangle is inherently also equiangular (with $60^{\circ}$ angles) and therefore always acute.

Special Relationships & Theorems

• Pythagorean Theorem (applicable only to right triangles):
  This fundamental theorem states that the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the two legs ($a$ and $b$).
  The formula is: $a^{2}+b^{2}=c^{2}$
  where $c$ represents the hypotenuse (the side opposite the right angle), and $a$ and $b$ represent the lengths of the two legs.

• Complementary acute angles in a right triangle - The two non-right angles (the acute angles) in any right triangle are always complementary, meaning their sum is $90^{\circ}$.

  • Example derivation: Since the sum of all interior angles in a triangle is $180^{\circ}$, for a right triangle with a $90^{\circ}$ angle and two acute angles, say $\angle x$ and $\angle y$, we have $90^{\circ}+ \angle x+ \angle y =180^{\circ}$. Subtracting $90^{\circ}$ from both sides gives $\angle x+ \angle y =90^{\circ}$ as required.

Vocabulary Drill: Adjacent & Opposite Sides

• Adjacent sides to an angle: These are the two sides that directly form that specific angle (i.e., they meet at the vertex of the angle). For example, for $\angle A$ in triangle $ABC$, the adjacent sides are $\overline{AB}$ and $\overline{AC}$.

• Side opposite an angle: This is the side that does not touch the vertex of the angle and is directly across from it. For example, the side opposite $\angle A$ in triangle $ABC$ is $\overline{BC}$.

Sample Numerical Illustrations (mirrors classroom board work)

1. Demonstrating interior-angle sum with specific measures:

   • In a classroom setup, given alternate-interior pairs $\angle f =50^{\circ}$ and $\angle g =50^{\circ}$ (these would be angles formed by a parallel line drawn through a vertex and the triangle's sides). When combined with a third interior angle, $\angle a + \angle f + \angle c = 180^{\circ}$ (forming a linear pair on the parallel line), students verified that the sum of the original interior angles ($\angle a + \angle b + \angle c$) equals $180^{\circ}$.

2. Exterior-angle computation example:

   • If interior $\angle A=86^{\circ}$ and the exterior angle $\angle D=128^{\circ}$ (remote to $\angle A$ and $\angle B$), then using the exterior angle theorem ($\angle D = \angle A + \angle B$):
     $\angle B = \angle D-\angle A = 128^{\circ} - 86^{\circ} = 42^{\circ}$

   • To find the third interior angle $\angle C$, use the interior-angle sum theorem:
     $\angle C = 180^{\circ} - (\angle A + \angle B) = 180^{\circ} - (86^{\circ}+42^{\circ}) = 180^{\circ} - 128^{\circ} = 52^{\circ}$.

   • (Verification: $\angle C$ and $\angle D$ form a linear pair; $52^{\circ} + 128^{\circ} = 180^{\circ}$, confirming accuracy).

3. Algebraic setup involving angle relationships:

   • Given an equation like $x+65 = 2x+10$ (this could arise from setting two angles equal, e.g., vertical angles, alternate interior angles, or parts of a more complex diagram).

   • To solve for $x$:
     Subtract $x$ from both sides: $65 = x + 10$
     Subtract $10$ from both sides: $x = 55^{\circ}$;

   • Check: Substituting $x=55^{\circ}$ gives $55+65 = 120^{\circ}$ and $2(55)+10 = 110+10 = 120^{\circ}$, verifying the equality.

4. Right-triangle acute-angle ratio:

   • In a right triangle, the two acute angles are complementary (their sum is $90^{\circ}$).

   • If one acute angle is twice the other, let the angles be $x$ and $2x$.

   • Set up the equation: $x+2x=90^{\circ}$ ($3x=90^{\circ}$).

   • Solve for $x$: $x=30^{\circ}$.

   • The two acute angles are $30^{\circ}$ and $2(30^{\circ}) = 60^{\circ}$.

5. Remote-interior practice:

   • If an exterior angle of a triangle is $105^{\circ}$ and one of its remote interior angles is $72^{\circ}$.

   • Using the exterior-angle theorem ($Exterior\,Angle = Sum\,of\,two\,Remote\,Interior\,Angles$):
     $105^{\circ} = 72^{\circ} + x$
     Solve for $x$: $x = 105^{\circ}-72^{\circ}=33^{\circ}$, which is the measure of the other remote interior angle.

6. Linear-pair / algebra combo:

   • Given an algebraic problem like $z-81 + z = z+30$ (this equation would typically represent some relationship between angles, not necessarily a linear pair sum of $180^{\circ}$, but an equation to solve for $z$).

   • Solve for $z$:
     Combine terms: $2z - 81 = z + 30$
     Subtract $z$ from both sides: $z - 81 = 30$
     Add $81$ to both sides: $z=111^{\circ}$.

7. Multi-step exterior problem:

   • If an exterior angle is $114^{\circ}$. The adjacent interior angle ($\angle C$) would be $180^{\circ} - 114^{\circ} = 66^{\circ}$.

   • If one of the remote interior angles is given as $47^{\circ}$ ($\angle B = 47^{\circ}$), find the other remote interior angle ($\angle A$, or $x$).

   • Using the exterior angle theorem: $114^{\circ} = \angle A + 47^{\circ}$. So, $\angle A = 114^{\circ} - 47^{\circ} = 67^{\circ}$.

   • The sum of interior angles can be confirmed: $67^{\circ} + 47^{\circ} + 66^{\circ} = 180^{\circ}$, which is correct.

Reasoning Principles Emphasized

• Deductive reasoning over memorization: Students should always seek a proof or logical explanation for mathematical facts (the "difference between fact & fake").

• Ask “Why?” for every rule; e.g., why interior angles sum to $180^{\circ}$, why remote angles add to an exterior angle, etc. Understanding the underlying principles leads to deeper learning.

Upcoming Topics (announced)

• Triangle inequalities: Rules governing the possible lengths of sides in a triangle.

• Triangle congruence criteria: Methods to prove that two triangles are identical in shape and size (e.g., SSS, SAS, ASA, AAS, HL).

Suggested Practice / Assignment (from board)

• Solve exterior-angle & remote-angle diagrams similar to figures shared.

• Deadline: “tomorrow.”