Classifying Triangles and Triangle Theorems Study Guide

Triangles are fundamentally classified based on two specific criteria: their internal angles and the lengths of their sides.
The first step in effectively classifying a triangle is mastering the associated vocabulary.

There are four primary terms used to classify triangles according to their internal angles:
- Acute Triangle: An acute triangle is defined as having three acute angles (angles measuring less than $90^\circ$ ).
- Right Triangle: The transcript refers to this as an "Awl qh + triangle," which is characterized by having exactly one right angle (measuring $90^\circ$ ) and two acute angles.
- Obtuse Triangle: An obtuse triangle contains exactly one obtuse angle (measuring greater than $90^\circ$ ) and two acute angles.
- Equiangular Triangle: Indicated in the transcript as an " © uwlangulac triangle," this refers to a triangle where all three internal angles are congruent (equal in measure).

There are three primary terms used to classify triangles by the relative lengths of their sides:
- Scolene Triangle: A Scolene triangle is defined as a triangle with no congruent sides (all sides have different lengths).
- SOSCeles Triangle: An SOSCeles triangle is defined as having at least two congruent sides.
- Equilateral Triangle: Referred to in the transcript as an "_@Qvi lake pak triangle," this classification describes a triangle where all three sides are congruent.

Case 1: A triangle with internal angles of $60^\circ$ , $60^\circ$ , and $60^\circ$ .
- Classification: Equicaguiar (Equiangular) and Equlateral (Equilateral).
Case 2: A triangle with angles of $90^\circ$ , $40^\circ$ , and $50^\circ$ .
- Classification: Right and Scolene (Scalene).
Case 3: A visual example of a triangle with one large angle and two equal sides.
- Classification: Olotuse (Obtuse) and SOSCeles (Isosceles).

The Triangle Sum Theorem states that the sum of the measures of the interior angles of any triangle is always exactly $180^\circ$ .
Fundamental formula for any triangle with angles $1$ , $2$ , and $3$ : $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$

Example 1: Finding a missing angle $m$
- Equation: $m + 35 + 81 = 180$
- Simplify: $m + 116 = 180$
- Solution: $m = 64$
Example 2: Finding a variable $x$ in a right triangle with angles $(5x + 12)$ , $(2x)$ , and $90^\circ$
- Equation: $5x + 12 + 2x + 90 = 180$
- Combine Like Terms: $7x + 102 = 180$
- Isolate Variable: $7x = 78$
- Solution: $x = \frac{78}{7}$

Practice 1: Finding the variable $w$ given angles $7w$ , $8w$ , and $(4w - 10)$ .
- Equation: $7w + 8w + 4w - 10 = 180$
- Simplify: $19w - 10 = 180$
- Combine: $19w = 190$
- Solution: $w = 10$
Practice 2: Finding variable $x$ with algebraic expressions:
- Equation: $x^2 + 3x + 10 = 180$ (interpreted from transcript logic and quadratic solution)
- Equation form: $x^2 + 3x - 70 = 0$
- Factoring: $(x + 10)(x - 7) = 0$
- Solutions: $x = -10$ and $x = 7$

Terminology:
- Legs: The two congruent (equal) sides of an isosceles triangle are specifically called its legs.
- Base: The third, non-necessarily congruent side is referred to as the base.
Angle Relationships: An isosceles triangle always contains congruent angles.
Primary Theorem: If two sides of a triangle are congruent, then the angles opposite those specific sides are also congruent.
Converse Theorem: If two angles of a triangle are congruent, then the sides opposite those angles must be congruent.

Variable Identification (Example A):
- Given values based on triangle properties: $c = 34$ and $b = 72$ .
Variable Identification (Example B):
- Given congruent properties: $k = y = 25$ .

Problem 1: Solving for $d$ in a triangle where the angles are related to the base angles of an isosceles structure.
- Setup based on the transcript's math summary: $4d + 2$ and $d$ .
- The equation provided in the work: $9d + 4 = 180$
- Subtracting constant: $9d = 176$
- Division results: $d = 19.5$ (rounded as per the transcript notation $a = t$ or similar scribble).