CHAPTER 4: Triangles and Congruence Course Notes

Triangle Sums and Angle Classification

• Triangle Classifications by Sides:

  • Scalene: A triangle where all three sides are distinct lengths.

  • Isosceles: A triangle where at least two sides are congruent.

  • Equilateral: A triangle where all three sides are congruent. By definition, an equilateral triangle is also an isosceles triangle.

• Triangle Classifications by Angles:

  • Right: A triangle containing exactly one right angle ($90^\circ$).

  • Equiangular: A triangle where all three angles are congruent.

  • Acute: A triangle where all three angles measure less than $90^\circ$.

  • Obtuse: A triangle where one angle is greater than $90^\circ$.

• Triangle Sum Theorem:

  • Statement: The interior angles of any triangle sum to $180^\circ$.

  • Proof Summary:

    1. Given $\triangle ABC$ with a line $AD$ parallel to side $BC$.

    2. $\angle 1 \cong \angle 4$ and $\angle 2 \cong \angle 5$ by the Alternate Interior Angles Theorem.

    3. $m\angle 4 + m\angle 3 + m\angle 5 = 180^\circ$ via the Linear Pair Postulate and Angle Addition Postulate.

    4. By Substitution, $m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ$.

• Theorems Derived from Triangle Sums:

  • Theorem 4-1: Each angle in an equiangular triangle measures exactly $60^\circ$.

    • Calculation: $3x = 180^\circ \rightarrow x = 60^\circ$.

  • Theorem 4-2: The acute angles in a right triangle are always complementary ($m\angle 1 + m\angle 2 = 90^\circ$).

• Triangle Classifications by Sides:

  • Scalene: A triangle where all three sides are distinct lengths.

  • Isosceles: A triangle where at least two sides are congruent.

  • Equilateral: A triangle where all three sides are congruent. By definition, an equilateral triangle is also an isosceles triangle.

• Triangle Classifications by Angles:

  • Right: A triangle containing exactly one right angle ($90^\circ$).

  • Equiangular: A triangle where all three angles are congruent.

  • Acute: A triangle where all three angles measure less than $90^\circ$.

  • Obtuse: A triangle where one angle is greater than $90^\circ$.

• Key Geometric Definitions:

  • Interior Angles (in polygons): The angles located inside a closed figure with straight sides.

  • Vertex: The specific point where the sides of a polygon meet. A triangle possesses three interior angles, three vertices, and three sides.

• Investigation 4-1: Triangle Tear-Up:

  • Procedure: Draw a triangle with three different angle sizes. Label the interior angles as $\angle 1, \angle 2, \text{ and } \angle 3$. Tear off the angles and align their points.

  • Outcome: The three angles fit together to form a straight line, representing a straight angle.

  • Conclusion: The interior angles of a triangle add up to $180^\circ$.

• Triangle Sum Theorem:

  • Statement: The interior angles of any triangle sum to $180^\circ$.

  • Proof Summary:

    1. Given $\triangle ABC$ with a line $AD$ parallel to side $BC$.

    2. $\angle 1 \cong \angle 4$ and $\angle 2 \cong \angle 5$ by the Alternate Interior Angles Theorem.

    3. $m\angle 4 + m\angle 3 + m\angle 5 = 180^\circ$ via the Linear Pair Postulate and Angle Addition Postulate.

    4. By Substitution, $m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ$.

• Theorems Derived from Triangle Sums:

  • Theorem 4-1: Each angle in an equiangular triangle measures exactly $60^\circ$.

    • Calculation: $3x = 180^\circ \rightarrow x = 60^\circ$.

  • Theorem 4-2: The acute angles in a right triangle are always complementary ($m\angle 1 + m\angle 2 = 90^\circ$).

Exterior Angles and Theorems

• Exterior Angle Definition: The angle formed by extending one side of a polygon. It is adjacent to an interior angle and forms a linear pair with it.

  • Vertical Angles: At each vertex, the two possible exterior angles are vertical angles and are therefore congruent.

• Exterior Angle Sum Theorem:

  • Statement: Each set of exterior angles (one at each vertex) of a polygon sums to $360^\circ$.

  • Example 6 Calculation: For exterior angles $92^\circ$ and $123^\circ$, the third exterior angle is $360^\circ - (92^\circ + 123^\circ) = 145^\circ$.

• Exterior Angle Theorem:

  • Definition of Remote Interior Angles: The two interior angles of a triangle that are not adjacent to the indicated exterior angle.

  • Theorem Statement: The sum of the two remote interior angles is equal to the measure of the non-adjacent exterior angle.

  • Proof Summary:

    1. $m\angle A + m\angle B + m\angle ACB = 180^\circ$ (Triangle Sum Theorem).

    2. $m\angle ACB + m\angle ACD = 180^\circ$ (Linear Pair Postulate).

    3. By Transitive Property, $m\angle A + m\angle B + m\angle ACB = m\angle ACB + m\angle ACD$.

    4. By Subtraction, $m\angle A + m\angle B = m\angle ACD$.

• Example 9 (Algebra Connection):

  • Interior angles: $(4x + 2)^\circ$ and $(2x - 9)^\circ$.

  • Exterior angle: $(5x + 13)^\circ$.

  • Equation: $(4x + 2) + (2x - 9) = 5x + 13 \rightarrow 6x - 7 = 5x + 13 \rightarrow x = 20$.

  • Angle Measures: Interior angles are $82^\circ$ and $31^\circ$; exterior angle is $113^\circ$. Check: $82 + 31 = 113$.

Congruent Figures and Properties

• Definition of Congruent Triangles: Two triangles are congruent ($\cong$) if and only if they have exactly the same size and shape. This requires all three pairs of corresponding angles and all three pairs of corresponding sides to be congruent.

• CPCTC: An acronym for Corresponding Parts of Congruent Triangles are Congruent. This principle is applied after proving two triangles are congruent.

• Congruence Statements:

  • Order is critical. For example, if $\triangle ABC \cong \triangle LMN$, then $A$ corresponds to $L$, $B$ to $M$, and $C$ to $N$.

  • A single congruence can be written in six ways (e.g., $\triangle ABC \cong \triangle LMN$, $\triangle BAC \cong \triangle MLN$, etc.).

• Third Angle Theorem:

  • If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also be congruent.

  • Note: While the angles are congruent, this theorem alone does not prove triangle congruence, as side lengths could differ.

• Congruence Properties (from Chapter 2):

  • Reflexive Property: Any shape is congruent to itself (e.g., $\triangle ABC \cong \triangle ABC$).

  • Symmetric Property: If $\triangle ABC \cong \triangle DEF$, then $\triangle DEF \cong \triangle ABC$.

  • Transitive Property: If $\triangle ABC \cong \triangle DEF$ and $\triangle DEF \cong \triangle GHI$, then $\triangle ABC \cong \triangle GHI$.

Proving Triangle Congruence (SSS and SAS)

• Side-Side-Side (SSS) Postulate:

  • If three sides in one triangle are congruent to three sides in another triangle, then the triangles are congruent.

  • Investigation 4-2 shows that given three specific side lengths, only one unique triangle can be constructed.

• Side-Angle-Side (SAS) Postulate:

  • If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the triangles are congruent.

  • Included Angle: The angle located between the two given sides.

  • Investigation 4-3 confirms that only one triangle can be created from two lengths and an included angle.

• SSS in the Coordinate Plane:

  • To prove congruence, use the Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

  • Calculate the length of all three sides for both triangles. If the side lengths match, the triangles are congruent by SSS.

• Real-World Application (Kitchen Triangles):

  • The triangle between the sink, refrigerator, and oven is ideally equilateral. In the "Know What?" scenario, the SSS postulate was used to determine the triangles were not congruent because one side (fridge to stove) was $4\,ft$ in one house and $4.5\,ft$ in another.

Triangle Congruence using ASA, AAS, and HL

• Angle-Side-Angle (ASA) Postulate:

  • If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the triangles are congruent.

  • Included Side: The side between the two given angles. Investigation 4-4 proves this produces a unique triangle.

• Angle-Angle-Side (AAS) Theorem:

  • If two angles and a non-included side in one triangle are congruent to two corresponding angles and a non-included side in another triangle, then the triangles are congruent.

  • This is a variation of ASA, validated by the Third Angle Theorem.

• Hypotenuse-Leg (HL) Congruence Theorem:

  • Applies only to right triangles.

  • If the hypotenuse and one leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, the triangles are congruent.

  • This relies on the Pythagorean Theorem: $(\text{leg})^2 + (\text{leg})^2 = (\text{hypotenuse})^2$, which ensures the third side must also be congruent.

• Non-Congruence Relationships:

  • AAA (Angle-Angle-Angle): All angles are congruent, but the triangles are similar, not necessarily congruent (different sizes).

  • SSA (Side-Side-Angle): Does not prove congruence. It is possible to draw two different triangles with the same two sides and a non-included angle.

Isosceles and Equilateral Triangles

• Isosceles Triangle Properties:

  • Legs: The two congruent sides.

  • Base: The third side.

  • Base Angles: The angles adjacent to the base.

  • Vertex Angle: The angle between the two legs.

• Base Angles Theorem:

  • The base angles of an isosceles triangle are congruent.

  • Converse: If two angles in a triangle are congruent, the opposite sides are congruent.

• Isosceles Triangle Theorem:

  • The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular bisector to the base.

  • Converse: The perpendicular bisector of the base is also the vertex angle bisector.

• Equilateral and Equiangular Triangles:

  • Equilateral Triangle Theorem: Every equilateral triangle is equiangular, with each angle measuring $60^\circ$.

  • Investigation 4-6: Constructing an equilateral triangle using a compass and ruler by maintaining consistent widths for arcs.

Review Questions & Discussion

• Bermuda Triangle Myth: The Bermuda Triangle is classified as an acute scalene triangle. While maps (flat) show specific measures, those measures change in reality because the Earth is curved.

• Quilt Patterns: A quilt pattern in the text is composed of 16 "A" triangles and 16 "B" triangles. The "A" triangles are $\frac{1}{32}$ of the total square, and "B" triangles are $\frac{1}{128}$ of the square. Both are right triangles, and all shared sets are congruent.

• Bathroom Tile Geometry: In a tile pattern of blue and green equilateral triangles forming a hexagon:

  • The dark blue outlined figure is an equiangular equilateral hexagon.

  • Each angle of the hexagon is $120^\circ$ (comprised of two $60^\circ$ angles).

  • The sum of angles in the hexagon is $720^\circ$.

  • Degrees around a central point where six triangles meet total $360^\circ$ ($6 \times 60^\circ$).

• Review Queue Answers Summary:

  1. A straight angle contains $180^\circ$.

  2. Triangle sum equations: $(5x+2) + (4x+3) + (3x-5) = 180^\circ \rightarrow 12x = 180 \rightarrow x = 15$.

  3. Distance formula examples: distance between ($-1, 5$) and ($4, 12$) is $\sqrt{74}$.