Chapter 4: Congruent Triangles

Classifications of Triangles by Angles
- Acute triangles has at least one acute angles.
- Equiangular triangles have 3 congruent acute angles.
- Obtuse triangles have 1 obtuse angle.
- Right triangle has 1 right angle.
Classifications of Triangles by Sides
- Equilateral triangle has 3 congruent sides.
- Isosceles triangles have AT LEAST 2 congruent sides.
- Scalene triangles has no congruent sides.

*An equilateral triangle is a special kind of isosceles triangle. You can also use the properties of isosceles and equilateral triangles to find missing values.*

Triangles are classified according to the number of congruent sides they have. To indicate that sides of a triangle are congruent, an equal number of hash marks is drawn on the corresponding sides.
Equilateral triangles are always equiangular triangles.
If the addition of 2 small sides of a triangle equals the hypotenuse then it is a right angle triangle.
If the addition of 2 small sides of a triangle is greater than the hypotenuse then it is an acute angle triangle.
If the addition of 2 small sides of a triangle is lesser than the hypotenuse then it is an obtuse triangle.

4.2 Angles of Triangles

Thanks to Lekhana Chennuru and Ishwari Veerkar

Triangle Angle Sum Theorem: The sum of the measure of the angles of a triangle is 180 degrees.
- The proof of the Triangle Angle Sum Theorem requires the use of an auxiliary line. An auxiliary line is an extra line or segment drawn in a figure to help analyze geometric relationships.
- In addition to its 3 interior angles, a triangle can have exterior angles formed by 1 side of the triangle and the extension of an adjacent side. Each exterior angle of a triangle has 2 remote interior angles that are not adjacent to the exterior angle.
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measure of the 2 remote interior angles.
A flow proof uses statements written in boxes and arrows to show the logical progression of an argument. Flow proofs can be written vertically or horizontally.
A corollary is a theorem with a proof that follows as a direct result of another theorem.
4.1 Corollary --> The acute angles of a right triangle are complementary.
4.2 Corollary --> There can be at most 1 right or obtuse angle in a triangle.

4.3 Congruent Triangles

Thanks to Lekhana Chennuru

If 2 geometric figures have exactly the same shape and size, they are congruent.
In 2 congruent polygons, all of the parts of 1 polygon are congruent to corresponding parts or matching parts of the other polygons. These corresponding parts include corresponding angles and corresponding sides.
Definition of congruent polygons: 2 polygons are congruent if and only if their corresponding are congruent.
"If and only if" means that both the conditional and its converse are true. For triangles, we say corresponding parts of congruent triangles are congruent or CPCTC.
Theorem 4.3 --> Third Angle Theorem: If 2 angles of 1 triangle are congruent to 2 angles of a second triangle, then the 3rd angles of the triangles are congruent.
When 2 triangles share a common side, use the reflexive property of congruence to establish that the common side is congruent to itself.
Theorem 4.4 --> Properties of Triangles Congruence: Reflexive Property-- Triangle ABC is congruent to triangle ABC. Symmetric Property -- If triangle ABC is congruent to triangle EFG, then triangle EFG is congruent to triangle ABC. Transitive Property -- If triangle ABC is congruent triangle EFG and triangle EFG is congruent to JKL, then triangle ABC is congruent to triangle JKL.

4.4 Proving Triangles Congruent -- SSS, SAS

Thanks to Lekhana Chennuru

SSS Congruence: If 3 sides of 1 triangle are congruent to 3 sides of a 2nd triangle, then the triangles are congruent.
SAS Congruence: If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of a 2nd triangle, then the triangles are congruent. The measures of 2 sides and a nonincluded angle are not sufficient to prove 2 triangles congruent.
Overlapping Figures: When triangles overlap, it can be helpful to draw each triangle separately and label the congruent parts.

4.5 Proving Triangles Congruent -- AAS, ASA

Thanks to Lekhana Chennuru

An included side is the side located between 2 consecutive angles of a polygon.
ASA (angle-side-angle) Congruence: If 2 angles and the included side of one triangle and the included side of another triangle are congruent, then the two triangles are congruent.
AAS (angle-angle-side) Congruence: If 2 angles and the non-included side of one triangle are congruent to the corresponding 2 angles and the side of a second triangle, then the 2 triangles are congruent.
You can use the congruent triangles to measure distances that are difficult to measure directly.
CONCEPT SUMMARY:

--> SSS; 3 pairs of corresponding sides are congruent.

--> SAS; 2 pairs of corresponding sides and their included angles are congruent.

--> ASA; 2 pairs of corresponding angles and their included sides are congruent.

--> AAS; 2 pairs of corresponding angles and the corresponding nonincluded sides are congruent.

4.6 Isosceles & Equilateral Triangles

Thanks to Lekhana Chennuru & Ishwari Veerkar

The 2 congruent sides are called the legs of an isosceles triangle.
The angle with sides that are legs is called the vertex angle.
The side of the triangle opposite to the vertex angle is called the base.
The 2 angles formed by the base and the congruent sides are called the base angles.
Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent.
Converse of Isosceles Triangle Theorem: If 2 angles of a triangle are congruent, then the sides opposite those angles are congruent.
A triangle is equilateral if and only if it is equiangular.

4.7 Congruence Transformations

Thanks to Lekhana Chennuru

A transformation is an operation that maps an original geometric figure, the preimage, onto a new figure called the image. A transformation can change the position, size or shape of a figure. A congruence transformation is noted by arrows.
A congruence transformation, also called a rigid transformation or an isometry, is one in which the position of the image may differ from that of the preimage but the 2 figures remain congruent.
KEY CONCEPT:

--> Reflection: A reflection or flip is a transformation over a line called the line of reflection. Each point of the preimage and its image are the same distance from the line of reflections.

--> Translation: A translation or slide is a transformation that moves all points of the original figure the same distance in the same direction.

--> Rotation: A rotation or a turn is a transformation around a forced point called the center fs rotation, through a specific angle and in a specific direction. Each point of the original and its image are the same distance from the center.

4.8 Triangles & Coordinate Proof

Thanks to Lekhana Chennuru

Coordinate proofs use figures in the coordinate plane and algebra to prove geometric concepts.
Right Angle --> The intersection of the x- and y-axis forms a right angle, so it is a convenient place to locate the right angle of a figure such as a right triangle.
After a triangle is placed on the coordinate plane and labeled, we can use coordinate proofs to verify properties and to prove theorems.
STEPS TO PLACE TRIANGLES ON COORDINATE PLANE:

--> Use the origin as a vertex or center of the triangle.

--> Place at least one side of a triangle on an axis.

--> Keep the triangle within the first quadrant if possible.

--> Use coordinates the make computations as simple as possible.