Grade 9 Triangles

Triangle: A polygon with three sides and three angles.

Scalene Triangle: A triangle in which all three sides have different lengths. All three angles also have different measures.

Isosceles Triangle: A triangle in which at least two sides are equal in length. The angles opposite the equal sides are also equal (base angles).

Equilateral Triangle: A triangle in which all three sides are equal in length. All three angles are also equal, each measuring 60 degrees. An equilateral triangle is also an isosceles triangle.

Acute Triangle: A triangle in which all three angles are acute angles (each measuring less than 90 degrees).

Right Triangle: A triangle in which one angle is a right angle (measuring exactly 90 degrees). The side opposite the right angle is called the hypotenuse.

Obtuse Triangle: A triangle in which one angle is an obtuse angle (measuring more than 90 degrees but less than 180 degrees).

Note: A triangle can have at most one right angle or one obtuse angle. The other two angles must be acute.

Property: The sum of the interior angles of any triangle is always 180 degrees.
If the angles of a triangle are Angle A, Angle B, and Angle C, then Angle A + Angle B + Angle C = 180 degrees.

Example Problem: In triangle ABC, Angle A = 50 degrees, Angle B = 60 degrees. Find Angle C.

Step 1: Use the Angle Sum Property: Angle A + Angle B + Angle C = 180 degrees.

Step 2: Substitute the given values into the equation.
50 degrees + 60 degrees + Angle C = 180 degrees.

Step 3: Simplify and solve for Angle C.
110 degrees + Angle C = 180 degrees.
Angle C = 180 degrees - 110 degrees.

Step 4: Calculate the value of Angle C.
Angle C = 70 degrees.

Result: The measure of Angle C is 70 degrees.

Exterior Angle: When a side of a triangle is extended, the angle formed outside the triangle is called an exterior angle.

Interior Opposite Angles: For an exterior angle, the two interior angles of the triangle that are not adjacent (next) to it are called interior opposite angles.

Property: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.
Example: If side BC of triangle ABC is extended to point D, forming exterior angle ACD, then Angle ACD = Angle BAC + Angle ABC.

Congruent Figures: Figures that have exactly the same shape and the same size are called congruent figures.

Congruent Triangles: Two triangles are congruent if and only if their corresponding sides and corresponding angles are equal.
If triangle ABC is congruent to triangle PQR (written as triangle ABC $\cong$ triangle PQR), then:
AB = PQ, BC = QR, AC = PR (Corresponding Sides)
Angle A = Angle P, Angle B = Angle Q, Angle C = Angle R (Corresponding Angles)

CPCTC: Corresponding Parts of Congruent Triangles are Congruent. This is a key principle used to prove parts of triangles are equal once the triangles are proven congruent.

Congruence Criteria: We don't need to check all six pairs of corresponding parts to prove congruence. There are specific criteria (like SSS, SAS, ASA, AAS) that use fewer parts.

SSS (Side-Side-Side) Congruence: If three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.

SAS (Side-Angle-Side) Congruence: If two sides and the included angle (the angle between the two sides) of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.

ASA (Angle-Side-Angle) Congruence: If two angles and the included side (the side between the two angles) of one triangle are equal to the two corresponding angles and the included side of another triangle, then the two triangles are congruent.

AAS (Angle-Angle-Side) Congruence: If two angles and a non-included side of one triangle are equal to the two corresponding angles and the corresponding non-included side of another triangle, then the two triangles are congruent. (Note: AAS is often considered a consequence of ASA and Angle Sum Property).