Triangle Congruence

Get Started. It's Free
or sign up with your email address
Triangle Congruence by Mind Map: Triangle Congruence

1. Coordinate Proofs

1.1. Setup

1.1.1. Coordinate Proofs should be set up where the shape has a side touching the x-axis, and where the shape also has a side touching the y-axis or is centered horizontally on the y-axis.

1.2. Angles

1.2.1. Interior Angles are formed by to sides of the triangle.

1.2.2. Exterior Angles are formed by one side of the triangle and an extension of another side.

1.3. Auxiliary Lines

1.3.1. Auxiliary Lines are lines that are drawn to prove something in a proof, but are not lines that are originally part of the question.

1.4. Note: In theoretical questions, the coordinates of vertices should be represented by variables (with coefficients).

2. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

2.1. When two triangles are congruent, their corresponding sides and angles are congruent.

3. Triangle Names/Types

3.1. Sides

3.1.1. Equilateral

3.1.1.1. If all sides are congruent, the triangle is equilateral.

3.1.1.2. If all angles are congruent, the triangle is equiangular, but it is also equilateral.

3.1.2. Isosceles

3.1.2.1. If at least two sides are congruent, the triangle is isosceles.

3.1.2.2. If at least two angles are congruent, the triangle is isosceles.

3.1.3. Scalene

3.1.3.1. If no sides are congruent, the triangle is scalene.

3.1.4. Note: All equilateral triangles are isosceles.

3.2. Angles

3.2.1. Acute

3.2.1.1. If there are only acute angles, the triangle is acute.

3.2.1.2. Acute angles are less than 90 degrees.

3.2.2. Right

3.2.2.1. If there is one right angle, the triangle is right.

3.2.2.2. Right angles are 90 degrees.

3.2.3. Obtuse

3.2.3.1. If there is one obtuse angle, the triangle is obtuse.

3.2.3.2. Obtuse angles are more than 90 degrees

3.2.4. Note: All other angles of the triangle will be acute.

4. Proving Triangle Congruence

4.1. Methods

4.1.1. SSS

4.1.1.1. If all sides are congruent, the triangles are congruent.

4.1.2. SAS

4.1.2.1. If two sides and the included angle are congruent, the triangles are congruent.

4.1.3. ASA

4.1.3.1. If two angles and the included side are congruent, the triangles are congruent.

4.1.4. AAS

4.1.4.1. If two sides and a non-included angle are congruent, the triangles are congruent.

4.1.5. HL

4.1.5.1. If the hypotenuse and a one leg of two right triangles are congruent, then the triangles are congruent.

5. Some Theorems

5.1. If a triangle is equilateral, then it is equiangular.

5.2. If a triangle is equiangular, then it is equilateral.

5.3. If a triangle is equilateral, then all of its angles are 60 degrees.

5.4. If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

5.5. If two angles of a triangle are congruent, then the sides opposite those angle are congruent.