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Corresponding Parts of Congruent Triangles are Congruent - CPCTC
Triangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. Once you know that two triangles are congruent, you know that all corresponding parts are congruent.
Identify Congruence Transformations
If two triangles are congruent, you can slide, flip, or turn one of the triangles and they will still be congruent. These are called congruence transformations because they do not change the size or shape of the figure. It is common to use prime symbols to distinguish between an original ABC and a transformed triangle A'B'C'.
Proving Congruence -- SSS, SAS, ASA, AAS, HL
Triangles have special properties that allow you to use shortcuts for proving triangles congruent.
Side - Side - Side Postulate SSS
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. The Side - Side - Side Postulate lets you show that two triangles are congruent if you only know that the sides of one triangle are congruent to the sides of the second triangle.
Side - Angle - Side Postulate SAS
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
Angle - Side - Angle Postulate ASA
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Angle - Angle - Side Theorem AAS
If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.
Hypotenuse - Leg Congruence Theorem HL
In order to use the Hypotenuse - Leg Congruence Theorem, you must first verify that that the two triangles are right triangles.
Isosceles Triangles
An isosceles triangle has two congruent sides. The angle formed by these sides is called the vertex angle. The other two angles are called base angles. Here are the Isosceles Triangle Theorem and its converse:
- If two sides of a triangle are congruent, then the angles opposite those sides are congruent (Isosceles Triangle Theorem).
- If two angles of a triangle are congruent, then the sides opposite those angles are congruent (Converse of Isosceles Triangle Theorem).
Properties of Equilateral Triangles
An equilateral triangle has three congruent sides.
The Isosceles Triangle Theorem can be used to prove two properties of equilateral triangles:
- A triangle is equilateral if and only if it is equiangular.
- Each angle of an equilateral triangle measures 60 degrees.
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Triangle Congruence
