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Name the postulate or theorem that you can use to show the triangles are congruent. Then explain why the statement is true.

a) The Angle-Angle-Side (AAS) Postulate states that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. This is true because when two angles and a non-included side are congruent, the corresponding parts of the triangles must be congruent, ensuring congruence of the whole triangles.

b) The Side-Angle-Side (SAS) Theorem states that 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. This is true because if the two sides and the included angle are congruent, the remaining parts of the triangles are forced to be congruent as well, making the entire triangles congruent.

c) The Hypotenuse-Leg (HL) Theorem applies specifically to right triangles. It states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. This is true because the hypotenuse and one leg uniquely define a right triangle, so if these parts are congruent, the entire right triangles must be congruent.

d) The Corresponding Angles Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar, not necessarily congruent. Therefore, this postulate does not apply to proving congruence.

Answer :

Final answer:

The AAS Postulate, SAS Theorem and HL Theorem are some tools used to determine if triangles are congruent by comparing their components. These work because congruent parts ensure congruence of entire triangles. The Corresponding Angles Postulate, however, only affirms triangle similarity.

Explanation:

In geometry, you can use postulates and theorems to determine if triangles are congruent. The Angle-Angle-Side (AAS) Postulate says if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, the triangles are congruent. This is because congruent parts indicate whole triangle congruence. The Side-Angle-Side (SAS) Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. This stems from the fact that congruent parts force the remaining parts, hence, the entire triangles to be congruent. Continuing, the Hypotenuse-Leg (HL) Theorem applies specifically to right triangles. It ascertains if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. This is ascribed to the unique definition of a right triangle by the hypotenuse and one leg. However, the Corresponding Angles Postulate, while indicating similarity between two triangles, does not authenticate triangle congruence.

Learn more about Triangle Congruence here:

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