Proving the Angle-Angle-Side Theorem

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In summary, the Angle-Angle-Side theorem, or AAS, states that two triangles are congruent if two angles and the side between them are congruent to the corresponding angles and side of the other triangle. This can be proven using the fact that two pairs of angles are congruent, and the corresponding side is also congruent, resulting in congruent triangles by the ASA postulate.
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pholee95
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Hello everyone. I need help on proofs. I have to proof the Angle-Angle-Side theorem. Can someone help me with this?

The AAS states : If triangles ABC and DEF are two triangles such that angle ABC is congruent to angle DEF, angle BCA is congruent to angle EFD, and segment AC is congruent to DF, then triangles ABC and DEF are congruent.

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pholee95 said:
Hello everyone. i have to proof the Angle-Angle-Side theorem.
Can someone help me with this?

The AAS states : If triangles ABC and DEF are two triangles such that angle ABC is congruent to angle DEF,
angle BCA is congruent to angle EFD, and segment AC is congruent to DF,
then triangles ABC and DEF are congruent.
Since two pairs of angles are congruent, the third pair is also congruent.
. . That is: [tex]\angle BAC = \angle EDF.[/tex]

We have: [tex]\angle ACB = \angle DFE.\;AC = DF,\;\angle BAC = \angle EDF.[/tex]

The triangles are congruent by ASA.

.
 

FAQ: Proving the Angle-Angle-Side Theorem

What is the Angle-Angle-Side Theorem?

The Angle-Angle-Side Theorem, also known as the AAS Theorem, states that if two angles of one triangle are congruent to two angles of another triangle, and the included side is also congruent, then the two triangles are congruent.

How do you prove the Angle-Angle-Side Theorem?

To prove the Angle-Angle-Side Theorem, you must show that the two triangles have the same three corresponding parts: two angles and the included side. This can be done through the use of congruence postulates, such as the Side-Angle-Side (SAS) Congruence Postulate or the Angle-Side-Angle (ASA) Congruence Postulate.

What is the difference between the Angle-Angle-Side Theorem and the Side-Angle-Side Theorem?

The Angle-Angle-Side Theorem and the Side-Angle-Side Theorem (SAS) are both methods for proving congruence between two triangles. However, the SAS Theorem requires that both triangles have two sides and the included angle congruent, while the AAS Theorem only requires two angles and the included side to be congruent.

Can the Angle-Angle-Side Theorem be used to prove all triangles congruent?

No, the Angle-Angle-Side Theorem can only be used to prove congruence between two triangles. It cannot be used to prove congruence between more than two triangles, or between other types of polygons.

What are some real-world applications of the Angle-Angle-Side Theorem?

The Angle-Angle-Side Theorem can be used in various fields, such as engineering, architecture, and surveying, to ensure that structures and measurements are accurate and congruent. It can also be used in navigation and mapmaking, as well as in computer graphics and animation to create realistic and proportional images.

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