Two or an ext triangles that have the very same size and shape are called congruent triangles.

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The 4 triangles room congruent with each various other regardless whether they are rotated or flipped. The congruence of two objects is frequently represented making use of the prize "≅". In the number below, △ABC ≅ △DEF.

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As shown in the figure above, the lengths of the equivalent sides and measures that the matching angles do not have to be explicitly shown to indicate congruence. One equal number of tick marks deserve to be supplied to display that sides room congruent. Similarly, one equal number of arcs can be supplied to show that angles room congruent.

The matching congruent angles are: ∠A≅∠D, ∠B≅∠E, ∠C≅∠F.The equivalent congruent political parties are: AB≅DE, BC≅EF, AC≅DF.

Also, the corresponding vertices that the 2 triangles must be composed in order. So, △ABC≅△DEF could additionally be composed as △CBA≅△FED but not △BCA≅△DEF.

Determining congruence because that triangles

Two triangles must have actually the very same size and also shape for all sides and angles to be congruent, any one that the following comparisons can be offered to confirm the congruence the triangles.

Side-Side-Side (SSS)

If 3 sides the one triangle space congruent to three sides of another triangle, the 2 triangles room congruent.

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In the number above, AB≅DE, BC≅EF, AC≅DF. Because of this △ABC≅△DEF.

Side-Angle-Side (SAS)

If two sides and also the contained angle the one triangle are congruent to 2 sides and the included angle of an additional triangle, the two triangles room congruent.

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In the number above, AB≅DE, AC≅DF, and also ∠A≅∠D. Therefore, △ABC≅△DEF.

Angle-Side-Angle (ASA)

If two angles and also the had side of one triangle room congruent to 2 angles and also the had side of another triangle, the two triangles space congruent.

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In the figure above, ∠A≅∠D, ∠B≅∠E, and also AB≅DE.Therefore,△ABC≅△DEF.

Angle-Angle-Side (AAS)

If two angles and the non-included side of one triangle are congruent to 2 angles and the non-included next of another triangle, the two triangles space congruent.

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In the figure above, ∠D≅∠A, ∠E≅∠B, and BC≅EF. Therefore, △DEF≅△ABC.

The Angle-Angle-Side organize is a sport of the Angle-Side-Angle theorem. In the figure, since ∠D≅∠A, ∠E≅∠B, and the 3 angles of a triangle always include to 180°, ∠F≅∠C. This then becomes one Angle-Side-Angle comparison because ∠E≅∠B, ∠F≅∠C, and also BC≅EF.

Hypotenuse-Leg congruence

If the hypotenuse and also a leg of one ideal triangle room congruent to the hypotenuse and also a foot of one more triangle, the 2 triangles space congruent.

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In the number above, AC≅DF, AB≅DE, ∠B and also ∠E are ideal angles. Therefore, △ABC≅△DEF.

Angle-Side-Side (ASS)

If 2 sides and also the non-included angle of one triangle room congruent to two sides and also the non-included edge of an additional triangle, the two triangles room not constantly congruent.

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In the number above, AC≅DF, BC≅EF, ∠A≅∠D, however △ABC is not congruent come △DEF.

Angle-Angle-Angle (AAA)

If three angles the one triangle are congruent to 3 angles of one more triangle, the 2 triangles are not constantly congruent. As presented in the figure below, the dimension of 2 triangles deserve to be various even if the 3 angles room congruent.

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Corresponding parts

When 2 triangles are congruent, every their matching angles and also corresponding political parties (referred come as equivalent parts) room congruent.

Once it deserve to be displayed that 2 triangles are congruent using one of the above congruence methods, we additionally know that all corresponding parts of the congruent triangles room congruent (abbreviated CPCTC).


Example:

State the congruence because that the two triangles as well as all the congruent equivalent parts.

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Since two angles the △ABC are congruent to two angles that △PQR, the third pair of angles must additionally be congruent, so ∠C≅∠R, and △ABC≅△PQR through ASA.

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The corresponding congruent angles are: ∠A≅∠P, ∠B≅∠Q, ∠C≅∠R.