The four triangles are congruent with each other regardless whether they are rotated or flipped. The congruence of two objects is often represented using the symbol "≅". In the figure below, △ABC ≅ △DEF. Show
 As shown in the figure above, the lengths of the corresponding sides and measures of the corresponding angles do not have to be explicitly shown to indicate congruence. An equal number of tick marks can be used to show that sides are congruent. Similarly, an equal number of arcs can be used to show that angles are congruent. The corresponding congruent angles are: ∠A≅∠D, ∠B≅∠E, ∠C≅∠F. Also, the corresponding vertices of the two triangles should be written in order. So, △ABC≅△DEF could also be written as △CBA≅△FED but not △BCA≅△DEF. Determining congruence for trianglesTwo triangles must have the same size and shape for all sides and angles to be congruent, Any one of the following comparisons can be used to confirm the congruence of triangles. Side-Side-Side (SSS)If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. In the figure above, AB≅DE, BC≅EF, AC≅DF. Therefore △ABC≅△DEF. Side-Angle-Side (SAS)If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the two triangles are congruent. In the figure above, AB≅DE, AC≅DF, and ∠A≅∠D. Therefore, △ABC≅△DEF. Angle-Side-Angle (ASA)If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the two triangles are congruent. In the figure above, ∠A≅∠D, ∠B≅∠E, and AB≅DE.Therefore,△ABC≅△DEF. Angle-Angle-Side (AAS)If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, the two triangles are congruent. In the figure above, ∠D≅∠A, ∠E≅∠B, and BC≅EF. Therefore, △DEF≅△ABC. The Angle-Angle-Side theorem is a variation of the Angle-Side-Angle theorem. In the figure, since ∠D≅∠A, ∠E≅∠B, and the three angles of a triangle always add to 180°, ∠F≅∠C. This then becomes an Angle-Side-Angle comparison since ∠E≅∠B, ∠F≅∠C, and BC≅EF. Hypotenuse-Leg congruenceIf the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another triangle, the two triangles are congruent. In the figure above, AC≅DF, AB≅DE, ∠B and ∠E are right angles. Therefore, △ABC≅△DEF. Angle-Side-Side (ASS)If two sides and the non-included angle of one triangle are congruent to two sides and the non-included angle of another triangle, the two triangles are not always congruent. In the figure above, AC≅DF, BC≅EF, ∠A≅∠D, but △ABC is not congruent to △DEF. Angle-Angle-Angle (AAA)If three angles of one triangle are congruent to three angles of another triangle, the two triangles are not always congruent. As shown in the figure below, the size of two triangles can be different even if the three angles are congruent. Corresponding partsWhen two triangles are congruent, all their corresponding angles and corresponding sides (referred to as corresponding parts) are congruent. Once it can be shown that two triangles are congruent using one of the above congruence methods, we also know that all corresponding parts of the congruent triangles are congruent (abbreviated CPCTC). Example: State the congruence for the two triangles as well as all the congruent corresponding parts. Since two angles of △ABC are congruent to two angles of △PQR, the third pair of angles must also be congruent, so ∠C≅∠R, and △ABC≅△PQR by ASA. Consider two triangles △ABC and △DEF, whose two pairs of corresponding sides are and the included angles are congruent. These triangles can be proven to be similar by identifying a that maps one triangle onto the other. First, △DEF can be with the k=DEAB about D, forming the new triangle △DE′F′. Because dilation is a similarity transformation, it can be concluded that △DE′F′ and △DEF are . Now, it has to be proven that a that maps △DE′F′ onto △ABC exists. The ratios of the corresponding side lengths of are the same and equal to the scale factor. In this case, the scale factor k is DEAB. Since AB and AC are proportional to DE and DF respectively, the scale factor can be expressed by any of the following ratios. Applying the , three equations can be formed and simplified.DEDE′=DEABDFDF′=DFAC⇒⇒DE′=ABDF′=AC These relations imply that the two sides of △DE′F′ are to the corresponding two sides of △ABC. Moreover, the included angles ∠A and ∠D are also congruent. Therefore, by the , the two triangles are congruent. Since congruent figures can be transformed into each other using rigid motions, and △ABC and △DE′F′ are congruent triangles, there is a rigid motion placing △DE′F′ onto △ABC.The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps △DEF onto △ABC. Is ABC congruent to Def?If in triangles ABC and DEF, angle A = angle D, angle B = angle E, and AB = DE, then triangle ABC is congruent to triangle DEF.
Which theorem shows that △ ABC ≅ △ def?By the SSS Congruence Theorem, △ABC ≅ △DEF.
Is given that ∆ ABC ≅ ∆ def Is it true to say that AB EF justify your answer?This is an Expert-Verified Answer
But, AB is not equal to EF because only same sides of triangle will be equal if triangles are congruent. Here AB is not on the same side of EF.
Is ABC is similar to DEF Why or why not?ABC will be similar to DEF if two sides along with their corresponding angle of ABC is equal to two sides and their corresponding angle of DEF. ABC will be similar to DEF if the corresponding ratios of all sides of both triangles are equal to each other.
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Is ABC is similar to DEF the sides of ABC must be congruent to the corresponding sides of Def?
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