Congruent triangles is a key point of junior high school knowledge. It often appears in the form of filling in the blanks, multiple-choice questions and solving problems in exams, which accounts for a large proportion of the scores. Therefore, congruent triangles's study is particularly important. There are five ways to judge congruent triangles * * *: SSS, SAS, ASA, AAS and HL. Under special circumstances, translation, rotation and folding will also constitute a congruent triangles.

A page from congruent triangles, a math textbook.

way

Congruent triangles Judgment Method 1: SSS (Side by Side), that is, two triangles corresponding to three sides are congruent.

For example, as shown below, AC=BD, AD=BC, and verification ∠ A = ∠ B.

Proof: in △ACD and △BDC {AC=BD, AD=BC, CD=CD.

∴△ACD≌△BDC.(SSS)

∴∠ A =∠ B. (The corresponding angles of congruent triangles are equal)

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Congruent triangles judgment method 2: SAS (Angle and Edge), that is, two sides of a triangle are equal, and the included angle of two sides also corresponds to the congruence of two equal triangles.

For example, as shown in the following figure, AB is divided by ∠CAD, AC=AD, and verification ∠ C = ∠ D. 。

Proof: ∫AB equal division ∠CAD.

∴∠CAB=∠BAD.

In △ACB and △ADB {AC=AD, ∠CAB=∠BAD, AB=AB.

∴△ACB≌△ADB.(SAS)

∴∠ C =∠ D. (congruent triangles's corresponding angles are equal)

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Congruent triangles judgment method 3: ASA (Angle and Angle), that is, two angles of a triangle are equal, and the sides of two corner clips are equal to two triangles.

For example, in the following figure, AB=AC, ∠B=∠C, proving △ Abe △ ACD.

Proof: in △ABE and △ACD {∠A=∠A, AB=AC, ∠ B = ∠ C.

∴△ABE≌△ACD.(ASA)

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Congruent triangles Judgment Method 4: AAS (Angular Edge), that is, two angles of a triangle are equal, and the edges corresponding to equal angles are also congruent with two equal triangles.

For example, as shown in the following figure, AB=DE, ∠A=∠E, and verification ∠ B = ∠ D. 。

Proof: in △ABC and △EDC, {∠A=∠E, ∠ACB=∠DCE, AB=DE.

∴△ABC≌△EDC.(AAS)

∴∠ B =∠ D. (The angles corresponding to congruent triangles are equal)

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Congruent triangles's judgment method 5: HL (hypotenuse and right-angled side), that is, one hypotenuse and one right-angled side in a right-angled triangle correspond to the congruence of two right-angled triangles.

For example, as shown in the following figure, Rt△ADC and Rt△BCD, AC=BD, and verification AD=BC.

Proof: in Rt△ADC and Rt△BCD {AC=BD, CD=CD.

∴Rt△ADC and Rt△BCD. (HL)

∴AD=BC. (The corresponding sides of congruent triangles are equal)

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Attachment: Two triangles that are translated, rotated or folded in half are congruent.

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Matters needing attention

SSS, SAS, ASA and AAS can be used for any triangle; HL is limited to right-angled triangles.

Note that SSA and AAA cannot judge congruent triangles.

When proving, we should pay attention to the use of theorems, such as: equality property, equivalent substitution, equal angle coincidence, equal edge, common angle, equal vertex angle, equal complementary angle or complementary angle of equal angle or same angle, the definition of bisector of angle, the definition of midpoint of line segment, etc.

Pay attention to the writing order when proving the congruent writing conditions.

Pay attention to the position of the corresponding vertex when writing congruence conclusion.

Sometimes congruent triangles will combine isosceles triangles to present propositions.