Are two right triangles with an angle of 60 ° congruent? Are two isosceles triangles congruent with an obtuse angle

Are two right triangles with an angle of 60 ° congruent? Are two isosceles triangles congruent with an obtuse angle

Two right triangles with 60 ° angle are not congruent triangles (because only three angles are equal and the sides are not necessarily equal, that is, AAA congruence does not have this proof method). The latter one is the same, only AAA does not have the condition of edge

(1) Write two true propositions about an isosceles triangle

(1) 1. The bisectors of corresponding angles of congruent triangles are equal; 2. The heights of corresponding sides of congruent triangles are equal
(2) 2. A triangle with two angles of 45 ° is an isosceles triangle

The inverse proposition of the following proposition is false: the two base angles of an isosceles triangle are equal, and the corresponding sides of a congruent triangle are equal C. The corresponding angles of congruent triangles are equal D. if a 2 ＞ B 2, then a is greater than B ·

The corresponding angles of congruent triangles are equal

Write a true proposition related to isosceles triangle

The two waists of an isosceles triangle are equal
The two base angles of an isosceles triangle are equal

In this paper, the proposition "the midline on the corresponding side of an congruent triangle is equal" is written in the form of "known" and "proved", and the proof is given

It is known that △ ABC ≌ △ a'b'c ', ad is the midline of △ ABC, and a'd' is the midline of △ a'b'c '
Confirmation: ad = a'd '
prove:
∵△ABC≌△A'B'C'
∴AB=A'B',BC=B'C',∠A=∠A'
∵ ad is the midline of ᙽ ABC, and a'd 'is the midline of ᙽ a'b'c'
∴BD=B'C'/2,BD=B'C'/2
∴BD=B'D'
∴△ABD≌△A'B'D'(SAS)
∴AD=A'D

Proof of judging theorem of similar triangle It is necessary to prove that the line parallel to one side of the triangle intersects the other two sides, and the triangle formed is similar to the original triangle. How can we prove that the three sides of the triangle are proportional and the triangle is equal? Do not use any other judgment theorem to prove, I want to use the axiom proof! Thank you! It would be better if we could give pictures

Judging theorem of similar triangles:
(1) If two angles of a triangle are equal to those of another triangle, then the two triangles are similar
(2) If the two sides of a triangle are proportional to the two sides of another triangle and the included angles are equal, then the two triangles are similar
(3) If the three sides of a triangle are proportional to the three sides of another triangle, then the two triangles are similar
Judging theorem of right triangle similarity:
(1) A right triangle is divided into two parts by the height of the hypotenuse
(2) If the hypotenuse and one right angle side of a right triangle are proportional to the hypotenuse and one right angle side of another right triangle, the two right triangles are similar
Property theorem of similar triangles
(1) The corresponding angles of similar triangles are equal
(2) The corresponding sides of similar triangles are proportional
(3) The ratio of the corresponding high line, the corresponding center line and the corresponding angle bisector of the similar triangle are all equal to the similarity ratio
(4) The perimeter ratio of similar triangles is equal to the similarity ratio
(5) The area ratio of a similar triangle is equal to the square of the similarity ratio
Transitivity of similar triangles
If △ ABC ∽ a1b1c1, △ a1b1c1 ∽ a2b2c2, then △ ABC ∽ a2b2c2

Proving similar triangle Theorem 3

Two triangles are similar if their angles are equal
It is proved that let △ ABC and △ def, ∠ a = ∠ D, ∠ B = ∠ E
∵ sum of internal angles of triangles = 180 ᙽ
∴∠C=180°-∠A-∠B=180°-∠D-∠E
And ∠ f = 180 ° - ∠ D - ∠ E
∴∠C=∠F
∵∠C=∠F,∠A=∠D,∠B=∠E
∴△ABC∽△DEF

Proving the preparatory theorem of similar triangle The theorem is proved only by the definition of similar triangle Preparation theorem for similar triangles: a straight line parallel to one side of the triangle and intersecting with the other two sides, the three sides of the triangle cut are proportional to the three sides of the original triangle Note: we can only use the definition of similar triangle to prove that it is similar. Of course, other criteria that are not similar triangles can also be used. On the first floor, it seems to use the judgment of similar triangle 1, and there is this preparatory theorem that can be proved in previous books.

Mathematical proof of similar triangle preparation theorem 20 - there are 14 days and 23 hours before the end of the problem. Only the definition of similar triangle is used to prove this theorem. Similar triangle preparation theorem: parallel to one side of the triangle and intersecting with other two sides, the three sides of the triangle cut are in proportion to the three sides of the original triangle

Proof of congruent triangle theorem Note: it is to prove the SSS, SAS and ASA theorems, not to use them to prove triangle congruence

For example, an edge angle can not uniquely determine a triangle, so it can not be used to prove congruence, unless the angle is a right angle

As shown in the figure, ab = AC = 10, BC = 12 in △ ABC

As shown in the figure, make ad ⊥ BC, if the vertical foot is D, then o must be on AD,
So ad=
102−62=8；
Let OA = R, ob2 = OD2 + BD2,
That is, R2 = (8-r) 2 + 62,
The solution is r = 25
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Answer: △ the radius of ABC circumscribed circle is 25
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