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How do you prove that two right triangles are congruent if the hypotenuse and an acute angle of one are equal respectively to the hypotenuse and acute angle of the other?
Wiki User
∙ 13y ago
Let ABC and DEF be triangles which are right angled at A and D, such that the hypotenuses BC and EF are equal and, without loss of generality, angle B = angle E.Then
Then by the sine rule, BC/sin(A) = AC/sin(B) and EF/sin(D) = DF/sin(E)
Since angle A = angle D = pi/2 radians, then sin(A) = sin(D) = 1
so that BC/sin(A) = BC while EF/sin(D) = EF
therefore, since the hypotenuses BC and EF are equal, the left hand sides of the two equations are equal.
Therefore, AC/sin(B) = DF/sin(E)
then, since angle B = angle E, then sin(B) = sin(E) so that AC = DF.
Also, angle C = pi/2 - angle B
and angle F = pi/2 - angle E
the right hand sides are equal so angle C = angle F.
Then in a manner similar to the above, we can show that AB = DE.
Thus all three pairs of corresponding sides are equal and all three pairs of corresponding angles are equal so that the two triangles are congruent.
Wiki User
∙ 7y ago
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What are four ways you can prove two right triangles are congruent?
1. The side angle side theorem, when used for right triangles is often called the leg leg theorem. it says if two legs of a right triangle are congruent to two legs of another right triangle, then the triangles are congruent. Now if you want to think of it as SAS, just remember both angles are right angles so you need only look at the legs.2. The next is the The Leg-Acute Angle Theorem which states if a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent. This is the same as angle side angle for a general triangle. Just use the right angle as one of the angles, the leg and then the acute angle.3. The Hypotenuse-Acute Angle Theorem is the third way to prove 2 right triangles are congruent. This one is equivalent to AAS or angle angle side. This theorem says if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, the two triangles are congruent. This is the same as AAS again since you can use the right angle as the second angle in AAS.4. Last, but not least is Hypotenuse-Leg Postulate. Since it is NOT based on any other rules, this is a postulate and not a theorem. HL says if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
What are the names of five triangles?
Equilateral ( all sides are congruent), Isosceles (2 sides are congruent, Scalene (no sides are congruent), Acute ( three acute angles), Obtuse ( one obtuse angle), Right ( one right angle).
If an acute angle of one right triangle is congruent to an angle of a second triangle then are the triangles similar?
true.
Triangles with all angles less than 90 degrees?
this is known as an acute triangle acute triangles-no angles congruent or greater than 90 degrees right-one angle equal to 90 degrees obtuse-triangle with one angle greater than 90 degrees
Does a obtuse angled isosceles triangle have congruent side?
All isosceles triangles must have two congruent sides otherwise it is not isosceles. Whether it is acute angled, right angled or obtuse angled is irrelevant.
What is the hypotenuse angle theorem?
The hypotenuse angle theorem, also known as the HA theorem, states that 'if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.'
Which of these best describes the hypotenuse-angle theorem?
The theorem is best described "If the hypotenuse and an acute angle of a right triangle are equal respectively to the corresponding parts of another right triangle, then the triangles are congruent."
What are the four congruence theorems for a right triangle?
The four congruence theorem for right triangles are:- LL Congruence Theorem --> If the two legs of a right triangle is congruent to the corresponding two legs of another right triangle, then the triangles are congruent.- LA Congruence Theorem --> If a leg and an acute angle of a right triangles is congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent.- HA Congruence Theorem --> If the hypotenuse and an acute angle of a right triangle is congruent to the corresponding hypotenuse and acute angle of another triangle, then the triangles are congruent.- HL Congruence Theorem --> If the hypotenuse and a leg of a right triangle is congruent to the corresponding hypotenuse and leg of another right triangle, then the triangles are congruent.
What does the hypotenuse-angle theorem say?
If the hypotenuse and an acute angle of a right triangle are congruent to the correspondingparts of another right triangle, then the triangles are congruent.
What is HA Congruence Theorem?
HA Congruence Theorem says: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two right triangles are congruent.
What are four ways you can prove two right triangles are congruent?
1. The side angle side theorem, when used for right triangles is often called the leg leg theorem. it says if two legs of a right triangle are congruent to two legs of another right triangle, then the triangles are congruent. Now if you want to think of it as SAS, just remember both angles are right angles so you need only look at the legs.2. The next is the The Leg-Acute Angle Theorem which states if a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent. This is the same as angle side angle for a general triangle. Just use the right angle as one of the angles, the leg and then the acute angle.3. The Hypotenuse-Acute Angle Theorem is the third way to prove 2 right triangles are congruent. This one is equivalent to AAS or angle angle side. This theorem says if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, the two triangles are congruent. This is the same as AAS again since you can use the right angle as the second angle in AAS.4. Last, but not least is Hypotenuse-Leg Postulate. Since it is NOT based on any other rules, this is a postulate and not a theorem. HL says if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Are acute-angled triangles similar?
Similar in the number of sides but not congruent
The LAtheorem says that two right triangles will be congruent if they have a congruent and a congruent leg?
The leg-angle congruence theorem says if one leg and an acute angle of one right triangle are congruent to one leg and an acute angle of another right triangle, then the two right triangles are congruent.
Does an acute triangle have two congruent sides?
Not every acute triangle has two congruent sides, although some do, in which case they are called isosceles triangles.
Two right triangles that are not similar still may have one congruent acute angle in common?
Not so. The two acute angles of a right triangle must add up to 90 degrees.So if the triangles have one congruent acute angle in common, they must alsohave the other acute angle in common, and then they're similar.
Two right triangles are similar if the acute angles of one triangle are congruent to the acute angles of the other triangle?
yes there similar
'True or false Two right triangles are similar if an acute angle of one triangle is congruent to an acute angle of the other triangle'?
Sounds true to me, all three angles are congruent...
