How many criteria are there to proof congruence of triangle Tell me with example.
Dear Student
Definition of Triangle Congruence
We say that triangle ABC is congruent to triangle DEF if
1.Side-Angle-Side (SAS)
This criterion for triangle congruence is one of our axioms. So we do not prove it but use it to prove other criteria.
Using words:
If two sides in one triangle are congruent to two sides of a second triangle, and also if the included angles are congruent, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, AB = DE, AC = DF, and angle A = angle D, then triangle ABC is congruent to triangle DEF.
2.Side- Side-Side (SSS)
Using words:
If 3 sides in one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, AB = DE, BC = EF, and CA = FD, then triangle ABC is congruent to triangle DEF.
3.Angle-Side-Angle (ASA)
Using words:
If two angle in one triangle are congruent to two angles of a second triangle, and also if the included sides are congruent, then the triangles are congruent.
Using labels:
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.
Proof: This proof was left to reading and was not presented in class. Again, one can make congruent copies of each triangle so that the copies share a side. Then one can see that AC must = DF.
4.Side-Side-Angle (SSA) not valid in general
Using labels:
SSA would mean for example, that in triangles ABC and DEF, angle A = angle D, AB = DE, and BC = EF.
With these assumptions it is not true that triangle ABC is congruent to triangle DEF. In general there are two sets of congruent triangles with the same SSA data.
5.Hypotenuse-Leg (HL) for Right Triangles
There is one case where SSA is valid, and that is when the angles are right angles.
Using words:
In words, if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
Using labels
If in triangles ABC and DEF, angle A = angle D = right angle, AB = DE (leg), and BC = EF (hypotenuse), then triangle ABC is congruent to triangle DEF.
Note:-For more example go to our website.
Regards
Definition of Triangle Congruence
We say that triangle ABC is congruent to triangle DEF if
- AB = DE
- BC = EF
- CA = FD
- Angle A = Angle D
- Angle B = Angle E
- Angle C = Angle F
Congruence Criteria
It turns out that knowing some of the six congruences of corresponding sides and angles are enough to guarantee congruence of the triangle and the truth of all six congruences.1.Side-Angle-Side (SAS)
This criterion for triangle congruence is one of our axioms. So we do not prove it but use it to prove other criteria.
Using words:
If two sides in one triangle are congruent to two sides of a second triangle, and also if the included angles are congruent, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, AB = DE, AC = DF, and angle A = angle D, then triangle ABC is congruent to triangle DEF.
2.Side- Side-Side (SSS)
Using words:
If 3 sides in one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent.
Using labels:
If in triangles ABC and DEF, AB = DE, BC = EF, and CA = FD, then triangle ABC is congruent to triangle DEF.
3.Angle-Side-Angle (ASA)
Using words:
If two angle in one triangle are congruent to two angles of a second triangle, and also if the included sides are congruent, then the triangles are congruent.
Using labels:
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.
Proof: This proof was left to reading and was not presented in class. Again, one can make congruent copies of each triangle so that the copies share a side. Then one can see that AC must = DF.
4.Side-Side-Angle (SSA) not valid in general
Using labels:
SSA would mean for example, that in triangles ABC and DEF, angle A = angle D, AB = DE, and BC = EF.
With these assumptions it is not true that triangle ABC is congruent to triangle DEF. In general there are two sets of congruent triangles with the same SSA data.
5.Hypotenuse-Leg (HL) for Right Triangles
There is one case where SSA is valid, and that is when the angles are right angles.
Using words:
In words, if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
Using labels
If in triangles ABC and DEF, angle A = angle D = right angle, AB = DE (leg), and BC = EF (hypotenuse), then triangle ABC is congruent to triangle DEF.
Note:-For more example go to our website.
Regards