PS = PK , AC = BD , m∠ABC=m∠BAD, ΔAOK ≅ ΔBOL are proved.
What are congruent triangles?
Congruent triangles are those of the same size and shape. This means that the corresponding sides are equal, as are the corresponding angles. We can determine whether two triangles are congruent without testing all of their sides and angles.
1) in ΔPQR we have ,
PQ = PR
IN, ∠PRQ = ∠PQR (Isosceles triangle)
180° - ∠PQR = 180° - ∠PQR
∠PQR = ∠PSQ
IN ΔPSQ = PRK
∠SPQ = ∠RPK
∠Q = ∠R IS GIVEN IN QUESTION SO,
∠SPQ = ∠RPQ
so, by ASA congruence we have ,
ΔPSQ ≅ ΔPRK
So, PS = PK
hence proved .
2) There are several ways two triangles can be congruent.
AC = BD congruent by SAS
ABC ≅ BAD congruent by the corresponding theorem
In ΔAOC and ΔBOD, we have the following observations
AO = DO ( O is the midpoint of line segment AD)
BO = CO ( Because O is the midpoint of line segment BC)
∠AOB = ∠COD (vertical angles are congruent)
∠AOC = ∠BOD ( vertical angles are congruent)
Using the SAS (side-angle-side) postulate, we have:
AC = BD
Using the corresponding theorem,
ΔABC ≅ ΔBAD
The above congruence equation is true because:
2 sides of both triangles are congruent
1 angle each of both triangles is equal
Corresponding angles are equal
Hence Proved.
3) There are several ways two triangles can be congruent.
ΔAOK ≅ ΔBOK
are congruent by SAS
In ΔAOL and ΔBOK, we have the following observations
AO = BO ( O is the midpoint of line segment AB)
∠AOL = ∠BOK ( vertical angles are congruent)
LO = KO ( O is the midpoint of line segment KL)
Using the SAS (side-angle-side) postulate, we have:
ΔAOK ≅ ΔBOK
The above congruence equation is true because:
2 sides of both triangles are congruent
1 angle each of both triangles is equal
Hence proved
To learn more about congruent triangles, refer to :
https://brainly.com/question/1582456
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