in a rhombus pqrs diagonal PR is equals to side QR find the measures of angles of a rhombus pqrs
Answers
ABCD is a Rhombus ⇒ AC = BC ------------ (1) [Given data]
BC = AB --------------- (2) [Sides of the Rhombus]
From equations (1) and (2)
AC = BC = AB
⇒ ΔABC is an equilateral triangle.
So, <ABC = 60°,
<BCA = 60° ------------------------- (3)
<CAB = 60° --------------------------(4)
Similarly, in ΔADC, AD = DC [Sides of a rhombus]
AD = BC
But BC = AC
∴ AD = AC
∴ AD = DC = AC
∴ DADC is an equilateral triangle.
⇒ <CAD = 60° ---------------- (5)
⇒ <ADC = 60°
⇒ <DCA = 60° ----------------- (6)
From equations (3) and (6), we get <BCA + <DCA = 60° + 60° = 120°
∴ <C = 120°
From equations (4) and (5), <CAB + <CAD = 60° + 60° = 120°
<A = 120°.
Therefore, four angles of the rhombus are 120°, 60°, 120°, 60°.
Answer:
Step-by-step explanation:
In rhombus PQRS the diagonal PR equals the side QR ...(GIVEN)
Now consider only Δ PQR
PR≅QR
hence its an isoceles triangle
thus,∠RPQ≅∠PRQ ......(1)
Now as the given quadrilateral is a rhombus all the side are congruent
thus QR ≅PS
Thus considering the ΔPRS
PS≅PR..........(∵QR≅PR And QR≅PS)
thus ∠PSR =PRQ......(2)
from 1 and 2
∠PSR ≅ ∠PQR
now in rhombus opposite angles are congruent
hence by angle addition rule of quadrilateral
∠P+∠Q+∠R+∠S =360
from 3
2∠P + 2∠Q = 360 (P≅R AND S≅Q)
Thus P+Q=180
but from given statement we get the values
as ∠P=120°
∠Q=60
∠R=120
∠S=60