Question

1) find the number of diagonal of octagine and with side 12 cm.

2) ABCD is a rhombus P, Q, R, and S are mid point of side AB, BC, CD, and AD respective show that PQRS is a parallelogram.

3) Prove that angle sum properties of triangle.​



ANSWER ALL QUESTIONS

Answer 1

Step-by-step explanation:

(2)Given- ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

To Prove-PQRS is a rectangle

Construction,

AC and BD are joined.

Proof,

In ΔDRS and ΔBPQ,

DS = BQ (Halves of the opposite sides of the rhombus)

∠SDR = ∠QBP (Opposite angles of the rhombus)

DR = BP (Halves of the opposite sides of the rhombus)

Thus, ΔDRS ≅ ΔBPQ by SAS congruence condition.

RS = PQ by CPCT --- (i)

In ΔQCR and ΔSAP,

RC = PA (Halves of the opposite sides of the rhombus)

∠RCQ = ∠PAS (Opposite angles of the rhombus)

CQ = AS (Halves of the opposite sides of the rhombus)

Thus, ΔQCR ≅ ΔSAP by SAS congruence condition.

RQ = SP by CPCT --- (ii)

Now,

In ΔCDB,

R and Q are the mid points of CD and BC respectively.

⇒ QR || BD

also,

P and S are the mid points of AD and AB respectively.

⇒ PS || BD

⇒ QR || PS

Thus, PQRS is a parallelogram.

also, ∠PQR = 90°

Now,

In PQRS,

RS = PQ and RQ = SP from (i) and (ii)

∠Q = 90°

Thus, PQRS is a rectangle.

(3)The angle sum property of a triangle states that the angles of a triangle always add up to 180°. Every triangle has three angles and whether it is an acute, obtuse, or right triangle, the angles sum to 180°. For example, in triangle ABC, angle A + angle B + angle C = 180°