Advertisements
Advertisements
प्रश्न
In the adjacent figure, if seg AB || seg PQ, seg AB ≅ seg PQ, seg AC || seg PR, seg AC ≅ seg PR then prove that, seg BC || seg QR and seg BC ≅ seg QR.
Advertisements
उत्तर
Given: seg AB || seg PQ, seg AB ≅ seg PQ,
seg AC || seg PR, seg AC ≅ seg PR
To prove: seg BC || seg QR, seg BC ≅ seg QR
Proof:
In `square`ABQP,
seg AB || seg PQ
seg AB ≅ seg PQ ...(Given)
∴ `square`ABQP is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ segAP || segBQ ...(i)
∴ seg AP ≅ seg BQ ...(ii) ...(Opposite sides of a parallelogram)
In `square`ACRP,
seg AC || seg PR
seg AC ≅ seg PR ...(Given)
∴ `square`ACRP is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ seg AP || seg CR ...(iii)
∴ seg AP ≅ seg CR ...(iv) ...(Opposite sides of a parallelogram)
In `square`BQRC,
seg BQ || seg CR ...[From (i) and (iii)]
seg BQ ≅ seg CR ...[From (ii) and (iv)]
∴ `square`BQRC is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ seg BC || seg QR
∴ seg BC ≅ seg QR ...(Opposite sides of parallelogram)
APPEARS IN
संबंधित प्रश्न
Consider the given parallelogram. Find the values of the unknowns x, y, z.
Consider the given parallelogram. Find the values of the unknowns x, y, z.
Consider the given parallelograms. Find the values of the unknowns x, y, z.
The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.
The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Find the measure of ∠P and ∠S, if `bar(SP) || bar(RQ)` in the following figure. (If you find m∠R, is there more than one method to find m∠P?).
Name the quadrilaterals whose diagonals bisect each other
In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that:
- AE = AD,
- DE bisects and ∠ADC and
- Angle DEC is a right angle.
In parallelogram ABCD, X and Y are midpoints of opposite sides AB and DC respectively. Prove that:
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram.
Use the information given in the alongside diagram to find the value of x, y, and z.
In the given figure, AB || EC, AB = AC and AE bisects ∠DAC. Prove that:
- ∠EAC = ∠ACB
- ABCE is a parallelogram.
Iron rods a, b, c, d, e, and f are making a design in a bridge as shown in the figure. If a || b, c || d, e || f, find the marked angles between d and e
In parallelogram PQRS, O is the mid point of SQ. Find ∠S, ∠R, PQ, QR and diagonal PR.
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.
In the following figure, FD || BC || AE and AC || ED. Find the value of x.
The following figure RUNS is parallelogram. Find x and y. (Lengths are in cm)
