Class 9th Mathematics Part Ii MHB Solution
- Diagonals of a parallelogram WXYZ intersect each other at point O. If ∠ XYZ = 135° then…
- In a parallelogram ABCD, If ∠A = (3x +12)°, ∠B = (2x -32) ° then find the value of x…
- Perimeter of a parallelogram is 150 cm. One of its sides is greater than the other side…
- If the ratio of measures of two adjacent angles of a parallelogram is 1 : 2, find the…
- Diagonals of a parallelogram intersect each other at point O. If AO = 5, BO = 12 and AB…
- In the figure 5.12, PQRS and ABCR are two parallelograms. If ∠P = 110° then find…
- In figure 5.13 ABCD is a parallelogram. Point E is on the ray AB such that BE = AB…
Practice Set 5.1
Question 1.Diagonals of a parallelogram WXYZ intersect each other at point O. If ∠ XYZ = 135° then what is the measure of ∠XWZ and ∠YZW? If l(OY)= 5 cm then l(WY)=?
Answer:
Given ZX and WY are the diagonals of the parallelogram
∠ XYZ = 135° ⇒ ∠ XWZ = 135° as the opposite angels of a parallelogram are congruent.
∠YZW + ∠ XWZ = 180° as the adjacent angels of the parallelogram are supplementary.
⇒ ∠YZW = 180° - 135° = 45°
Length of OY = 5 cm then length of WY = WO + OY = 5+5 = 10 cm
(diagonals of the parallelogram bisect each other. So, O is midpoint of WY)
Question 2.
In a parallelogram ABCD, If ∠A = (3x +12)°, ∠B = (2x -32) ° then find the value of x and then find the measures of ∠C and ∠D.
Answer:
∠A = (3x +12)°
∠B = (2x -32) °
∠A + ∠B = 180° (supplementary angles of the ∥gram)
(3x +12) + (2x -32) = 180
5x – 20 = 180°
5x = 200°
∴ x= 40°
∠A = (3 × 40) +12 = 120 + 12
= 132
⇒ ∠C = 132° (opposite ∠s are congruent)
Similarly, ∠B = 2× 40 – 32
= 80 - 32°
= 48°
⇒ ∠D = 48°(opposite ∠s are congruent)
Question 3.
Perimeter of a parallelogram is 150 cm. One of its sides is greater than the other side by 25 cm. Find the lengths of all sides.
Answer:
perimeter of parallelogram = 150cm
Let the one side of parallelogram be x cm then
Acc. To the given condition
Other side is (x+25) cm
Perimeter of parallelogram = 2(a+b)
150 = 2( x+x+25)
150 = 2(2x+25)
One side is 25cm and the other side is 50cm.
Question 4.
If the ratio of measures of two adjacent angles of a parallelogram is 1 : 2, find the measures of all angles of the parallelogram.
Answer:
Given that the ratio of measures of two adjacent angles of a parallelogram= 1 : 2
If one ∠ is x other would be 180 – x as the adjacent ∠s of a parallelogram are supplementary.
Other ∠ is 120° .
The measure of all the angles are 60 °, 120 °, 60 ° and 120 ° where 60 ° and 120 ° are adjacent ∠s and 60 ° and 60 ° are congruent opposite angles.
Question 5.
Diagonals of a parallelogram intersect each other at point O. If AO = 5, BO = 12 and AB = 13 then show that ABCD is a rhombus.
Answer:
The figure is given below:
Given AO =5, BO = 12 and AB = 13
In Δ AOB, AO2 + BO2 = AB2
∵ 52 + 122 = 132
252 + 1442 = 1692
so by the Pythagoras theorem
Δ AOB is right angled at ∠ AOB.
But ∠ AOB + ∠ AOD forms a linear pair so the given parallelogram is rhombus whose diagonal bisects each other at 90°.
Question 6.
In the figure 5.12, PQRS and ABCR are two parallelograms. If ∠P = 110° then find the measures of all angles of ABCR.
Answer:
given PQRS and ABCR are two ∥gram.
∠P = 110° ⇒ ∠R = 110°
(opposite ∠s of parallelogram are congruent)
Now if , ∠R = 110° ⇒ ∠ B = 110°
∠B + ∠A = 180°
(adjacent ∠s of a parallelogram are supplementary)
⇒ ∠A = 70° ⇒ ∠C = 70°
(opposite ∠s of parallelogram are congruent)
Question 7.
In figure 5.13 ABCD is a parallelogram. Point E is on the ray AB such that BE = AB then prove that line ED bisects seg BC at point F.
Answer:
Given, ABCD is a parallelogram
And BE = AB
But AB = DC (opposite sides of the parallelogram are equal and parallel)
⇒ DC = BE
In Δ BEF and ∠DCF
∠DFC = ∠BFE (vertically opposite angles)
∠DFC = ∠ BFE (alternate ∠s on the transversal BC with AB and DC as ∥ )
And BE = AB (given)
Δ BEF ≅ ∠DCF (by AAS criterion)
⇒ BF =FC (corresponding parts of the congruent triangles)
⇒ F is mid-point of the line BC. Hence proved.
