QUADRILATERALS : Ganita Prakash | Grade 8 | Chapter-4 | Page 102 | Solutions

1. Find the remaining angles in the following quadrilaterals (assuming they are rectangles):

The diagonals of a rectangle bisect each other and are equal in length. If the diagonals AC and BD intersect at O, then OA=OB=OC=OD. This means the four triangles formed at the intersection point (△OAB,△OBC,△OCD,△ODA) are isosceles.

Case 1.1: Rectangle P Q S R (Angle between diagonals 110∘)

Let the intersection point be O, and assume ∠QOR=110∘.

Step 1: Angles at the intersection (O)

• ∠POS=∠QOR=110∘ (Vertically opposite angles).
• ∠POQ=180∘−110∘=70∘ (Angles on a straight line).
• ∠ROS=∠POQ=70∘ (Vertically opposite angles).

Step 2: Angles in △QOR

• Since OQ=OR, △QOR is isosceles.
• ∠OQR=∠ORQ
• $2 \times \angle O Q R = 180^\circ - 110^\circ = 70^\circ$
• ∠OQR=∠ORQ=35∘

Step 3: Angles in △POQ

• Since OP=OQ, △POQ is isosceles.
• ∠OPQ=∠OQP
• $2 \times \angle O P Q = 180^\circ - 70^\circ = 110^\circ$
• ∠OPQ=∠OQP=55∘

Step 4: Remaining angles (by symmetry)

• In △POS: ∠OPS=∠OSP=35∘ (Base angles opposite to 110∘).
• In △ROS: ∠ORS=∠OSR=55∘ (Base angles opposite to 70∘).

Summary of Angles along the sides:

• ∠OPQ=55∘
• ∠OQP=55∘
• ∠OQR=35∘
• ∠ORQ=35∘
• ∠ORS=55∘
• ∠OSR=55∘
• ∠OSP=35∘
• ∠OPS=35∘

(Verification: ∠PQR=55∘+35∘=90∘)

---

Case 1.2: Rectangle R A P E (Angle ∠PAO=40∘)

Let the diagonals intersect at O. Assume ∠OAP=40∘.

Step 1: Angles in △OAP

• Since OA=OP, △OAP is isosceles.
• ∠OPA=∠OAP=40∘
• Angle at intersection ∠AOP=180∘−(40∘+40∘)=100∘

Step 2: Angles at the intersection (O)

• ∠EOR=∠AOP=100∘ (Vertically opposite angles).
• ∠POE=180∘−100∘=80∘ (Angles on a straight line).
• ∠AOR=∠POE=80∘ (Vertically opposite angles).

Step 3: Angles in △POE

• Since OP=OE, △POE is isosceles.
• ∠OPE=∠OEP
• $2 \times \angle O P E = 180^\circ - 80^\circ = 100^\circ$
• ∠OPE=∠OEP=50∘

Step 4: Remaining angles (by symmetry)

• In △EOR: ∠OER=∠ORE=40∘ (Base angles opposite to 100∘).
• In △AOR: ∠ORA=∠OAR=50∘ (Base angles opposite to 80∘).

Summary of Angles along the sides:

• ∠OAP=40∘
• ∠OPA=40∘
• ∠OPE=50∘
• ∠OEP=50∘
• ∠OER=40∘
• ∠ORE=40∘
• ∠ORA=50∘
• ∠OAR=50∘

(Verification: ∠PAR=40∘+50∘=90∘)

---

Case 1.3: Rectangle U V X W (Angle ∠OVW=30∘)

Let the diagonals intersect at O. Assume ∠OVW=30∘.

Step 1: Angles in △OVW

• Since OV=OW, △OVW is isosceles.
• ∠OWV=∠OVW=30∘
• Angle at intersection ∠VOW=180∘−(30∘+30∘)=120∘

Step 2: Angles at the intersection (O)

• ∠UOX=∠VOW=120∘ (Vertically opposite angles).
• ∠WOX=180∘−120∘=60∘ (Angles on a straight line).
• ∠VOU=∠WOX=60∘ (Vertically opposite angles).

Step 3: Angles in △WOX

• Since OW=OX, △WOX is isosceles.
• ∠OWX=∠OXW
• $2 \times \angle O W X = 180^\circ - 60^\circ = 120^\circ$
• ∠OWX=∠OXW=60∘. (Note: △WOX is equilateral).

Step 4: Remaining angles (by symmetry)

• In △UOX: ∠OUX=∠OXU=30∘ (Base angles opposite to 120∘).
• In △VOU: ∠OUV=∠OVU=60∘ (Base angles opposite to 60∘).

Summary of Angles along the sides:

• ∠OVW=30∘
• ∠OWV=30∘
• ∠OWX=60∘
• ∠OXW=60∘
• ∠OXU=30∘
• ∠OUX=30∘
• ∠OUV=60∘
• ∠OVU=60∘

(Verification: ∠UVW=60∘+30∘=90∘)

---

2. Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140∘.

A parallelogram is a quadrilateral where the diagonals bisect each other.

Construction Steps:

1. Draw the first diagonal (AC): Draw a line segment AC=7 cm.
2. Find the midpoint (O): Locate the midpoint O of AC. AO=OC=3.5 cm.
3. Draw the angle: At point O, draw a line segment (a construction line) such that it makes an angle of 140∘ with AC. Let this line represent the second diagonal (BD).
4. Mark the second diagonal (BD): The second diagonal BD is 5 cm long and must be bisected by O. Therefore, mark points B and D on the construction line such that OB=OD=5/2=2.5 cm.
   • Mark B 2.5 cm from O along the 140∘ angle.
   • Mark D 2.5 cm from O along the extension line (the angle adjacent to 140∘ is 180∘−140∘=40∘).
5. Complete the parallelogram: Join the endpoints A, B, C, and D to form the parallelogram ABCD.

---

3. Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.

A rhombus is a quadrilateral where the diagonals bisect each other at right angles (90∘).

Construction Steps:

1. Draw the first diagonal (AC): Draw a line segment AC=5 cm.
2. Draw the perpendicular bisector: Construct the perpendicular bisector of AC. This bisector passes through the midpoint O of AC, meaning AO=OC=2.5 cm.
3. Mark the second diagonal (BD): The second diagonal BD is 4 cm long and lies along the perpendicular bisector, bisected by O. Therefore, mark points B and D on the perpendicular bisector such that OB=OD=4/2=2 cm.
4. Complete the rhombus: Join the endpoints A, B, C, and D to form the rhombus ABCD.