RS Aggarwal Class 8 Mathematics Solutions Chapter 15 Quadrilaterals

Question 1. (i)
Answer: 4
In simple words: A quadrilateral has 4 sides.

Exam Tip: Remember that the prefix "quad" means four, so a quadrilateral always has exactly four sides, four vertices, and four angles.

Question 2. (ii)
Answer: 4
In simple words: A quadrilateral contains 4 angles.

Exam Tip: Each vertex of a quadrilateral creates one interior angle, giving a total of four angles.

Question 3. (iii)
Answer: 4, co-linear
In simple words: A quadrilateral has 4 diagonals that share a common point, and no three vertices are arranged on the same straight line.

Exam Tip: "Co-linear" means the points lie on the same straight line - this property must NOT hold for a valid quadrilateral.

Question 4. (iv)
Answer: 2
In simple words: A quadrilateral has 2 diagonals.

Exam Tip: The two diagonals connect opposite vertices - one links the first and third vertices, while the other connects the second and fourth.

Question 5. (v)
Answer: opposite
In simple words: Two diagonals of a quadrilateral connect the opposite corners.

Exam Tip: Opposite vertices are those that do not share a common side - they are diagonally across from each other.

Question 6. (vi)
Answer: 360°
In simple words: When you add up all four interior angles of any quadrilateral, you always get 360 degrees.

Exam Tip: This is a key property - the sum of interior angles is constant for all quadrilaterals, regardless of their shape or size.

Question 7. A quadrilateral ABCD has the following properties: (i) There are four pairs of adjacent sides, namely (AB,BC), (BC,CD), (CD,DA) and (DA,AB). (ii) There are two pairs of opposite sides, namely (AB,DC) and (AD,BC). (iii) There are four pairs of adjacent angles, namely \( \angle A, \angle B \), \( \angle B, \angle C \), \( \angle C, \angle D \) and \( \angle D, \angle A \). (iv) There are two pairs of opposite angles, namely \( \angle A, \angle C \) and \( \angle B, \angle D \). (v) There are two diagonals, namely AC and BD.
Answer: The statement correctly identifies all key features of a quadrilateral. Four pairs of adjacent sides share a common vertex. Two pairs of opposite sides do not share any common vertex. Four pairs of adjacent angles sit next to each other around the quadrilateral. Two pairs of opposite angles are situated across from one another. The two diagonals are the line segments connecting non-adjacent vertices.
In simple words: Adjacent means next to each other, while opposite means across from each other. Every quadrilateral has these five categories of properties.

Exam Tip: Use clear labeling of vertices in order (A, B, C, D going around) to identify which pairs are adjacent and which are opposite - this prevents mixing them up.

Question 8. Prove that the sum of all the angles of a quadrilateral is 360°.
Answer: Let ABCD be a quadrilateral. Draw diagonal AC joining vertices A and C. This diagonal splits the quadrilateral into two triangles: triangle ABC and triangle ADC. Since the sum of angles in any triangle is 180°, we have for triangle ABC: \( \angle 2 + \angle 4 + \angle B = 180° \) and for triangle ADC: \( \angle 1 + \angle 3 + \angle D = 180° \). Adding these two equations together: \( (\angle 1 + \angle 2 + \angle 3 + \angle 4) + \angle B + \angle D = 360° \) or \( \angle A + \angle B + \angle C + \angle D = 360° \). Therefore, the sum of all the angles of a quadrilateral is 360°.
In simple words: Draw a line from one corner to the opposite corner. This creates two triangles inside. Each triangle's angles add up to 180°, so together they give 360°.

Exam Tip: Always show the diagonal construction clearly and label all angle parts - the examiner needs to see that you understand how the diagonal divides the quadrilateral into two triangles.

Question 9. The sum of all four angles of a quadrilateral is 360°. If the unknown angle is x°, find its measure. Given: 76 + 54 + 108 + x = 360.
Answer: The four angles of the quadrilateral must add up to 360°. Given that three angles measure 76°, 54°, and 108°, we can set up the equation: 76 + 54 + 108 + x = 360. Adding the known angles: 238 + x = 360. Solving for x: x = 360 - 238 = 122. The fourth angle measures 122°.
In simple words: Add up the three known angles, then subtract from 360° to find the missing angle.

Exam Tip: Always verify your answer by checking that all four angles sum to exactly 360° - this is a quick way to catch arithmetic mistakes.

Question 10. The measures of the angles of a quadrilateral are (3x)°, (5x)°, (7x)°, and (9x)°. Find the measure of each angle.
Answer: Since the sum of all four angles of a quadrilateral is 360°, we can write: (3x) + (5x) + (7x) + (9x) = 360. Combining like terms: 24x = 360. Solving for x: x = 15. Now we can find each angle by substituting x = 15 back into the expressions: First angle: 3(15) = 45°, Second angle: 5(15) = 75°, Third angle: 7(15) = 105°, Fourth angle: 9(15) = 135°. Therefore, the four angles measure 45°, 75°, 105°, and 135°.
In simple words: Set up an equation where all four angle expressions add to 360°, solve for x, then plug that value back into each expression to get all four angles.

Exam Tip: Verify by adding: 45 + 75 + 105 + 135 = 360 - always double-check that your final angles sum correctly.

Question 11. Three angles of a quadrilateral are equal. Find the measure of each angle if the fourth angle measures 120°.
Answer: Let each of the three equal angles measure x°. The sum of all angles in a quadrilateral is 360°. Therefore: x + x + x + 120 = 360. Simplifying: 3x + 120 = 360. Solving for x: 3x = 240, so x = 80. Each of the three equal angles measures 80°.
In simple words: Three angles are the same size, and one angle is 120°. Add them all to get 360°, then divide by 3 to find each equal angle.

Exam Tip: Check: 80 + 80 + 80 + 120 = 360 - this confirms all four angles add up correctly.

Question 12. In quadrilateral ABPD, the sum of angles A and B is 60° and 100°. One side is 100° and \( \angle APB = 12\angle A + \angle B + \angle P = 180° \). Using the given information, find \( \angle APB \).
Answer: Given that \( \angle A + \angle B = 60° + 100° = 160° \), and from triangle APB we know that \( 12\angle A + \angle B + \angle P = 180° \). Since \( \angle A + \angle B = 160° \), we can substitute: \( 100° + \angle P = 80° \). Therefore, \( \angle APB = 80° \).
In simple words: Use the sum of the two known angles and apply the triangle angle sum property to solve for the third angle.

Exam Tip: When working with angles in triangles formed within quadrilaterals, always identify which triangle you are working with and apply the 180° sum rule correctly.