Mathematics
In a quadrilateral ABCD; angles A, B, C and D are in the ratio 3 : 2 : 1 : 4. Prove that AD is parallel to BC.
Answer
Given: ABCD is the quadrilateral such that angles A, B, C and D are in the ratio 3 : 2 : 1 : 4.
To prove: AD is parallel to BC.
Proof: Let the angles of the quadrilateral ABCD be 3a, 2a, 1a and 4a, respectively.
The sum of the angles in any quadrilateral is 360°. Therefore,
3a + 2a + 1a + 4a = 360°
10a = 360°
a = 36°
∠A = 3a = 3 x 36° = 108°
∠B = 2a = 2 x 36° = 72°
∠C = a = 1 x 36° = 36°
∠D = 4a = 4 x 36° = 144°
The sum of consecutive interior angles is:
⇒ ∠A + ∠B = 108° + 72° = 180°
⇒ ∠C + ∠D = 36° + 144° = 180°
Since the consecutive interior angles add up to 180°, lines AD and BC are parallel.
Hence, AD is parallel to BC.
Related Questions
The ratio between the number of sides of two regular polygons is 3 : 4 and ratio between the sum of their interior angles is 2 : 3. Find the number of sides in each polygon.
If the difference between an interior angle of a regular polygon of (n + 1) sides and an interior angle of a regular polygon of n sides is 4°; find the value of n.
Also, state the difference between their exterior angles.
In a quadrilateral ABCD, AB = CD and ∠B = ∠C. Prove that:
(i) AC = DB,
(ii) AD is parallel to BC.
In the diagram below; P and Q are midpoints of sides BC and AD respectively of the parallelogram ABCD. If side AB = diagonal BD; prove that the quadrilateral BPDQ is a rectangle.
