Question

If the angles of a quad (taken in order) are in the ratio 1:2:3:4, find the measure of all the angles

of the quad. Hence show that the quadrilateral is a trapezium.​

Answer 1

Diagram:

$\setlength{\unitlength}{1.2 cm}\begin{picture}(0,0)\linethickness{0.5mm}\qbezier(0,0)(0,0)(1,3)\qbezier(5,0)(5,0)(4,3)\qbezier(1,3)(1,3)(4,3)\qbezier(3,0)(8.2,0)(0,0)\put(-0.5,-0.3){ \sf A }\put(5.3,-0.3){ \sf B }\put(4.2,3.1){ \sf C }\put(0.6,3.1){ \sf D }\end{picture}$

Given:

• ∠A = x
• ∠B = 2x
• ∠C = 3x
• ∠D = 4x

To find:

1. Measure of each angles
2. Show that Quadrilateral is a Trapezium.

Method:

First let's understand the concept;

Concept used:

Sum of all angles in a quadrilateral is 360°

x + 2x + 3x + 4x = 360°

⇒ 10x = 360°

⇒ x = 360° ÷ 10

⇒ x = 36°

For finding the measure of all angles;

∠A = x = 36°

∠B = 2x = 2(36) = 72°

∠C = 3x = 3(36) = 108°

∠D = 4x = 4(36) = 144°

Now the 4 angles are 36°, 72°, 108° and 144°

To show that ABCD is a trapezium;

By corresponding angles,

∠D + ∠A = 180°

⇒ 144 + 36 = 180°

⇒ 180° = 180°

LHS = RHS

By converse of Corresponding angles,

AB || DC

Similarly By Corresponding angles,

∠C + ∠D = 180°

⇒ 108° + 144° ≠ 180°

By Converse of Corresponding Angles,

AD ∦ BC

2 sides are parallel but other 2 sides are not parallel.

So ABCD is a Trapezium

Answer 2

Answer:

Given:

∠A = x

∠B = 2x

∠C = 3x

∠D = 4x

To find:

Measure of each angles

Show that Quadrilateral is a Trapezium.

Method:

First let's understand the concept;

Concept used:

Sum of all angles in a quadrilateral is 360°

x + 2x + 3x + 4x = 360°

⇒ 10x = 360°

⇒ x = 360° ÷ 10

⇒ x = 36°

For finding the measure of all angles;

∠A = x = 36°

∠B = 2x = 2(36) = 72°

∠C = 3x = 3(36) = 108°

∠D = 4x = 4(36) = 144°

Now the 4 angles are 36°, 72°, 108° and 144°

To show that ABCD is a trapezium;

By corresponding angles,

∠D + ∠A = 180°

⇒ 144 + 36 = 180°

⇒ 180° = 180°

LHS = RHS

By converse of Corresponding angles,

AB || DC

Similarly By Corresponding angles,

∠C + ∠D = 180°

⇒ 108° + 144° ≠ 180°

By Converse of Corresponding Angles,

AD ∦ BC

2 sides are parallel but other 2 sides are not parallel.

So ABCD is a Trapezium