Question

Match the reasons with the statements given in the proof. Note that the figure may not be drawn to scale.

Given:
WX > XY
Prove:
1 > 4

Answer 1

Answer with explanation:

It is given that, in the triangle ,X Y Z, ∠1, is exterior, ∠2,∠3 , ∠4 are interior angles.Also,Ray,Y Z is produced.

W X > X Y.

To Prove: ⇒∠1 > ∠4

Proof with Reason

1. W X > X Y⇒⇒[Given]

2.∠3>∠4⇒⇒Angle Opposite to longer side is longer.

3.∠1=∠2 +∠3→Exterior angle of a triangle is equal to sum of two interior opposite angles.

4.≡Since,∠1 is sum of two angles,which are,∠2 and ∠3.So, ∠1 >∠2 and ∠1>∠3.→→If, a=b+c, and, c>0, then , a > b.

From, inequality , (2) and (4),that is transitive Property

5.∠1=∠4→→ [Transitive Property]

Answer 2

`\begin{array}{lll}&\text{Statement}&\text{Reason}\\1.&WX>XY&\text{Given}\\2.&\angle 3>\angle 4&\text{angle opposite longer side is larger angle}\\3.&\angle 1=\angle 3+\angle 2&\text{exterior angle is equal to the sum of remote interior angles}\\4.&\angle 1>\angle 3&\text{If a=b+c and } c>0, \text{then } a>b\\5.&\angle 1>\angle 4&\text{transitive property}\end{array}`

Explanation:

1. Given WX>XY.

2. Then angle 3 has larger measure than angle 4, because angle opposite longer side is larger angle. This means that

`\angle 3>\angle 4.`

3. Consider exterior angle 1. By the exterior angle theorem, the measure of the exterior angle is equal to the sum of the measures of remote interior angles. Then

`\angle 1=\angle 3+\angle 2.`

4. Note that angle 2 has positive measure and then

`\angle 1>\angle 3.`

5. Since
`\angle 1>\angle 3` and
`\angle 3>\angle 4,` by the transitive property,

`\angle 1>\angle 4.`