Question

Can someone help me on this Proofs problems? It's just number one and two.

Answer 1

See Explanation

Explanation:

1. By interior angle sum Postulate of a triangle.

`m\angle a+ m\angle y +m\angle z= 180\degree.... (1)\\\\</p><p>m\angle a+ m\angle w +m\angle x= 180\degree.... (2)\\\\`

From equations (1) & (2), we find:

`m\angle a+ m\angle y +m\angle z= m\angle a+ m\angle w +m\angle x\\\\</p><p>m\angle y +m\angle z= m\angle w +m\angle x\\`

Hence proved

2. In
`\triangle PQS, \: \angle QSR` is exterior angle.

Therefore, by remote interior angle theorem, we have:

`m\angle QPS + m\angle PQS= m\angle QSR\\\\</p><p>x + m\angle PQS= 2x\\\\</p><p>m\angle PQS= 2x-x\\\\</p><p>m\angle PQS = x.... (1)\\\\</p><p>m\angle QPS = x.... (given).... (2)\\`

From equations (1) & (2), we find:

`m\angle PQS = m\angle QPS\\\\</p><p>\therefore \angle PQS \cong \angle QPS\\\\</p><p>\therefore PS \cong QS\\(sides\: opposite \: to\: congruent \:\angle s) \\\\`

`\therefore \triangle PQS` is an isosceles triangle.

Thus proved.