Question

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

Answer 1

See Explanation

Step-by-step explanation:

1. By interior angle sum Postulate of a triangle.

$m\angle a+ m\angle y +m\angle z= 180\degree.... (1)\\\\m\angle a+ m\angle w +m\angle x= 180\degree.... (2)$

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

$m\angle a+ m\angle y +m\angle z= m\angle a+ m\angle w +m\angle x\\\\m\angle y +m\angle z= m\angle w +m\angle x$

Hence proved

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

Therefore, by remote interior angle theorem, we have:

$m\angle QPS + m\angle PQS= m\angle QSR\\\\x + m\angle PQS= 2x\\\\m\angle PQS= 2x-x\\\\m\angle PQS = x.... (1)\\\\m\angle QPS = x.... (given).... (2)$

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

$m\angle PQS = m\angle QPS\\\\\therefore \angle PQS \cong \angle QPS\\\\\therefore PS \cong QS\\(sides\: opposite \: to\: congruent \:\angle s)$

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

Thus proved.