Question

Prove the following statement by proving its contrapositive. Begin by writing what is given and what is to be proved.

If $m\angle A+m\angle B\ne 180°$then $m\angle D+m\angle C\ne 180°$.

Short answer

Hence proved that if$m\angle A+m\angle B\ne 180°$ then $m\angle D+m\angle C\ne 180°$.

Step-by-step solution

Step 1. Define contrapositive statement

If the given statement is ‘’If p then q’’, the contrapositive is ‘’If not q then not p’’.

The given statement is if$m\angle A+m\angle B\ne 180°$ then $m\angle D+m\angle C\ne 180°$.

Step 2. Let the given statement and find its contrapositive

The contrapositive of the given statement is if $m\angle D+m\angle C=180°$then $m\angle A+m\angle B=180°$.

Step 3. Prove the statement

Since the sum of all angles of a quadrilateral is 360, therefore following will hold from the given figure as:

$\begin{array}{c}m\angle A+m\angle B+m\angle D+m\angle C=360°\\ m\angle A+m\angle B+{180}^o=360°\\ m\angle A+m\angle B=360°-180°\\ m\angle A+m\angle B=180°\end{array}$