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Q21.

Expert-verified
Found in: Page 212

Geometry

Book edition Student Edition
Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen
Pages 227 pages
ISBN 9780395977279

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 $m\angle D+m\angle C\ne 180°$.

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

See the 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:

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