Chapter 1, Problem 27RP

To determine

To prove: In the given figure $\angle ABD\cong \angle FEH$ i.e. both the angles are congruent

Explanation of Solution

Given information: Two angles $\angle ABC$ and $\angle FEH$ as shown in figure (i) and (ii) .

  $\angle ABC$ is a right angle which means $\angle ABC=90°$ also $\angle \text{DBC=20}°\text{,}\angle \text{FEG=40}°\text{ , }\angle GEH=30°$

Formula used: Rule of adjacent angles and definition of congruency is used.

Proof:

Statement	Reason
$\angle ABC=90°\text{ and }\angle \text{DBC=20}°$	Given in the question
$\angle ABD=\angle ABC-\angle DBC$	$\angle ABD\text{ and }\angle DBC$ adjacent angles as they have a common arm BD and non-common arms AB and BC have same vertex i.e. B
$\angle ABD=90°-20°$	Substituting given values of angles $\angle ABC\text{ and }\angle DBC$
$\angle ABD=70°$	Solving calculation for previous step
$\angle FEG=40°\text{ and }\angle \text{GEH=30}°$	Given in the question
$\angle FEH=\angle FEG+\angle GEH$$$	$\angle FEG\text{ , }\angle \text{GEH}$ are adjacent angles as they have a common arm EG and non-common arms FE and EH have same vertex i.e. E
$\angle FEH=40°+30°$	Substituting given values of angles $\angle FEG\text{ , }\angle GEH$
$\angle FEH=70°$	Solving calculation for previous step
$\angle ABD=\angle FEH$	As both have same values
$\angle ABD\cong \angle FEH$	Using definition of congruency of angles which says angles of same measurement are congruent