Given:  $p \| q$.To do: We have to find the unknown angles.Solution:$\angle e$ and $125^o$ form a linear pair.Therefore, $\angle e+125^o=180^o$$\angle e=180^{\circ}-125^{\circ}$$=55^{\circ}$$\angle e=\angle f$                [Vertically opposite angles]Therefore, $\angle f=55^{\circ}$$\angle a=\angle f=55^{\circ}$      [Alternate interior angles]$\angle c=\angle a=55^{\circ}$      [Vertically opposite angles]$\angle d=125^{\circ}$  ... Read More

To do:We have to identify(i) the pairs of corresponding angles.(ii) the pairs of alternate interior angles.(iii) the pairs of interior angles on the same side of the transversal.(iv the vertically opposite angles.Solution:(i) The pairs of corresponding angles are $\angle 1$ and $\angle 5, \ \angle 2$ and $\angle 6, \ ... Read More

To do:We have to state the property that is used in each of the given statements.Solution:(i) Given $a \| b$We know that pairs of corresponding angles are equal if two lines are parallel.Therefore, $\angle1=\angle5$Corresponding angles property is used in the given statement.(ii) Given $\angle 4=\angle 6$If the pair of alternate ... Read More

To do:We have to fill in the given blanks.Solution:(i) If two angles are complementary, then the sum of their measures is $90^{\circ}$.(ii) If two angles are supplementary, then the sum of their measures is $180^{\circ}$.(iii) Two angles forming a linear pair are Supplementary.(iv) If two adjacent angles are supplementary, they ... Read More

To do:We have to name (i) Obtuse vertically opposite angles(ii) Adjacent complementary angles(iii) Equal supplementary angles(iv) Unequal supplementary angles(v) Adjacent angles that do not form a linear pairSolution:(i) Angles greater than $90^o$ and form a pair of vertically opposite angles are called Obtuse vertically opposite angles.Therefore, $\angle BOC$ and $\angle AOD$ are ... Read More

To do:We have to indicate which pairs of angles are:(i) Vertically opposite angles.(ii) Linear pairs.Solution:Vertically opposite angles are the angles that are formed by the intersection of two straight lines.A linear pair is defined as two adjacent angles that add up to $180^o$.Therefore, (i) Vertically opposite angles are $\angle 1$ and ... Read More

To do:We have to find whether two angles be supplementary if both of them are:(i) Acute(ii) Obtuse(iii) RightSolution:(i) We know that, Acute angle $< 90^{\circ}$Therefore, Acute angle $+$ Acute angle $< 90^{\circ}+90^{\circ}$This implies, Acute angle $+$ Acute angle $< 180^{\circ}$Therefore, two acute angles cannot be supplementary angles.(ii) We know that, ... Read More

Given:An angle is greater than $45^{\circ}$. To do:We have to find whether its complementary angle is greater than $45^{\circ}$ or equal to $45^{\circ}$ or less than $45^{\circ}$.Solution:Given angle is greater than $45^{\circ}$Let the given angle be $x^{\circ}$This implies, $x^{\circ} > 45^{\circ}$Complement of $x=90^{\circ}-x^{\circ}$Therefore, Complement of $x< 45^{\circ}$      [Since $x^{\circ} ... Read More

To do:We have to answer the given questions by observing the given figure.Solution:(i) Yes, $\angle 1$ and $\angle 2$ are adjacent angles.$\angle 1$ and $\angle 2$ have a common vertex $O$, a common arm $OC$ and $OA$ and $OE$ are on both sides of the common arm $OC$.(ii) No, $\angle ... Read More

To do:We have to identify which of the given pairs of angles are complementary and which are supplementary.Solution:Complementary angles:Two angles are said to be complementary if the sum of their measures is $90^o$.Supplementary angles:Two angles are said to be supplementary if the sum of their measures is $180^o$.Therefore, (i) $65^{\circ}+115^{\circ}=180^{\circ}$They ... Read More