Explanation
We know, two angles are complementary when they add up to $$90^o.$$
Given, measure of one complementary angle is $$24^o.$$
$$\Rightarrow$$ Measure of other complementary angle $$=90^o-24^o$$ $$=66^o.$$
$$\therefore$$ Measure of a complementary angle of $$ 24^{o}=66^o$$.
Therefore, option $$A$$ is correct.
Draw a line $$XY$$ parallel to $${{PQ}}\parallel {{ST}}$$.
It is known that the sum of interior angles on the same side of the transversal is $$180^\circ $$.
So, $$\angle {{PQR}} + \angle {{QRX}} = 180^\circ $$
$$110^\circ + \angle {{QRX}} = 180^\circ $$
$$\angle {{QRX}} = 70^\circ $$
Similarly,
$$\angle {{RST}} + \angle {{SRY}} = 180^\circ $$
$$130^\circ + \angle {{SRY}} = 180^\circ $$
$$\angle {{SRY}} = 50^\circ $$
Now, by property of linear pair,
$$\angle {{QRX}} + \angle {{QRS}} + \angle {{SRY}} = 180^\circ $$
$$70^\circ + \angle {{QRS}} + 50^\circ = 180^\circ $$
$$\angle {{QRS}} = 60^\circ $$
Given, measure of one complementary angle is $$20^o.$$
$$\Rightarrow$$ Measure of other complementary angle $$=90^o-20^o$$ $$=70^o.$$
$$\therefore$$ Measure of a complementary angle of $$ 20^{o}=70^o$$.
Given, measure of one complementary angle is $$48^o.$$
$$\Rightarrow$$ Measure of other complementary angle $$=90^o-48^o$$ $$=42^o.$$
$$\therefore$$ Measure of a complementary angle of $$ 48^{o}=42^o$$.
Given, measure of one complementary angle is $$35^o.$$
$$\Rightarrow$$ Measure of other complementary angle $$=90^o-35^o$$ $$=55^o.$$
$$\therefore$$ Measure of a complementary angle of $$ 35^{o}=55^o$$.
Given, measure of one complementary angle is $$64^o.$$
$$\Rightarrow$$ Measure of other complementary angle $$=90^o-64^o$$ $$=26^o.$$
$$\therefore$$ Measure of a complementary angle of $$ 64^{o}=26^o$$.
Hence, the given statement is true.
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