question_answer

In the given figure/ PQ, RS, and UT are parallel lines.

(a) If \[c\text{ }=\text{ }57{}^\circ \text{ }and\text{ }a\text{ }=\frac{c}{3}\], then find the value of d.

(b) If \[c\text{ }=\text{ }75{}^\circ \text{ }and\text{ }a\text{ }=\frac{2}{5}c\], then find the value of b.

Answer:

Given, PQ||RS||UT

(a) Given, \[c=57{}^\circ \]and a =\[\frac{c}{3}\]

\[\because PQ||UT\]

\[\therefore \angle UTP=\angle QPT\]

[alternate interior angles]

\[\angle c=\angle a+\angle b\,\,[\because QPT=a+b]\]

\[57{}^\circ =\frac{57{}^\circ }{3}=\angle b\]

\[57{}^\circ 19{}^\circ =\angle b\]

\[\angle b=38{}^\circ \]

\[\therefore \angle b+\angle d=180{}^\circ \]

\[\angle d=180{}^\circ 38{}^\circ =142{}^\circ \]

(b)        \[Given,\text{ }c=75{}^\circ \text{ }and\text{ }a=\frac{2}{5}c\]

c = 75° and a = \[\frac{2}{5}\times 75{}^\circ =30{}^\circ \]

\[\therefore \angle c=\angle a+\angle b\]

[alternate interior angles]

\[75{}^\circ =30{}^\circ +\angle b\]

\[75{}^\circ 30{}^\circ =\angle b\]

\[\angle b=45{}^\circ \]