Two lines \(\overleftrightarrow{PS}\) and \(\overleftrightarrow{QT}\)

1.	Two lines \(\overleftrightarrow{PS}\) and \(\overleftrightarrow{QT}\) intersect at M. Observe the figure and find x.
Answer» In the given figure \(\overleftrightarrow{PS}\) and \(\overleftrightarrow{QT}\) intersecting at M. ∠PMQ = ∠TMS (Vertically opposite angles) ∠QMS = ∠PMT (Vertically opposite angles) ∠QMR + ∠RMS = ∠PMT (we know ∠QMS = ∠QMR + ∠RMS) But, given, ∠QMR = 40°, ∠RMS = x° and ∠PMT = 105° ⇒ 40° + x° – 40° = 105° – 40° ∴ x° = 65° According to question, PS and QT intersect at M. So, angles QMS and PMT are equal ( Vertically opposite angles) Angle QMS =40+x° Angle PMT= 105 ° According to question, 40+x°=105° x=105-40= 65° Thus, the value of x=65°

Two lines \(\overleftrightarrow{PS}\) and \(\overleftrightarrow{QT}\) intersect at M. Observe the figure and find x.

Answer»

In the given figure \(\overleftrightarrow{PS}\) and \(\overleftrightarrow{QT}\) intersecting at M.

∠PMQ = ∠TMS (Vertically opposite angles)

∠QMS = ∠PMT (Vertically opposite angles)

∠QMR + ∠RMS = ∠PMT (we know ∠QMS = ∠QMR + ∠RMS)

But, given, ∠QMR = 40°, ∠RMS = x° and

∠PMT = 105°

⇒ 40° + x° – 40° = 105° – 40°

∴ x° = 65°

According to question,

PS and QT intersect at M.

So, angles QMS and PMT are equal ( Vertically opposite angles)

Angle QMS =40+x°

Angle PMT= 105 °

According to question,

40+x°=105°

x=105-40= 65°

Thus, the value of x=65°