**Question :In the given figure, line RP || line MS and line DK is their transversal. ∠DHP = 85°. Find the measures of following angles.**

**i. ∠RHDii. ∠PHGiii. ∠HGSiv. ∠MGKSolution:**i. ∠DHP = 85° …..(i)

∠DHP + ∠RHD = 180° [Angles in a linear pair]

85° + ∠RHD = 180°

∴ ∠RHD = 180°- 85°

∴ ∠RHD = 95° …..(ii)

ii. ∠PHG = ∠RHD [Vertically opposite angles]

∴ ∠PHG = 95° [From (ii)]

iii. line RP || line MS and line DK is their transversal. [Corresponding angles]

∴ ∠HGS = ∠DHP …..(iii) [From (i)]

iv. ∠HGS = 85° [Vertically opposite angles]

∴ ∠MGK = ∠HGS ∠MGK = 85° [From (iii)]

**Question 2.In the given figure line p line q and line l and line m are tranversals.Measures of some angles are shown. Hence find the measures of ∠a, ∠b, ∠c, ∠d.Solution:**

i. 110 + ∠a = 180° [Angles in a linear pair]

∴ ∠a = 180° – 110°

∴ ∠a = 70°

ii. consider ∠e as shown in the figure line p || line q, and line lis their transversal.

∠e + 110° = 180° [Interior angles]

∴ ∠e = 180° – 110°

∴ ∠e = 70°

But, ∠b = ∠e [Vertically opposite angles]

∴ ∠b = 70°

iii. line p || line q, and line m is their transversal.

∴ ∠c = 115° [Corresponding angles]

iv. 115° + ∠d = 180° [Angles in a linear pair]

∴ ∠d = 180° – 115°

∴ ∠d = 65°

**Question 3.In the given figure, line 11| line m and line n || line p. Find ∠a, ∠b, ∠c from the given measure of an angle.Solution:**

i. consider ∠d as shown in the figure

line l || line m, and line p is their transversal.

∴ ∠d = 45° [Corresponding angles]

Now, ∠d + ∠b = 180° [Angles in a linear pair]

∴ 45° +∠b = 180°

∴ ∠b = 180° – 45°

∴ ∠b = 135° …..(i)

ii. ∠a = ∠b [Vertically opposite angles]

∴ ∠a = 135° [From (i)]

iii. line n || line p, and line m is their transversal.

∴ ∠c = ∠b [Corresponding angles]

∴ ∠c = 135° [From (i)]

**Question 4.In the given figure, sides of ∠PQR and ∠XYZ are parallel to each other. Prove that, ∠PQR ≅ ∠XYZ.Given: Ray YZ || ray QRandray YX || ray QPTo prove: ∠PQR ≅ ∠XYZConstruction: Extend ray YZ in the opposite direction. It intersects ray QP at point S.Solution:**

Proof:

Ray YX || ray QP [Given]

Ray YX || ray SP and seg SY is their transversal [P-S-Q]

∴ ∠XYZ ≅ ∠PSY ……(i) [Corresponding angles]

ray YZ || ray QR [Given]

ray SZ || ray QR and seg PQ is their transversal. [S-Y-Z]

∴ ∠PSY ≅ ∠SQR [Corresponding angles]

∴ ∠PSY ≅ ∠PQR …….. (ii) [P-S-Q]

∴ ∠PQR ≅ ∠XYZ [From (i) and (ii)]

**Question 5.In the given figure, line AB || line CD and line PQ is transversal. Measure of one of the angles is given. Hence find the measures of the following angles.i. ∠ARTii. ∠CTQiii. ∠DTQiv. ∠PRBSolution:**

i. ∠BRT = 105° ….(i)

∠ART + ∠BRT = 180° [Angles in a linear pair]

∴ ∠ART + 105° = 180°

∴ ∠ART = 180° – 105°

∴ ∠ART = 75° …(ii)

ii. line AB || line CD and line PQ is their transversal.

∴ ∠CTQ = ∠ART [Corresponding angles]

∴ ∠CTQ = 75° [From (ii)]

iii. line AB || line CD and line PQ is their transversal.

∴ ∠DTQ = ∠BRT [Corresponding angles]

∴ ∠DTQ = 105° [From (i)]

iv. ∠PRB = ∠ART [Vertically opposite angles]

∴ ∠PRB = 75° [From (ii)]

**Intext Questions and Activities**

**Question 1.Angles formed by two lines and their transversal. (Textbook pg, no. 13)When a transversal (line n) intersects two lines (line l and m) in two distinct points, 8 angles are formed as shown in the figure. Pairs of angles formed out of these angles are as follows:**Pairs of corresponding angles

i. ∠d, ∠h

ii. ∠a, ∠e

iii. ∠c, ∠g

iv. ∠b, ∠f

Pairs of alternate interior angles

i. ∠c, ∠e

ii. ∠b, ∠h

Pairs of alternate exterior angles

i. ∠d, ∠f

ii. ∠a, ∠g

Pairs of interior angles on the same side of the transversal

i. ∠c, ∠h

ii. ∠b, ∠e

Some important properties:

1. When two lines intersect, the pairs of vertically opposite angles formed are congruent.

Example:

In the given diagram,

line l and m intersect at point P.

The pairs of vertically opposite angles that are congruent are:

i. ∠a ≅ ∠b

ii. ∠c ≅ ∠d

2. The angles in a linear pair are supplementary.

Example:

For the given diagram,

∠a and ∠c are in linear pair

∴ ∠a + ∠c = 180°

Also, ∠d and ∠b are in linear pair

∴ ∠d + ∠b = 180°

3. When one pair of corresponding angles is congruent, then all the remaining pairs of corresponding angles are congruent.

Example:

In the given diagram,

If ∠a ≅ ∠b

then ∠e ≅ ∠f, ∠c ≅ ∠d and ∠g ≅ ∠h

4. When one pair of alternate angles is congruent, then all the remaining pairs of alternate angles are congruent.

Example:

For the given diagram,

If ∠e ≅ ∠d, then ∠g ≅ ∠b

Also, ∠a ≅ ∠h and ∠c ≅ ∠f

5. When one pair of interior angles on one side of the transversal is supplementary, then the other pair of interior angles is also supplementary.

Example:

For the given diagram,

If ∠e + ∠b = 180°, then ∠g + ∠d = 180°.