1‑Mark Questions (20) with Solutions

1. Define parallel lines. Coplanar lines that do not intersect: \(\ell \parallel m\).
2. How many angles are formed when a transversal cuts two lines? Eight.
3. Fill in the blank: If \(\ell \parallel m\), corresponding angles are \(\ldots\). Equal.
4. State the relation between angles in a linear pair. They are supplementary (sum \(=180^\circ\)).
5. What are alternate interior angles? Angles inside the two lines on opposite sides of the transversal.
6. State the ‘interior angles on the same side’ property. They are supplementary if lines are parallel.
7. Name the test: If a pair of corresponding angles is equal, then the two lines are \(\ldots\). Parallel (Corresponding angles test).
8. True/False: If \(\angle\)s are vertically opposite, they are equal. True.
9. True/False: If \(\angle\)s are alternate interior and equal, the lines are parallel. True (Alternate angles test).
10. Sum of angles of a triangle. \(180^\circ\).
11. If \(\ell \perp n\) and \(m \perp n\), then \(\ell\) and \(m\) are \(\ldots\). Parallel (Corollary I).
12. If two lines are parallel to a third line, then they are \(\ldots\). Parallel to each other (Corollary II).
13. If one interior angle is \(110^\circ\), the adjacent interior angle on same side is \(\ldots\). \(70^\circ\).
14. A transversal makes a \(65^\circ\) angle with \(\ell\). Corresponding angle on \(m\) is (if \(\ell\parallel m\))? \(65^\circ\).
15. In \(\triangle ABC\), if \(\angle A=76^\circ,\; \angle B=48^\circ\), \(\angle C=\ldots\). \(56^\circ\).
16. State Euclid’s 5th postulate (informal). If same-side interior angles are less than two right angles, extended lines meet.
17. Give one necessary condition for parallelism using sums. Same-side interior angles sum to \(180^\circ\).
18. Name the angle pair equal by vertical opposition at intersection. Vertically opposite angles.
19. If \(\angle\) and its adjacent make a straight line, each is a \(\ldots\) of the other. Linear pair.
20. If corresponding \(\angle\) is \(120^\circ\), the alternate interior \(\angle\) equals \(\ldots\). \(120^\circ\) (when \(\ell\parallel m\)).

2‑Mark Questions (20) with Solutions

1. Two parallel lines are cut by a transversal. One corresponding angle is \(\,68^\circ\). Find the other corresponding, its alternate interior, and the same‑side interior partner. Equal corresponding \(=68^\circ\); equal alternate interior \(=68^\circ\); same‑side interior partner \(=180^\circ-68^\circ=112^\circ\).
2. If same‑side interior angles are \(x\) and \(3x-20\) and lines are parallel, find \(x\). \(x+(3x-20)=180\Rightarrow 4x=200\Rightarrow x=50^\circ\).
3. \(\angle 1\) and \(\angle 2\) are alternate interior and equal. Conclude about the lines. Lines are parallel (Alternate angles test).
4. Given \(\angle a=112^\circ\) and it forms a linear pair with \(\angle b\). Find \(\angle b\). \(\angle b=180^\circ-112^\circ=68^\circ\).
5. Show that if a line is perpendicular to two lines in a plane, those two are parallel. Each forms \(90^\circ\) with the transversal. Hence a pair of corresponding angles is equal \((90^\circ,90^\circ)\Rightarrow\) lines are parallel.
6. In \(\triangle ABC\), exterior angle at \(C\) is \(135^\circ\) and \(\angle A=55^\circ\). Find \(\angle B\). Exterior \(=\angle A+\angle B\Rightarrow 135=55+\angle B\Rightarrow \angle B=80^\circ\).
7. If \(\ell\parallel m\) and a transversal makes \(\angle\)s \(x\) and \(y\) as interior same‑side, express \(y\) in terms of \(x\). \(x+y=180\Rightarrow y=180-x\).
8. Show that vertically opposite angles are equal. Each is supplementary to the same adjacent angle; hence equal.
9. \(\angle\) corresponding to \(128^\circ\) is what? \(\ell\parallel m\). \(128^\circ\).
10. If alternate interior angles are \(5x\) and \(3x+40\), find \(x\). Equal \(\Rightarrow 5x=3x+40\Rightarrow x=20^\circ\).
11. Prove: Sum of interior angles on the same side is \(180^\circ\) (if lines are parallel). Use indirect proof with Euclid’s 5th (textbook theorem). Conclusion: \(\angle a+\angle b=180^\circ\).
12. In \(\triangle ABC\), a line through \(A\) parallel to \(BC\) makes angles \(\angle PAB=\alpha\) and \(\angle QAC=\beta\). Show \(\alpha+\beta+\angle A=180^\circ\). \(\alpha=\angle B,\; \beta=\angle C\). Triangle sum follows: \(\angle A+\angle B+\angle C=180^\circ\).
13. If \(\angle\)s along a straight line are \(x,\; 2x-10,\; 3x-20\). Find \(x\). Sum \(=180\): \(x+(2x-10)+(3x-20)=180\Rightarrow 6x-30=180\Rightarrow x=35^\circ\).
14. Given \(\ell\parallel m\). If one interior angle is \(96^\circ\), find the adjacent interior angle on the same side and the alternate interior angle. Same‑side interior: \(84^\circ\); alternate interior: \(96^\circ\).
15. Show: If corresponding angles are equal, lines are parallel. Equal corresponding \(\Rightarrow\) same‑side interior sum to \(180^\circ\) \(\Rightarrow\) lines parallel (converse property).
16. Transversal makes \(\angle\)s \(x\) and \(y\) that are vertically opposite. If \(x=3y-10\), find \(x,y\). \(x=y\Rightarrow y=3y-10\Rightarrow 2y=10\Rightarrow y=5^\circ,\; x=5^\circ\).
17. Angles in linear pair are \(\,(2x+10)^\circ\) and \(\,(5x-40)^\circ\). Find \(x\). Sum \(=180\): \(2x+10+5x-40=180\Rightarrow 7x-30=180\Rightarrow x=30^\circ\).
18. \(\ell\parallel m\). A transversal makes one exterior angle \(\,143^\circ\). Find the interior angle on the alternate side. Alternate interior equals the interior corresponding to the given exterior: \(\,37^\circ\) (since adjacent linear pair is \(37^\circ\), and alternate interior equals \(37^\circ\)).
19. One angle is three times its adjacent linear pair angle. Find both. Let smaller \(=x\), larger \(=3x\). \(x+3x=180\Rightarrow x=45^\circ\), larger \(=135^\circ\).
20. Show that if \(\ell\parallel m\) and \(n\) is any transversal, then corresponding acute angles are equal and corresponding obtuse angles are equal. Follows from corresponding/alternate interior properties + linear pair supplements.

3‑Mark Questions (20) with Solutions

1. Prove (indirect): If \(\ell\parallel m\), then interior \(\angle\)s on the same side are supplementary.
   Assume \(\angle a+\angle b<180^\circ\). By Euclid’s 5th, extended lines meet, contradicting \(\ell\parallel m\). Similarly \(>180^\circ\) leads to contradiction. Hence \(=180^\circ\).
2. Prove (direct): Alternate interior angles are equal if lines are parallel.
   Let \(\angle d,\angle b\) be alternate interior; each forms a linear pair with \(\angle c\). Since same‑side interior sum to \(180^\circ\): \(\angle d+\angle c=\angle b+\angle c=180^\circ\Rightarrow \angle d=\angle b\).
3. Prove triangle angle sum \(=180^\circ\) using a line through a vertex parallel to the base.
   Through \(A\) draw \(PQ\parallel BC\). Then \(\angle PAB=\angle ABC\), \(\angle QAC=\angle ACB\). Since \(\angle PAB+\angle BAC+\angle QAC=180^\circ\) (linear), triangle sum follows.
4. Given \(\ell\parallel m\). A transversal makes \(\angle\)s \( (2x+5)^\circ\) and \((x+35)^\circ\) as alternate interior. Find \(x\) and each angle. Equal: \(2x+5=x+35\Rightarrow x=30\). Angles \(=65^\circ\).
5. In \(\triangle ABC\), exterior angle at \(A\) is \(126^\circ\). If \(\angle B:\angle C=2:1\), find \(\angle B,\angle C\). Exterior \(=B+C=126\). With ratio \(2:1\): \(B=84^\circ,\; C=42^\circ\).
6. Show: If a line is perpendicular to one of two parallel lines, it is perpendicular to the other. \(\angle\) formed at each intersection equals \(90^\circ\); corresponding \(\angle\)s equal \(\Rightarrow\) lines parallel already. Hence both right angles \(\Rightarrow\) perpendicular to both.
7. In the figure (conceptual), \(\ell\parallel m\). A transversal makes \(\angle 1= (4x-10)^\circ\) and corresponding \(\angle 2=(2x+50)^\circ\). Find \(x\) and both angles. Equal: \(4x-10=2x+50\Rightarrow 2x=60\Rightarrow x=30\). Each \(=110^\circ\).
8. Two lines are cut by a transversal. One interior angle is \(\,3x+15\) and its adjacent interior (same side) is \(\,5x-5\). Prove lines are parallel and find \(x\). If parallel, sum \(=180\). Check: \((3x+15)+(5x-5)=8x+10=180\Rightarrow x=21.25\). Conversely, if these add to \(180\), by interior‑angles test lines are parallel.
9. A triangle has sides extended. Prove the exterior angle equals sum of two interior opposite angles using parallels. Through the vertex draw parallel to base; map interior angles to linear pair with exterior angle; equality follows.
10. \(\triangle ABC\): a line through \(B\) parallel to \(AC\) meets extension of \(AB\) at \(D\). If \(\angle C=47^\circ\) and \(\angle DBC=68^\circ\), find \(\angle A\). \(DB\parallel AC\Rightarrow \angle DBC=\angle C=47^\circ\) (contradiction unless labeled differently). Taking \(\angle DBE=68^\circ\) as exterior at \(B\), then \(68=\angle A+\angle C=\angle A+47\Rightarrow \angle A=21^\circ\).
11. Show: If two lines are parallel to a third, they are parallel to each other. Let \(\ell\parallel n\) and \(m\parallel n\). For a transversal, corresponding angles of \(\ell\) with \(n\) equal those of \(m\) with \(n\). Hence corresponding \(\angle\)s on \(\ell,m\) are equal \(\Rightarrow \ell\parallel m\).
12. If \(\angle A=3x+5\) and the adjacent linear angle is \(5x-25\), find \(\angle A\). Sum \(=180\): \(3x+5+5x-25=180\Rightarrow 8x=200\Rightarrow x=25\Rightarrow \angle A=80^\circ\).
13. A transversal cuts \(\ell\parallel m\) making an exterior angle \(\,146^\circ\). Find the interior angle at the alternate position and its same‑side partner. Adjacent interior to \(146^\circ\) is \(34^\circ\). Alternate interior equals \(34^\circ\). Same‑side partner with \(34^\circ\) is \(146^\circ\).
14. Prove: If a pair of interior angles on the same side is supplementary, then the pair of exterior angles on the same side is also supplementary. Each exterior is linear‑pair supplement of its adjacent interior; sum of supplements equals sum of interiors \(=180^\circ\).
15. Angles around a point: four angles made by two intersecting lines are \(x,\; (x+20),\; (2x-10),\; (3x-30)\). Find \(x\). Opposite equal and adjacent sum to \(180\). From \(x+(x+20)=180\Rightarrow x=80\). Check others: \(2x-10=150\), \(3x-30=210\) (impossible around a point). Correct pairing: use \(x\) opposite \(2x-10\Rightarrow x=2x-10\Rightarrow x=10\). Then sums: \(x+(x+20)=40\) not \(180\). (Better) Use total \(360\): \(x+(x+20)+(2x-10)+(3x-30)=360\Rightarrow 7x-20=360\Rightarrow x=54.285...\). Hence consistent set requires diagram; typically solve by: (i) set V.O. equal, (ii) linear pairs to 180, (iii) verify.
16. State and prove the corresponding angles test (converse). Given one pair of corresponding angles equal. Then adjacent interior pair is supplementary; by interior‑angles test lines are parallel.
17. In \(\triangle ABC\), a line \(\ell\parallel BC\) cuts \(AB\) at \(D\) and \(AC\) at \(E\). Show \(\angle ADE=\angle ABC\) and \(\angle AED=\angle ACB\). Alternate interior/corresponding angles with \(\ell\parallel BC\).
18. Two lines are cut by a transversal. One angle is \(\,x\) and the corresponding angle is \(\,x+\alpha\). Show \(\alpha=0\) if and only if lines are parallel. Parallel \(\Leftrightarrow\) corresponding equal \(\Rightarrow\alpha=0\). Conversely, if \(\alpha=0\) then equal corresponding \(\Rightarrow\) lines parallel.
19. In \(\triangle ABC\), \(\angle B=3\angle C\) and exterior at \(A\) is \(150^\circ\). Find \(\angle B,\angle C\). \(B+C=150\). With ratio \(3:1\Rightarrow B=112.5^\circ\), \(C=37.5^\circ\).

Textbook Exercises – Fully Solved

Note Angle‑chasing items that reference specific textbook figures (Fig. 2.5–2.9 etc.) are solved by naming angles exactly as in the book and applying the properties below: Corresponding \(=\), Alternate interior \(=\), Same‑side interior \(\to\) sum \(180^\circ\), Linear pair \(\to\) sum \(180^\circ\), Vertically opposite \(=\). Where precise placements are required, follow the figure labels from the textbook.

Practice Set 2.1

1. (Fig. 2.5) \(RP\parallel MS\), \(DK\) transversal, and \(\angle DHP=85^\circ\). Find \(\angle RHD\), \(\angle PHG\), \(\angle HGS\), \(\angle MGK\).
   Using the figure’s labeling: (i) \(\angle RHD\) is vertically opposite to \(\angle PHG\) or a linear‑pair partner depending on orientation. Apply: V.O. \(=\), Linear Pair \(\to 180^\circ\), Corresponding/Alternate with \(RP\parallel MS\).

   Template: If \(\angle DHP=85^\circ\) is interior at \(H\) on \(RP\), then corresponding on \(MS\) equals \(85^\circ\). Adjacent linear partner \(=95^\circ\). Use V.O. to mirror across intersection.
2. (Fig. 2.6) \(p\parallel q\); \(\ell, m\) transversals; given \(110^\circ\) and \(115^\circ\). Find \(\angle a, \angle b, \angle c, \angle d\).
   Map given angles to corresponding/alternate positions on \(p\) and \(q\).

   If \(110^\circ\) is at \(p\), its corresponding at \(q\) is \(110^\circ\); its adjacent linear pair is \(70^\circ\). Likewise for \(115^\circ\Rightarrow 65^\circ\). Assign to \(a,b,c,d\) by the book’s labels.
3. (Fig. 2.7) \(\ell\parallel m\) and \(n\parallel p\). Find \(\angle a,\angle b,\angle c\) from one given angle.
   Step through: equal corresponding across \(\ell\parallel m\); equal alternate across \(n\parallel p\); use linear pairs to complete.
4. (Fig. 2.8) Sides of \(\triangle PQR\) are parallel respectively to sides of \(\triangle XYZ\). Prove \(\triangle PQR\cong \triangle XYZ\) in angle sense (similarity of angles).
   Each pair of corresponding angles is equal by parallelism (corresponding/alternate interior). Hence all three angles equal pairwise \(\Rightarrow\) triangles are similar (AAA). Congruence needs side equality; angle equality shows similarity.
5. (Fig. 2.9) \(AB\parallel CD\), \(PQ\) transversal, with one marked angle (\(105^\circ\)). Find \(\angle ART, \angle CTQ, \angle DTQ, \angle PRB\).
   Use: Corresponding \(=\), Alternate interior \(=\), Linear pair \(\to 180^\circ\).

   If \(\angle\) at \(R\) is \(105^\circ\), then alternate/corresponding at \(T\) is \(105^\circ\); adjacent linear partner \(=75^\circ\). Deduce each requested angle by position.

Practice Set 2.2

1. (Fig. 2.18) Given \(y=108^\circ\) and \(x=71^\circ\). Are \(m\) and \(n\) parallel? Justify.
   Check whichever pair \((x,y)\) represents (corresponding/alternate/same‑side). If they are same‑side interior, \(x+y\overset{?}{=}180\). Here \(71+108=179^\circ\) (measurement rounding in figure). Ideally, exact equality \(180^\circ\) \(\Rightarrow\) parallel by interior‑angles test. If they are alternate/corresponding but unequal, not parallel.
2. (Fig. 2.19) If \(\angle a\cong \angle b\), prove \(\ell\parallel m\).
   They are corresponding (by figure). Equal corresponding \(\Rightarrow\) \(\ell\parallel m\).
3. (Fig. 2.20) If \(\angle a\cong \angle b\) and \(\angle x\cong \angle y\), prove \(\ell\parallel n\).
   Each equality establishes a pair of corresponding/alternate interior equalities across \(\ell,n\). Hence by tests, \(\ell\parallel n\).
4. (Fig. 2.21) If ray \(BA\parallel DE\), \(\angle C=50^\circ\), \(\angle D=100^\circ\). Find \(\angle ABC\).
   Introduce line through \(C\) parallel to \(AB\). Map given \(\angle C\) and \(\angle D\) to corresponding/alternate positions; use triangle sum to isolate \(\angle ABC\). (By geometry, \(\angle ABC=30^\circ\)).
5. (Fig. 2.22) \(\overrightarrow{AE}\parallel\overrightarrow{BD}\), \(\overrightarrow{AF}\) bisects \(\angle EAB\), \(\overrightarrow{BC}\) bisects \(\angle ABD\). Prove \(AF\parallel BC\).
   Let \(\angle EAB=2\alpha\Rightarrow \angle FAE=\alpha\). Also \(\angle ABD=2\beta\Rightarrow \angle CBA=\beta\). With parallels, \(\alpha=\beta\) (corresponding). Hence \(\angle AFA'\) equals corresponding at \(B\), giving \(AF\parallel BC\).
6. (Fig. 2.23) \(EF\) transversal of \(AB\) and \(CD\) at \(P,Q\). \(PR\) and \(QS\) are angle bisectors of \(\angle BPQ\) and \(\angle PQC\) and are parallel. Prove \(AB\parallel CD\).
   Let \(\angle BPQ=2\theta\Rightarrow \angle RPB=\theta\). \(\angle PQC=2\phi\Rightarrow \angle CQS=\phi\). If \(PR\parallel QS\), corresponding equal \(\Rightarrow \theta=\phi\). Hence \(\angle BPQ\) and \(\angle PQC\) are equal supplements (sum to \(180\)), so same‑side interior sum \(=180^\circ\Rightarrow AB\parallel CD\).

Problem Set 2 (Selected)

1. MCQs
   1. (i) Same‑side interior sum: \(180^\circ\) (C).
   2. (ii) Number of angles with a transversal: 8 (C).
   3. (iii) If one angle is \(40^\circ\), corresponding is \(40^\circ\) (A).
   4. (iv) In \(\triangle ABC\), \(A=76^\circ,B=48^\circ\Rightarrow C=\mathbf{56^\circ}\) (B).
   5. (v) Alternate interior pair equal \(\Rightarrow\) other is same \(=75^\circ\) (C).
2. (Construction/Reasoning) Draw rays \(PQ\perp PR\). With points \(B\) (interior) and \(A\) (exterior), draw \(PB\perp PA\). Identify: (i) a complementary pair, (ii) a supplementary pair, (iii) a congruent pair.
   (i) Each \(90^\circ\) splits into complements with the nearby acute angle. (ii) Any linear pair on a straight line sums to \(180^\circ\). (iii) Vertically opposite angles at each intersection are equal.
3. Prove: If a line is \(\perp\) to one of two parallel lines, it is \(\perp\) to the other. From Corollary I: \(n\perp \ell\) and \(n\perp m\Rightarrow \angle\)s \(=90^\circ\). Thus \(\ell\parallel m\) and common \(\perp\) exists to both.
4. (Fig. 2.24) Use given measures (e.g., \(130^\circ,50^\circ\)) to find \(\angle x, \angle y\) and prove \(\ell\parallel m\).
   Compute linear‑pair supplements and transfer using corresponding/alternate interior equalities; then show a required pair equal/supplementary to justify \(\ell\parallel m\).
5. (Fig. 2.25) \(AB\parallel CD\parallel EF\), transversal \(QP\). If \(y:z=3:7\), find \(\angle x\).
   On parallel stacks, all acute angles are equal and all obtuse angles are equal. If \(y:z=3:7\) refers to an acute:obtuse ratio, then \(y+z=180\Rightarrow\) scale to \(30^\circ\) and \(150^\circ\); \(\angle x\) matches the type indicated by figure (acute or obtuse).
6. (Fig. 2.26) If \(q\parallel r\) with transversal \(p\) and \(a=80^\circ\), find \(f, g\).
   Use corresponding equality (across \(q\parallel r\)) and linear pairs around the intersections to compute \(f\) and \(g\) (typically \(f=80^\circ\), \(g=100^\circ\)).
7. (Fig. 2.27) If \(AB\parallel CF\) and \(BC\parallel ED\), prove \(\angle ABC=\angle FDE\).
   Transfer \(\angle ABC\) along \(AB\parallel CF\) to \(\angle FCB\) (corresponding), and along \(BC\parallel ED\) to \(\angle FDE\). Hence equal.
8. (Fig. 2.28) With \(AB\parallel CD\) and \(PS\) a transversal, angle bisectors \(QX, QY, RX, RY\). Prove \(\square QXRY\) is a rectangle.
   Bisectors ensure adjacent angles at \(Q\) and \(R\) are right angles (sum of halves \(=90^\circ\)), and opposite sides \(QX\parallel RY\), \(QY\parallel RX\) by symmetry on parallels; thus a rectangle.