7. A Tale of Three Intersecting Lines​

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Chapter 7 • A Tale of Three Intersecting Lines • 1-Mark Q&A

Chapter 7: A Tale of Three Intersecting Lines 1-Mark Q&A

  1. Write the symbol used to denote a triangle.
    \( \triangle \)
  2. Name the three angles of \( \triangle ABC \).
    \( \angle A, \angle B, \angle C \) (i.e., \( \angle CAB, \angle ABC, \angle BCA \)).
  3. Define an equilateral triangle.
    All three sides are equal; each angle is \(60^\circ\).
  4. Define an isosceles triangle.
    Exactly two sides are equal (base angles equal).
  5. Define a scalene triangle.
    All three sides are unequal.
  6. State the angle-sum property of a triangle.
    \( \angle A+\angle B+\angle C=180^\circ \).
  7. What is an altitude of a triangle?
    A perpendicular from a vertex to the opposite side (or its extension).
  8. What is the included angle between sides \(AB\) and \(AC\)?
    \( \angle A \).
  9. State the triangle inequality in words.
    Each side is less than the sum of the other two.
  10. Check if \(3,4,8\) can form a triangle.
    No, \(8 \not< 3+4\).
  11. Check if \(4,5,8\) can form a triangle.
    Yes, \(8<4+5=9\).
  12. Define a right-angled triangle.
    A triangle with one angle \(=90^\circ\).
  13. What is an exterior angle at vertex \(C\)?
    The angle formed by extending one side at \(C\) and the adjacent side, e.g., \( \angle ACD \).
  14. Exterior angle theorem (statement).
    An exterior angle equals the sum of its two remote interior angles.
  15. If \( \angle A=50^\circ\) and \( \angle B=60^\circ\), find \( \angle C\).
    \( \angle C=70^\circ\).
  16. If \( \angle A=70^\circ\) and \( \angle B=70^\circ\), can all angles be \(70^\circ\)?
    No, \( \angle C=40^\circ\); equal-angles triangle must have \(60^\circ\) each.
  17. Name the construction with two sides and included angle known.
    S-A-S (two Sides and included Angle).
  18. Name the construction with two angles and the included side known.
    A-S-A (two Angles and included Side).
  19. Altitude in an obtuse triangle may fall where?
    Outside the triangle (on the extension of the base).
  20. Can an equilateral triangle be right-angled?
    No (each angle is \(60^\circ\)).
Chapter 7 • A Tale of Three Intersecting Lines • 2-Mark Q&A

Chapter 7: A Tale of Three Intersecting Lines 2-Mark Q&A

  1. Check triangle: \(2,2,5\).
    No; \(5\not<2+2\).
  2. Check triangle: \(3,4,6\).
    Yes; \(6<3+4=7\).
  3. Check triangle: \(2,4,8\).
    No; \(8\not<2+4\).
  4. Check triangle: \(5,5,8\).
    Yes; \(8<5+5=10\).
  5. Check triangle: \(10,20,25\).
    Yes; \(25<10+20=30\).
  6. Check triangle: \(10,20,35\).
    No; \(35\not<10+20\).
  7. Check triangle: \(24,26,28\).
    Yes; each side is less than sum of other two.
  8. Find \( \angle C\) if \( \angle A=36^\circ, \angle B=72^\circ\).
    \( \angle C=180^\circ-108^\circ=72^\circ\).
  9. Find \( \angle A\) if \( \angle B=150^\circ, \angle C=15^\circ\).
    \( \angle A=180^\circ-165^\circ=15^\circ\).
  10. Exterior angle at \(C\) if \( \angle A=50^\circ, \angle B=60^\circ\).
    Exterior \(= \angle A+\angle B=110^\circ\).
  11. In \( \triangle ABC\), \( \angle B=\angle C\) and \( \angle A=50^\circ\). Find \( \angle B,\angle C\).
    \( \angle B=\angle C=\dfrac{180^\circ-50^\circ}{2}=65^\circ\).
  12. State whether two angles \(90^\circ\) and \(85^\circ\) with an included side form a triangle.
    No; sum \(=175^\circ\) & third angle \(=5^\circ\) exists, but the two given angles leave very small angle—triangle exists; however if both base angles \( \ge 90^\circ\), lines won’t meet. Here it does meet, so triangle possible.
  13. Decide: with two base angles \( \ge 90^\circ\) and a base, can a triangle be drawn?
    No; rays are divergent and never meet.
  14. Given \(AB=5\text{ cm}, AC=4\text{ cm}, \angle A=45^\circ\). Name the construction type.
    S-A-S (two sides & included angle).
  15. Given \(AB=5\text{ cm}, \angle A=45^\circ, \angle B=80^\circ\). Name construction type.
    A-S-A (two angles & included side).
  16. Altitude from vertex \(A\) to side \(BC\) in an obtuse triangle falls where?
    On extension of \(BC\) (outside the triangle).
  17. Complete: If two angles are \(60^\circ\) and \(70^\circ\), the third angle is ___.
    \(50^\circ\).
  18. Can a triangle have angles \(50^\circ, 50^\circ, 90^\circ\)?
    No; sum \(=190^\circ\) (must be \(180^\circ\)).
  19. Can a side of a triangle be an altitude?
    Yes, in a right triangle the legs are altitudes to each other.
  20. State one necessary & sufficient test for triangle existence using lengths \(a\le b\le c\).
    Triangle exists iff \(c
Chapter 7 • A Tale of Three Intersecting Lines • 3-Mark Q&A

Chapter 7: A Tale of Three Intersecting Lines 3-Mark Q&A

  1. Using parallel lines, prove the angle-sum property of a triangle.
    Through \(A\) draw \(XY\parallel BC\). Then \( \angle XAB=\angle B\) and \( \angle YAC=\angle C\) (alternate angles). Straight angle at \(A\): \( \angle XAB+\angle BAC+\angle YAC=180^\circ\Rightarrow \angle B+\angle A+\angle C=180^\circ.\)
  2. Explain why \(10,15,30\) cannot form a triangle without construction.
    Triangle inequality fails: \(30\not<10+15\). Geometrically, circles of radii \(10,15\) on base \(30\) do not intersect internally.
  3. Construct \( \triangle ABC\) if \(AB=5\text{ cm}, AC=4\text{ cm}, \angle A=45^\circ\) (S-A-S).
    Draw \(AB=5\text{ cm}\). At \(A\), construct \(45^\circ\). On this ray mark \(AC=4\text{ cm}\). Join \(BC\).
  4. Construct \( \triangle ABC\) if \(AB=5\text{ cm}, \angle A=45^\circ, \angle B=80^\circ\) (A-S-A).
    Draw \(AB=5\text{ cm}\). At \(A\) draw \(45^\circ\); at \(B\) draw \(80^\circ\). Their intersection is \(C\).
  5. For \( \triangle ABC\), \( \angle B=50^\circ, \angle C=70^\circ\). Find exterior angle at \(C\).
    Interior at \(C\) is \(70^\circ\). Exterior \(=180^\circ-70^\circ=110^\circ\). Also \(= \angle A+\angle B=60^\circ+50^\circ=110^\circ\).
  6. Show that with two base angles \( \ge 90^\circ\), a triangle cannot be formed on a given base.
    Rays at ends of base open away; internal angles on same side \( \ge 180^\circ\), so sides never meet.
  7. Altitude placement: Sketch and explain altitudes in acute, right and obtuse triangles.
    Acute: all altitudes inside. Right: both legs are altitudes; third altitude from right-angle vertex to hypotenuse. Obtuse: altitude from obtuse-angle vertex falls outside on base extension.
  8. Given sides \(4,5,8\). Show circles intersect internally.
    Base \(AB=8\). Circles with radii \(4\) and \(5\) around \(A,B\) satisfy \(4+5>8\Rightarrow\) two intersection points ⇒ triangle exists.
  9. Find third angle quickly: \( \angle A=75^\circ, \angle B=45^\circ\).
    \( \angle C=180^\circ-120^\circ=60^\circ\).
  10. Equal angles & sides: If \( \angle B=\angle C\) in \( \triangle ABC\), show \(AB=AC\).
    Base-angles theorem (converse): In an isosceles triangle equal angles subtend equal opposite sides ⇒ \(AB=AC\).
  11. State precise triangle-existence test for \(a\le b\le c\).
    Triangle exists iff \(ca+b\) ⇒ impossible.
  12. Can a triangle with sides \(6,8,14\) exist? If yes, classify by sides and by angles.
    No: \(14\not<6+8\). (So classification not applicable.)
  13. Construct an altitude from \(A\) to \(BC\) using ruler & set-square (steps).
    Align ruler on \(BC\). Place set-square right angle on ruler; slide until vertical edge passes through \(A\); draw along that edge.
  14. Explain why equilateral \(=\) acute-angled.
    All sides equal ⇒ all angles equal; angle-sum \(180^\circ\Rightarrow60^\circ\) each (all acute).
  15. If two angles are \(60^\circ\) and \(40^\circ\), will third angle depend on side lengths?
    No; \( \angle =180^\circ-(60^\circ+40^\circ)=80^\circ\), independent of sides.
  16. On a base \(AB\), for which smallest \( \angle B\) (with \( \angle A=40^\circ\)) will sides not meet?
    When line at \(B\) is parallel to line at \(A\); internal same-side sum \(=180^\circ\Rightarrow \angle B=140^\circ\). Any \( \ge 140^\circ\): no triangle.
  17. Right triangle with hypotenuse \(AC=5\) cm and \( \angle B=90^\circ\): how many such triangles?
    Infinitely many; \(B\) lies on a semicircle with diameter \(AC\) (Thales).
  18. Prove exterior angle equals sum of two remote interior angles.
    At \(C\): \( \angle ACB+\angle ACD=180^\circ\). Also \( \angle A+\angle B+\angle ACB=180^\circ\). Subtract to get \( \angle ACD=\angle A+\angle B\).
  19. Shortest path on a box from one corner to opposite corner (surface path idea).
    Unfold to a rectangle. If box has \(l,w,h\), the shortest surface path is a straight line of length \( \sqrt{(l+w)^2+h^2}\) (choose best unfolding).
  20. Classify: (i) \(5,5,8\) (ii) \(7,10,12\) by sides & angles.
    (i) Isosceles; angle type depends on measures (here \(5^2+5^2 \;?\; 8^2\Rightarrow50<64\) ⇒ obtuse). (ii) Scalene; \(7^2+10^2=149>12^2=144\) ⇒ acute.
Chapter 7 • A Tale of Three Intersecting Lines • Textbook Exercises (Solved)

Chapter 7: A Tale of Three Intersecting Lines Textbook Exercises — Perfect Solutions

7.1 Equilateral Triangles & Constructing by Sides

E1. Construct an equilateral triangle of side \(4\text{ cm}\).

Draw \(AB=4\text{ cm}\). From \(A\) and \(B\), draw arcs radius \(4\text{ cm}\). Intersection is \(C\). Join \(AC,BC\).

E2. Construct triangles with side lengths (in cm): (a) \(4,4,6\) (b) \(3,4,5\) (c) \(1,5,5\) (d) \(4,6,8\) (e) \(3.5,3.5,3.5\).

All satisfy triangle inequality ⇒ all constructible. (a) isosceles; (b) scalene; (c) isosceles; (d) scalene; (e) equilateral.
Method (SSS): take longest as base; draw arcs of the other two radii; intersect to get third vertex; join.

E3. Figure it Out: Use points on a circle and the centre to form isosceles triangles.

Any triangle with two vertices on the circle and the centre as third has two radii as sides ⇒ isosceles.

E4. Figure it Out: With equal circles, use centres and points to form isosceles/equilateral.

If centres \(A,B\) and a point \(P\) on either circle, triangles like \( \triangle ABP\) often have two equal radii ⇒ isosceles. If three centres of congruent circles form \( \triangle ABC\), equal centre-to-centre distances can give equilateral when spacing is same.

7.2 Triangle Inequality & Existence

E5. Why triangles do not exist for \(3,4,8\) and \(2,3,6\) without constructing?

Longest \(8\not<3+4\); longest \(6\not<2+3\). Triangle inequality fails ⇒ impossible.

E6. Decide existence for: (a) \(10,10,25\) (b) \(5,10,20\) (c) \(12,20,40\).

(a) No; \(25\not<10+10\). (b) No; \(20\not<5+10\). (c) No; \(40\not<12+20\).

E7. Which sets can be sides of a triangle? (a) \(2,2,5\) (b) \(3,4,6\) (c) \(2,4,8\) (d) \(5,5,8\) (e) \(10,20,25\) (f) \(10,20,35\) (g) \(24,26,28\).

(a) No (b) Yes (c) No (d) Yes (e) Yes (f) No (g) Yes.

E8. Prove: If lengths satisfy triangle inequality, circles (with smaller lengths as radii on longest base) intersect internally.

Let base \(AB=c\) (largest), radii \(a,b\) with \(a\le b\le c\). If \(a+b>c\) (triangle inequality), two circles intersect at two points ⇒ triangle exists.

E9. Give examples where circles: (a) touch internally (b) do not intersect.

(a) \(3,4,7\) (sum equals base). (b) \(3,4,9\) (sum less than base). All same units.

E10. Final rule (conclusion).

A triangle exists iff each length is less than the sum of the other two (i.e., with \(a\le b\le c\): \(c

7.3 Constructing with Sides & Angles (S-A-S / A-S-A)

E11. Construct \( \triangle ABC\) with \(AB=5\text{ cm}, AC=4\text{ cm}, \angle A=45^\circ\) (included angle given).

Draw \(AB\). At \(A\), construct \(45^\circ\). Mark \(AC=4\) cm on that ray. Join \(BC\).

E12. Construct triangles: (a) \(3\text{ cm}, 75^\circ, 7\text{ cm}\) (b) \(6\text{ cm}, 25^\circ, 3\text{ cm}\) (c) \(3\text{ cm}, 120^\circ, 8\text{ cm}\) with angle included.

S-A-S in each. All possible as long as the chosen angle is included between the two given sides; draw base, set angle at an endpoint, mark second side on that arm, join.

E13. Is there any S-A-S data where triangle not possible?

If the “given angle” is not the included angle between the two given sides (misplacement), or if the two given sides with angle force the third side to be impossible (degenerate case). With proper included angle and positive lengths, S-A-S yields a triangle.

E14. Construct \( \triangle ABC\) with \(AB=5\text{ cm}, \angle A=45^\circ, \angle B=80^\circ\) (A-S-A).

Draw \(AB\). At \(A\): \(45^\circ\), at \(B\): \(80^\circ\). Their intersection gives \(C\).

E15. Construct: (a) \(75^\circ, 5\text{ cm}, 75^\circ\) (b) \(25^\circ, 3\text{ cm}, 60^\circ\) (c) \(120^\circ, 6\text{ cm}, 30^\circ\).

A-S-A each. Triangle exists when sum of given angles \(<180^\circ\). Cases (a,b,c): sums are \(150^\circ,85^\circ,150^\circ\) ⇒ all possible.

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