1. If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
(A) an isosceles triangle (B) an obtuse triangle (C) an equilateral triangle (D) a right triangle
2. An exterior angle of a triangle is 105° and its two interior opposite angles are equal. Each of these equal angles is
(A) 37+ 1/2° (B)52+ 1/2° (C) 72+ 1/2° (D) 75°
3. The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is
(A) an acute angled triangle (B) an obtuse angled triangle (C) a right triangle (D) an isosceles triangle
4 . If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be
(A) 50° (B) 65° (C) 145° (D) 155°
5. The sum of two angles of a triangle is equal to its third angle. Find the third angles.
(a) 900 (b) 450 (c) 600 (d) 700 Section B
1. If two lines intersect, prove that the vertically opposite angles are equal.
2. Bisectors of interior ∠B and exterior ∠ACD of a D ABC intersect at the point T. Prove that < BTC =1/2 <∠ BAC.
3. A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.
4. Prove that through a given point, we can draw only one perpendicular to a given line. [Hint: Use proof by contradiction].
5. Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other.
6. In DABC , ∠Q > ∠R, PA is the bisector of ∠QPR and PM ^QR. Prove that <∠APM = 1/2(∠< Q – ∠<R).
7. A triangle ABC is right angled at A. L is a point on BC such that AL ^ BC. Prove that ∠ < BAL = ∠ < ACB
(A) an isosceles triangle (B) an obtuse triangle (C) an equilateral triangle (D) a right triangle
2. An exterior angle of a triangle is 105° and its two interior opposite angles are equal. Each of these equal angles is
(A) 37+ 1/2° (B)52+ 1/2° (C) 72+ 1/2° (D) 75°
3. The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is
(A) an acute angled triangle (B) an obtuse angled triangle (C) a right triangle (D) an isosceles triangle
4 . If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be
(A) 50° (B) 65° (C) 145° (D) 155°
5. The sum of two angles of a triangle is equal to its third angle. Find the third angles.
(a) 900 (b) 450 (c) 600 (d) 700 Section B
1. If two lines intersect, prove that the vertically opposite angles are equal.
2. Bisectors of interior ∠B and exterior ∠ACD of a D ABC intersect at the point T. Prove that < BTC =1/2 <∠ BAC.
3. A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.
4. Prove that through a given point, we can draw only one perpendicular to a given line. [Hint: Use proof by contradiction].
5. Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other.
6. In DABC , ∠Q > ∠R, PA is the bisector of ∠QPR and PM ^QR. Prove that <∠APM = 1/2(∠< Q – ∠<R).
7. A triangle ABC is right angled at A. L is a point on BC such that AL ^ BC. Prove that ∠ < BAL = ∠ < ACB
8. Q is a point on the side SR of a Δ PSR such that PQ = PR. Prove that PS > PQ.
9. S is any point on side QR of a Δ PQR. Show that: PQ + QR + RP > 2 PS.
10. D is any point on side AC of a Δ ABC with AB = AC. Show that CD < BD.
11. l || m and M is the mid-point of a line segment AB. Show that M is also the mid-point of any line segment CD, having its end points on l and m, respectively.
12. Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Prove that ∠MOC =∠ABC.
13. Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC.
14. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR. {Produce QS to intersect PR at T}
15. Prove that in a right triangle, hypotenuse is the longest (or largest) side.
14. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR. {Produce QS to intersect PR at T}
15. Prove that in a right triangle, hypotenuse is the longest (or largest) side.
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