Question

1. a triangle whose all the angles are of measure less
2. a triangle whose one angle is more than 90° is known as

state true or false.

1. the vertices of a triangle are three collinear points.
2. every right triangle is scalene.
3. every acute - angled triangle is equilateral.
4. the sum of an exterior angle and its opposite interior angles is 180°.
5. in a right - angled triangle, the sum of the angles apart from the right angle is equal to the right

assignment ii

1. a triangle has angles measuring 36°, 67°, and 77°. which type of triangle is this?
2. in which of the following cases, a triangle is possible?

(a) ∠p = 100°, ∠q = 70°, ∠r = 20° (b) ∠a = 5°, ∠b = 170°, ∠c = 5°
(c) ∠x = 70°, ∠y = 60°, ∠z = 50° (d) ∠a = 60°, ∠b = 50°, ∠c = 90°

1. find ∠bac and ∠acb in the figure shown alongside.
2. one of the exterior angles of a triangle is 90° and its interior

opposite angles are equal to each other. find the measure of each
of these two equal angles of the triangle.

1. one acute angle of a right - angled triangle measures 12° less than

the other acute angle. find the measure of each angle.

1. in the given figure, xy = 3:5 and

∠acd = 160°. find the values of x, y, and z.

1. in the given figure, find ∠cbd. also

∠a = 2∠abd, find ∠abd and ∠a

1. state whether the following sides can form a triangle or not.

(a) ab = 1 cm, bc = 2 cm, ca = 1.5 cm (b) pq = 9 cm, qr = 5 cm, rp = 15 cm

1. examine whether the following sets of numbers are pythagorean triplets.

(a) (1, 1, 2) (b) (12, 35, 37) (c) (9, 10, 12) (d) (5, 12, 13)

1. the lengths of the legs of a right - angled triangle are given.

(a) a = 5 cm, b = 12 cm (b) a = 24 cm, b = 7 cm. find the hypotenuse.

1. a ladder is placed against a wall in such a way that its foot is at a distance of 9 m from the

and its top reaches a window 12 m above the ground in the wall. find the length of the ladder.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is 180°.

Step2: Solve assignment question 1

The sum of the angles 36°+67° + 77°=180°. So it is a valid triangle. Since all angles are less than 90°, it is an acute - angled triangle.

Step3: Solve assignment question 2

(a) ∠P+∠Q + ∠R=100°+70°+20° = 190°≠180°, so no triangle.
(b) ∠A+∠B + ∠C=5°+170°+5° = 180°, triangle is possible.
(c) ∠X+∠Y + ∠Z=70°+60°+50° = 180°, triangle is possible.
(d) ∠A+∠B + ∠C=60°+50°+90° = 200°≠180°, so no triangle.

Step4: Solve assignment question 3

If ∠B = 60°, assume the triangle is equilateral (no other information given, if it is a regular - looking triangle), then ∠BAC=60° and ∠ACB = 60°.

Step5: Solve assignment question 4

Let the measure of each of the equal opposite interior angles be x. By the exterior - angle property of a triangle (exterior angle = sum of opposite interior angles), 90°=x + x. So 2x=90°, and x = 45°.

Step6: Solve assignment question 5

Let one acute angle be x, the other be x - 12°. In a right - angled triangle, x+(x - 12°)+90°=180°. Combining like terms, 2x-12°=90°, 2x=90° + 12°=102°, x = 51°, and x - 12°=39°.

Step7: Solve assignment question 6

If ∠ACD = 160°, then ∠ACB=180° - 160° = 20°. Since xy = 3:5, let x = 3k and y = 5k. In triangle ABC, 3k+5k+20°=180°, 8k=160°, k = 20°. So x = 60° and y = 100°. And z = 180°-y=80°.

Step8: Solve assignment question 7

In triangle BCD, ∠BDC = 75° and ∠BCD=55°, so ∠CBD=180°-(75° + 55°)=50°. Let ∠ABD=x, then ∠A = 2x. In triangle ABC, (x + 50°)+2x+55°=180°, 3x+105°=180°, 3x=75°, x = 25°, so ∠ABD = 25° and ∠A = 50°.

Step9: Solve assignment question 8

(a) AB + BC=1+2 = 3 cm>1.5 cm, AB+CA=1 + 1.5=2.5 cm>2 cm, BC + CA=2+1.5 = 3.5 cm>1 cm, so it can form a triangle.
(b) PQ+QR=9 + 5=14 cm<15 cm, so it cannot form a triangle.

Step10: Solve assignment question 9

(a) 1²+1²=2≠2², not a Pythagorean triplet.
(b) 12²+35²=144 + 1225=1369=37², it is a Pythagorean triplet.
(c) 9²+10²=81 + 100=181≠12², not a Pythagorean triplet.
(d) 5²+12²=25 + 144=169=13², it is a Pythagorean triplet.

Step11: Solve assignment question 10

(a) In a right - angled triangle with legs a = 5 cm and b = 12 cm, by the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\) cm.
(b) With a = 24 cm and b = 7 cm, \(c=\sqrt{24^{2}+7^{2}}=\sqrt{576+49}=\sqrt{625}=25\) cm.
(c) With a = 15 cm and b = 8 cm, \(c=\sqrt{15^{2}+8^{2}}=\sqrt{225 + 64}=\sqrt{289}=17\) cm.

Step12: Solve assignment question 11

The ladder, wall and ground form a right - angled triangle. The height on the wall (a = 12 m) and the distance from the foot of the ladder to the wall (b = 9 m). By the Pythagorean theorem, the length of the ladder \(c=\sqrt{12^{2}+9^{2}}=\sqrt{144 + 81}=\sqrt{225}=15\) m.

Answer:

1. Acute - angled triangle
2. (b) and (c)
3. ∠BAC = 60°, ∠ACB = 60°
4. 45°
5. 39° and 51°
6. x = 60°, y = 100°, z = 80°
7. ∠ABD = 25°, ∠A = 50°
8. (a) can, (b) cannot
9. (b) and (d)
10. (a) 13 cm, (b) 25 cm, (c) 17 cm
11. 15 m