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 triangle - angle sum property

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

Step2: Solve Assignment 1

For the angles 36°, 67°, 77°, since \(36 + 67+77=180\), and all angles are less than 90°, it is an acute - angled triangle.

Step3: Solve Assignment 2

(a) \(\angle P+\angle Q+\angle R=100 + 70+20 = 190
eq180\), so a triangle is not possible.
(b) \(\angle A+\angle B+\angle C=5 + 170+5=180\), a triangle is possible.
(c) \(\angle X+\angle Y+\angle Z=70 + 60+50 = 180\), a triangle is possible.
(d) \(\angle A+\angle B+\angle C=60 + 50+90=200
eq180\), so a triangle is not possible.

Step4: Solve Assignment 3

(No figure provided, unable to solve this part completely. But if we assume some basic triangle - angle relationships, we would use angle - sum property and exterior - angle property).

Step5: Solve Assignment 4

Let the measure of each of the equal opposite interior angles be \(x\). By the exterior - angle property of a triangle, \(90=x + x\), so \(2x=90\), and \(x = 45^{\circ}\).

Step6: Solve Assignment 5

Let one acute angle be \(x\) and the other be \(x - 12\). In a right - angled triangle, \(x+(x - 12)+90 = 180\). Combining like terms gives \(2x-12=90\), then \(2x=102\), and \(x = 51^{\circ}\), and the other angle is \(x - 12=39^{\circ}\).

Step7: Solve Assignment 6

(No figure provided, unable to solve this part completely. But we would use angle - sum property and given ratios to find the angles).

Step8: Solve Assignment 7

(No figure provided, unable to solve this part completely. But we would use angle - sum property and given angle relationships to find the angles).

Step9: Solve Assignment 8

(a) For \(AB = 1\mathrm{cm}\), \(BC = 2\mathrm{cm}\), \(CA = 1.5\mathrm{cm}\), since \(1+1.5>2\), \(1 + 2>1.5\), and \(2+1.5>1\), these sides can form a triangle.
(b) For \(PQ = 9\mathrm{cm}\), \(QR = 5\mathrm{cm}\), \(RP = 15\mathrm{cm}\), since \(9 + 5=14<15\), these sides cannot form a triangle.

Step10: Solve Assignment 9

(a) \(1^{2}+1^{2}=2
eq2^{2}\), not a Pythagorean triplet.
(b) \(12^{2}+35^{2}=144 + 1225=1369=37^{2}\), it is a Pythagorean triplet.
(c) \(9^{2}+10^{2}=81 + 100=181
eq12^{2}\), not a Pythagorean triplet.
(d) \(5^{2}+12^{2}=25 + 144=169=13^{2}\), it is a Pythagorean triplet.

Step11: Solve Assignment 10

(a) Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), for \(a = 5\mathrm{cm}\), \(b = 12\mathrm{cm}\), \(c=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\mathrm{cm}\).
(b) For \(a = 24\mathrm{cm}\), \(b = 7\mathrm{cm}\), \(c=\sqrt{24^{2}+7^{2}}=\sqrt{576+49}=\sqrt{625}=25\mathrm{cm}\).
(c) (Value of \(b\) is not clear in the question. Assuming \(b\) is some value, we would use \(c=\sqrt{a^{2}+b^{2}}\)).

Step12: Solve Assignment 11

Using the Pythagorean theorem, if the height on the wall is \(a = 12\mathrm{m}\) and the distance of the foot from the wall is \(b = 9\mathrm{m}\), the length of the ladder \(c=\sqrt{12^{2}+9^{2}}=\sqrt{144 + 81}=\sqrt{225}=15\mathrm{m}\).

Answer:

1. Acute - angled triangle
2. (b) and (c)
3. (No figure, cannot answer)
4. \(45^{\circ}\)
5. \(51^{\circ}\) and \(39^{\circ}\)
6. (No figure, cannot answer)
7. (No figure, cannot answer)
8. (a) Can form a triangle; (b) Cannot form a triangle
9. (b) and (d) are Pythagorean triplets; (a) and (c) are not
10. (a) \(13\mathrm{cm}\); (b) \(25\mathrm{cm}\); (c) (Incomplete information)
11. \(15\mathrm{m}\)