12. a triangle whose one angle is 13. the vertices of a triangle are three collinear points state true or false. 14. every right triangle is scalene. 15. every acute - angled triangle is equilateral 16. the sum of an exterior angle and its opposite interior angles is 180° 17. 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°. what 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° 3. find ∠bac and ∠acb in the figure shown alongside. 4. 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. 5. one acute angle of a right - angled triangle measures 12° less than the other acute angle. find the measure of each angle. 6. in the given figure, x:y = 3:5 and ∠acd = 160°. find the values of x, y, and z. 7. in the given figure, find ∠cbd. if ∠a = 2∠abd, find ∠abd and ∠a 8. 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 9. 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) 10. the lengths of the legs of a right - angled triangle are given. find the hypotenuse. (a) a = 5 cm, b = 12 cm (b) a = 24 cm, b = 7 cm 11. a ladder is placed against a wall in such a way that its foot is at a distance of 9 m from the wall and its top reaches a window 12 m above the ground. find the length of the ladder.

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Accepted Answer

1. **Question 1**:
   - **Explanation**:
     - **Step1: Recall the definition of triangle - type by angles**
       - An acute - angled triangle has all angles less than $90^{\circ}$. Given angles are $36^{\circ},67^{\circ},77^{\circ}$, all of which are less than $90^{\circ}$.
   - **Answer**: Acute - angled triangle.
2. **Question 2**:
   - **Explanation**:
     - **Step1: Recall the angle - sum property of a triangle ($\angle A+\angle B+\angle C = 180^{\circ}$)**
       - For (a): $\angle P+\angle Q+\angle R=100^{\circ}+70^{\circ}+20^{\circ}=190^{\circ}
eq180^{\circ}$, so a triangle is not possible.
       - For (b): $\angle A+\angle B+\angle C = 5^{\circ}+170^{\circ}+5^{\circ}=180^{\circ}$, so a triangle is possible.
       - For (c): $\angle X+\angle Y+\angle Z=70^{\circ}+60^{\circ}+50^{\circ}=180^{\circ}$, so a triangle is possible.
       - For (d): $\angle A+\angle B+\angle C=60^{\circ}+50^{\circ}+90^{\circ}=200^{\circ}
eq180^{\circ}$, so a triangle is not possible.
   - **Answer**: (b) and (c).
3. **Question 4**:
   - **Explanation**:
     - **Step1: Recall the exterior - angle property of a triangle (exterior angle = sum of interior opposite angles)**
       - Let the two equal interior opposite angles be $x$. Given exterior angle $=90^{\circ}$. Then $x + x=90^{\circ}$ (since the two interior opposite angles are equal).
       - **Step2: Solve for $x$**
       - $2x = 90^{\circ}$, so $x=\frac{90^{\circ}}{2}=45^{\circ}$.
   - **Answer**: $45^{\circ}$.
4. **Question 5**:
   - **Explanation**:
     - **Step1: Let one acute angle be $x$ and the other be $x + 12^{\circ}$ in a right - angled triangle**
       - In a right - angled triangle, one angle is $90^{\circ}$, and by the angle - sum property of a triangle ($90^{\circ}+x+(x + 12^{\circ})=180^{\circ}$).
       - **Step2: Simplify the equation**
       - $90^{\circ}+2x+12^{\circ}=180^{\circ}$, $2x+102^{\circ}=180^{\circ}$, $2x=180^{\circ}-102^{\circ}=78^{\circ}$.
       - **Step3: Solve for $x$**
       - $x = 39^{\circ}$, and the other acute angle is $x + 12^{\circ}=39^{\circ}+12^{\circ}=51^{\circ}$.
   - **Answer**: $39^{\circ}$ and $51^{\circ}$.
5. **Question 6**:
   - **Explanation**:
     - **Step1: Use the exterior - angle property ($\angle ACD=x + y$)**
       - Given $x:y = 3:5$, so let $x = 3k$ and $y = 5k$. Also, $\angle ACD = 160^{\circ}$, then $3k+5k=160^{\circ}$, $8k=160^{\circ}$, $k = 20^{\circ}$.
       - So $x=3\times20^{\circ}=60^{\circ}$ and $y = 5\times20^{\circ}=100^{\circ}$.
       - **Step2: Find $z$**
       - $z=180^{\circ}-\angle ACD=180^{\circ}-160^{\circ}=20^{\circ}$.
   - **Answer**: $x = 60^{\circ}$, $y = 100^{\circ}$, $z = 20^{\circ}$.
6. **Question 8 (a)**:
   - **Explanation**:
     - **Step1: Recall the triangle - inequality theorem (sum of any two sides of a triangle must be greater than the third side)**
       - $AB + BC=1+2 = 3\mathrm{cm}\gt1.5\mathrm{cm}$, $AB+CA=1 + 1.5=2.5\mathrm{cm}\gt2\mathrm{cm}$, $BC + CA=2+1.5 = 3.5\mathrm{cm}\gt1\mathrm{cm}$.
   - **Answer**: Yes.
7. **Question 9 (b)**:
   - **Explanation**:
     - **Step1: Recall the Pythagorean theorem ($a^{2}+b^{2}=c^{2}$ for a right - angled triangle with sides $a,b,c$ where $c$ is the hypotenuse)**
       - For $(12,35,37)$, $12^{2}+35^{2}=144 + 1225=1369$ and $37^{2}=1369$.
   - **Answer**: Yes.
8. **Question 10 (a)**:
   - **Explanation**:
     - **Step1: Apply the Pythagorean theorem ($c=\sqrt{a^{2}+b^{2}}$ for a right - angled triangle with legs $a$ and $b$ and hypotenuse $c$)**
       - Given $a = 5\mathrm{cm}$, $b = 12\mathrm{cm}$, then $c=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\mathrm{cm}$.
   - **Answer**: $13\mathrm{cm}$.
9. **Question 11**:
   - **Explanation**:
     - **Step1: Consider the right - angled triangle formed by the ladder, the wall, and the ground**
       - The height of the window above the ground ($a = 12\mathrm{m}$) and the distance of the foot of the ladder from the wall ($b = 9\mathrm{m}$).
       - **Step2: Apply the Pythagorean theorem ($c=\sqrt{a^{2}+b^{2}}$)**
       - $c=\sqrt{12^{2}+9^{2}}=\sqrt{144 + 81}=\sqrt{225}=15\mathrm{m}$.
   - **Answer**: $15\mathrm{m}$.

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