Geometry Exercises

Triangle – Mixed Format Practice Questions

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1. Find the Missing Angle

Q1. In triangle $ABC$, if $\angle A = 50^\circ$ and $\angle B = 65^\circ$, find $\angle C$.

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Solution: $\angle C = 180^\circ - (50^\circ + 65^\circ) = 65^\circ$

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2. Use a Formula

Q2. Find the area of a triangle with base $b = 12$ cm and height $h = 5$ cm.

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Formula: $\text{Area} = \dfrac{1}{2} \cdot b \cdot h$

Solution: $\text{Area} = \dfrac{1}{2} \cdot 12 \cdot 5 = 30 \text{ cm}^2$

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3. Ratio of Sides in Similar Triangles

Q3. In $\triangle ABC \sim \triangle DEF$, if $AB = 6$, $DE = 9$, find the ratio of similarity.

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Solution: $\dfrac{AB}{DE} = \dfrac{6}{9} = \dfrac{2}{3}$

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4. Relationship Among Sides and Angles

Q4. In triangle $XYZ$, side $XY = 10$ cm, $YZ = 8$ cm, $XZ = 6$ cm. Which angle is largest?

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Solution: Largest angle is opposite the longest side → $\angle Z$ is largest.

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5. Find Missing Side Using Pythagoras

Q5. In a right triangle, if legs are $a = 5$ cm and $b = 12$ cm, find hypotenuse $c$.

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Formula: $c^2 = a^2 + b^2$

Solution: $c^2 = 5^2 + 12^2 = 25 + 144 = 169 \Rightarrow c = 13$ cm

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6. Use Angle-Side Relationship

Q6. In triangle $PQR$, if $\angle P = 40^\circ$, $\angle Q = 60^\circ$, which side is longest?

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Solution: Longest side is opposite largest angle

→ $\angle R = 80^\circ$ → side $PQ$ is longest.

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7. Area Using Heron’s Formula

Q7. Find area of triangle with sides $a = 7$, $b = 8$, $c = 9$.

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Steps:- Semi-perimeter: $s = \dfrac{a + b + c}{2} = \dfrac{24}{2} = 12$

• Area: $\sqrt{s(s - a)(s - b)(s - c)} = \sqrt{12(5)(4)(3)} = \sqrt{720} \approx 26.83$

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8. Find Side Using Similarity Ratio

Q8. In similar triangles, if $AB = 4$ cm, $DE = 6$ cm, and $AC = 5$ cm, find $DF$.

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Solution: Ratio: $\dfrac{AB}{DE} = \dfrac{4}{6} = \dfrac{2}{3}$

So: $\dfrac{AC}{DF} = \dfrac{2}{3} \Rightarrow DF = \dfrac{3}{2} \cdot 5 = 7.5$ cm

Triangle – True or False Questions

Q1. The sum of the interior angles of any triangle is $180^\circ$.

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Answer: True

Q2. In a right-angled triangle, the Pythagoras theorem is $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

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Answer: True

Q3. An equilateral triangle has all angles equal to $90^\circ$.

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Answer: False

Q4. The area of a triangle is given by $\dfrac{1}{2} \cdot \text{base} \cdot \text{height}$.

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Answer: True

Q5. In any triangle, the longest side is opposite the smallest angle.

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Answer: False

Q6. A triangle can have two right angles.

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Answer: False

Q7. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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Answer: True

Q8. In a triangle, the sum of any two sides is always less than the third side.

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Answer: False

Q9. A triangle with sides in the ratio $3:4:5$ is a right-angled triangle.

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Answer: True

Q10. Similar triangles have equal corresponding angles and proportional corresponding sides.

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Answer: True