Q1. Which theorem can be used to prove that two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle?
Solution:
The Side-Angle-Side (SAS) theorem can be used to prove the congruence of the triangles.
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Q2. In triangle XYZ, if XY = 8 cm, XZ = 6 cm, and YZ = 10 cm, which side is the longest?
Solution:
YZ is the longest side since 10 cm is greater than both 8 cm and 6 cm.
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Q3. In triangle PQR, if angle P = 30 degrees and angle Q = 90 degrees, what is angle R?
Solution:
Angle R = 180 - (30 + 90) = 60 degrees.
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Q4. If triangle ABC is similar to triangle DEF, and the ratio of their corresponding sides is 2:3, what is the ratio of their areas?
Solution:
The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2/3)² = 4/9.
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Q5. If the lengths of the sides of triangle ABC are 3 cm, 4 cm, and 5 cm, what type of triangle is it?
Solution:
Triangle ABC is a right triangle because it satisfies the Pythagorean theorem: 3² + 4² = 5².
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Q6. Which of the following triangles is congruent to triangle DEF if DE = 5 cm, EF = 7 cm, and DF = 5 cm?
Solution:
Triangles DEF and GHI are congruent by the Side-Side-Side (SSS) congruence criterion since they have the same side lengths.
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Q7. In triangle PQR, if angle P = 30 degrees and angle Q = 45 degrees, what is the length of side PR if PQ = 10 cm?
Solution:
Using the sine rule, PR = PQ * (sin(Q) / sin(P)) = 10 * (sin(45) / sin(30)) = 10 * (√2/2 / 1/2) = 10 * √2 = 8.66 cm.
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Q8. Which of the following statements is true for congruent triangles?
Solution:
Congruent triangles have the same shape and size, which implies they also have the same area and perimeter.
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Q9. In triangle JKL, if JK = 12 cm, KL = 16 cm, and JL = 20 cm, what is the perimeter of the triangle?
Solution:
Perimeter = JK + KL + JL = 12 + 16 + 20 = 48 cm.
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Q10. If triangle ABC is isosceles with AB = AC, which of the following is true?
Solution:
In an isosceles triangle, the angles opposite the equal sides are also equal.
