MCQ
Lines & angles, congruence of triangles; trigonometric ratios, heights and distances MCQ - Practice Questions with Answers
Solve 15 Lines & angles, congruence of triangles; trigonometric ratios, heights and distances questions for RAS/RPSC preparation.
Practice questions
Q1A person observes the top of a tower from eye level 1.5 m above the ground. The observation point is 20 m from the tower base, and the angle of elevation is 30°. What is the tower's total height?
The right triangle starts at the observer's eye level, not at the ground. The horizontal distance is 20 m and the angle of elevation is 30°. If the vertical difference from eye level to the tower top is h, then tan 30° = h/20. Since tan 30° = 1/√3, h = 20/√3 = 20√3/3 m. The tower's total height includes the 1.5 m eye level, so the full height is 1.5 + 20√3/3 m.
Q2Which statement about congruence of triangles is correct?
The note separates congruence from similarity. Congruent triangles match in both shape and actual size, so valid tests must fix corresponding side lengths and angles strongly enough. At CET level, SSS, SAS, ASA/AAS and RHS are accepted tests. AAA is a common trap because it fixes shape but not size, so it proves similarity, not congruence. RHS is also restricted to right triangles, and SAS needs the included angle between the two equal sides.
Q3In a right triangle, for angle θ, the opposite side is 5 cm and the adjacent side is 12 cm. Which pair of trigonometric ratios is correct?
The right triangle has legs 5 cm and 12 cm, so the hypotenuse is 13 cm by the 5-12-13 triangle. For the chosen angle, sine is opposite over hypotenuse, so sin θ = 5/13. Tangent is opposite over adjacent, so tan θ = 5/12. The hypotenuse is never the adjacent side, and the roles of the two legs depend on the chosen angle.
Q4Which statement is incorrect for a right triangle when one acute angle θ is chosen?
The hypotenuse is always the side opposite the right angle and is the longest side of a right triangle. It never becomes the adjacent side. What changes with the selected acute angle is the role of the two legs: one leg becomes opposite the chosen angle and the other becomes adjacent to it. The basic ratios remain fixed for the selected angle: sin uses opposite and hypotenuse, cos uses adjacent and hypotenuse, and tan uses opposite and adjacent.
Q5For an n-sided rectilinear figure, the interior-angle sum is (n - 2) × 180°. What is the interior-angle sum of a hexagon?
A hexagon has 6 sides, so put n = 6 in the rectilinear-figure formula from the note. The interior-angle sum is (6 - 2) × 180° = 4 × 180° = 720°. The reason for n - 2 is that a polygon can be divided into triangles from one vertex. For CET, the main trap is to multiply the number of sides directly by 180°, which would overcount the angle sum.
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More questions
6Four angles around a point are 70°, 95°, 110° and x°. What is x?
7Assertion: If two triangles have all three corresponding angles equal, they must be congruent. Reason: Equal corresponding angles give the same shape, but the actual side lengths may still be different.
8A student observes the top of a tower from eye level 1.5 m above the ground. The horizontal distance from the tower base is 15√3 m, and the angle of elevation is 30°. What is the tower's height?
9At a point, four angles are x°, 2x°, 3x° and 4x° with no gap between them. What is x?
10Match the angle situation with the correct rule. 1. Angles around a point 2. Linear pair 3. Vertically opposite angles 4. Co-interior angles on parallel lines
11Three angles around a point are 70°, 95° and 110°. What is the fourth angle?
12In a right triangle, for angle θ, tan θ = 3/4. What is sin θ?
13A closed rectilinear figure has an interior-angle sum of 720°. How many sides does it have?
14Assertion: From the top of a 30 m building, if the angle of depression of a ground point is 60°, the horizontal distance of the point is 10√3 m. Reason: The angle of depression can be treated as the equal angle of elevation, and tan 60° = √3.
15Match the condition with the correct conclusion. 1. SSS 2. AAA 3. RHS 4. SAS P. Valid only for right triangles Q. Gives similarity, not congruence R. Three corresponding sides are equal S. Two sides and the included angle are equal