Lines, Angles, and Triangles MCQ Quiz in বাংলা - Objective Question with Answer for Lines, Angles, and Triangles - বিনামূল্যে ডাউনলোড করুন [PDF]

An isosceles right triangle has two equal angles and one \( 90^\circ \) angle. Since the sum of angles in any triangle is \( 180^\circ \), the two equal angles must satisfy the equation \( 90 + 2x = 180 \). Solving for \( x \) gives \( 2x = 90 \), so \( x = 45 \). Therefore, each of the equal angles measures \( 45^\circ \) (Option 2). The other options do not satisfy this requirement.

In similar triangles, corresponding angles are congruent. This means that if angle \( A \) in \( \triangle ABC \) is \( 40^\circ \), then the corresponding angle \( D \) in \( \triangle DEF \) is also \( 40^\circ \). Option 4 is correct because it reflects the congruency of corresponding angles in similar triangles. Other options suggest incorrect measures that do not align with the property of similar triangles.

Given that triangles \(ABC\) and \(DEF\) are congruent, corresponding angles are equal. We know angles \(A\), \(B\), and \(C\) in triangle \(ABC\) must sum to \(180^\circ\). With \(A = 55^\circ\) and \(B = 90^\circ\), we find \(C\):

\(55^\circ + 90^\circ + C = 180^\circ\)

Solving for \(C\):

\(C = 180^\circ - 55^\circ - 90^\circ = 35^\circ\)

So, angle \(F\), which corresponds to angle \(C\), is \(35^\circ\). Thus, the correct answer is \(35^\circ\). Option 2 (55°) corresponds to angle \(A\). Option 3 (90°) corresponds to angle \(B\). Option 4 (125°) is incorrect as it does not satisfy the angle sum property of triangles.

In congruent triangles, all corresponding angles are equal. This means that if one triangle has angles of \(50^\circ\), \(60^\circ\), and \(70^\circ\), the other triangle must have angles of exactly the same measures: \(50^\circ\), \(60^\circ\), and \(70^\circ\). The sum of the angles in any triangle is \(180^\circ\). If we add \(50^\circ + 60^\circ + 70^\circ\), we indeed get \(180^\circ\), which is consistent with the angle sum property. Option 2 (60°, 70°, 80°), Option 3 (70°, 80°, 90°), and Option 4 (80°, 90°, 100°) do not match the measures of the angles in the first triangle, and thus they cannot be the correct answer.

In congruent triangles \(KLM\) and \(NOP\), corresponding angles are equal. The angles in triangle \(KLM\) are \(K\), \(L\), and \(M\), and they sum to \(180^\circ\). Given \(K = 45^\circ\) and \(L = 90^\circ\), we find angle \(M\):

\(45^\circ + 90^\circ + M = 180^\circ\)

Solving for \(M\):

\(M = 180^\circ - 45^\circ - 90^\circ = 45^\circ\)

Thus, angle \(P\), which corresponds to angle \(M\), is \(45^\circ\). Therefore, the correct answer is \(45^\circ\). 

Since triangles \(ABC\) and \(DEF\) are congruent, all corresponding angles in these triangles must be equal. In triangle \(ABC\), the angles \(A\), \(B\), and \(C\) must sum to \(180^\circ\). Given angle \(A\) is \(30^\circ\) and angle \(B\) is \(90^\circ\), we calculate angle \(C\):

\(30^\circ + 90^\circ + C = 180^\circ\)

Solving for \(C\):

\(C = 180^\circ - 30^\circ - 90^\circ = 60^\circ\)

Therefore, the measure of angle \(F\), which corresponds to angle \(C\), is \(60^\circ\). Thus, the correct answer is \(60^\circ\). Option 1 (30°) corresponds to angle \(A\). Option 3 (90°) corresponds to angle \(B\). Option 4 (120°) is incorrect as it exceeds the possible values for a triangle's angle.

Similar triangles have the same shape but not necessarily the same size, and their corresponding angles are equal. In triangle \(RST\), the angles \(R\), \(S\), and \(T\) sum to \(180^\circ\). Given \(R = 40^\circ\) and \(S = 90^\circ\), we calculate angle \(T\):

\(40^\circ + 90^\circ + T = 180^\circ\)

Solving for \(T\):

\(T = 180^\circ - 40^\circ - 90^\circ = 50^\circ\)

Therefore, angle \(W\), which corresponds to angle \(T\), is \(50^\circ\). Thus, the correct answer is \(50^\circ\). Option 1 (40°) corresponds to angle \(R\). Option 3 (90°) corresponds to angle \(S\). Option 4 (130°) is incorrect as it does not match the properties of similar triangles.

Since MN is parallel to KL, triangles JKL and JMN are similar. The length of KL can be found using the Pythagorean theorem: \(JL^2 + KL^2 = JK^2\)

\(54^2 + KL^2 = 90^2\)

\(2916 + KL^2 = 8100\)

\(KL^2 = 5184\)

\(KL = 72\)

The ratio of KL to MN is \(\frac{72}{18} = 4\). Therefore, the length of segment JM can be found using this ratio: \(JM = \frac{JL}{4} = \frac{54}{4} = 13.5\). However, we need to calculate the full length using the hypotenuse ratio: \(\frac{90}{JM} = 4\) \(JM = \frac{90}{4} = 22.5\).

To find IJ, the altitude from I to GH, we need to determine the length of GI using the Pythagorean theorem: \(GI^2 + HI^2 = GH^2\)

\(GI^2 + 80^2 = 100^2\)

\(GI^2 + 6400 = 10000\)

\(GI^2 = 3600\)

\(GI = 60\)

The area of triangle GHI is \(\frac{1}{2} \times HI \times GI = \frac{1}{2} \times 80 \times 60 = 2400\).

Using the altitude IJ, the area can also be expressed as \(\frac{1}{2} \times GH \times IJ = 2400\).

Solving for IJ gives \(50 \times IJ = 2400\)

\(IJ = 48\).

Therefore, the length of IJ is 60 units.

The sum of a triangle's angles is 180°. With a right triangle having one angle of 90° and another of 45°, their sum is 135°. The third angle can be found by subtracting from 180°: 180° - 135° = 45°. Thus, option 2 is correct. Option 1 is too low, option 3 exceeds the possible angle sum, and option 4 repeats the right angle, which is impossible for the third angle.