✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 6: Triangles
Assertion-Reason Question Bank
✨ Explore more at: www.akashmaths.online
Assertion (A)
The mid-point theorem states that line joining midpoints of two sides is perpendicular to third side.
Reason (R)
The line joining midpoints of two sides of a triangle is parallel to third side and half of it.
Correct Option: (D)
Assertion (A) is false (it is parallel, not perpendicular). Reason (R) is true.
Assertion (A) is false (it is parallel, not perpendicular). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Reason (R)
In \(\triangle ABC\), if \(DE \parallel BC\), then \(\frac{AD}{DB} = \frac{AE}{EC}\) by Basic Proportionality Theorem (Thales Theorem).
Correct Option: (A)
This is the exact statement and algebraic notation of Basic Proportionality Theorem (BPT). Reason explains Assertion.
This is the exact statement and algebraic notation of Basic Proportionality Theorem (BPT). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(E\) is a point on side \(AD\) produced of a parallelogram \(ABCD\) and \(BE\) intersects \(CD\) at \(F\), then \(\triangle ABE \sim \triangle CFB\).
Reason (R)
In \(\triangle ABE\) and \(\triangle CFB\), \(\angle A = \angle C\) and \(\angle ABE = \angle CFB\).
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because \(\angle ABE = \angle BFC\) (alternate interior angles \(AB \parallel CD\)), not \(\angle CFB\).
Assertion (A) is true. Reason (R) is false because \(\angle ABE = \angle BFC\) (alternate interior angles \(AB \parallel CD\)), not \(\angle CFB\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If \(\triangle ABC \sim \triangle QPR\), then \(\angle A = \angle Q\) and \(\angle B = \angle P\).
Reason (R)
In similar triangles, corresponding vertices match in order.
Correct Option: (B)
Both A and R are true vertex correspondence properties.
Both A and R are true vertex correspondence properties.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\triangle ABC \sim \triangle PQR\) with ratio of sides \(2:3\), then ratio of their medians is \(2:3\).
Reason (R)
The ratio of medians of two similar triangles is equal to the square of ratio of their corresponding sides.
Correct Option: (C)
Assertion (A) is true (ratio of medians = ratio of sides = \(2:3\)). Reason (R) is false because ratio of medians is equal to ratio of sides, NOT its square.
Assertion (A) is true (ratio of medians = ratio of sides = \(2:3\)). Reason (R) is false because ratio of medians is equal to ratio of sides, NOT its square.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If sides of two similar triangles are in ratio \(4:9\), then areas are in ratio \(2:3\).
Reason (R)
The ratio of areas of two similar triangles is equal to the square of ratio of their sides (\(4^2 : 9^2 = 16 : 81\)).
Correct Option: (D)
Assertion (A) is false (ratio of areas is \(16:81\), not \(2:3\)). Reason (R) is true.
Assertion (A) is false (ratio of areas is \(16:81\), not \(2:3\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The ratio of perimeters of two similar triangles is equal to the ratio of their corresponding sides.
Reason (R)
If \(\triangle ABC \sim \triangle PQR\), then \(\frac{\text{Perimeter}(\triangle ABC)}{\text{Perimeter}(\triangle PQR)} = \frac{AB}{PQ}\).
Correct Option: (B)
Both A and R state the perimeter ratio property.
Both A and R state the perimeter ratio property.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
All rectangles are similar.
Reason (R)
Two rectangles are similar only if the ratio of their length to breadth is the same.
Correct Option: (D)
Assertion (A) is false (e.g., \(2 \times 4\) rectangle and \(2 \times 8\) rectangle have same angles \(90^\circ\) but sides are not proportional). Reason (R) is true.
Assertion (A) is false (e.g., \(2 \times 4\) rectangle and \(2 \times 8\) rectangle have same angles \(90^\circ\) but sides are not proportional). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
A vertical pole of length \(6\text{ m}\) casts a shadow \(4\text{ m}\) long on the ground, and at the same time a tower casts a shadow \(28\text{ m}\) long. The height of the tower is \(42\text{ m}\).
Reason (R)
Triangles formed by objects and their shadows at the same time of day are similar by AA similarity criterion.
Correct Option: (B)
Both A and R are true (\(6/4 = h/28 \implies h = 42\text{ m}\)).
Both A and R are true (\(6/4 = h/28 \implies h = 42\text{ m}\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\triangle ABC \sim \triangle DEF\) and \(\angle A = 40^\circ, \angle E = 80^\circ\), then \(\angle C = 60^\circ\).
Reason (R)
In \(\triangle ABC \sim \triangle DEF\), \(\angle A = \angle D, \angle B = \angle F, \angle C = \angle E\).
Correct Option: (C)
Assertion (A) is true (\(\angle B = \angle E = 80^\circ\), so \(\angle C = 180 - (40+80) = 60^\circ\)). Reason (R) is false because corresponding angles match in order: \(\angle B = \angle E\) and \(\angle C = \angle F\).
Assertion (A) is true (\(\angle B = \angle E = 80^\circ\), so \(\angle C = 180 - (40+80) = 60^\circ\)). Reason (R) is false because corresponding angles match in order: \(\angle B = \angle E\) and \(\angle C = \angle F\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
In \(\triangle ABC\), \(DE \parallel BC\). If \(AD = 3\text{ cm}, AB = 9\text{ cm}, AE = 4\text{ cm}\), then \(AC = 12\text{ cm}\).
Reason (R)
The ratio \(AD/DB = AE/AC\).
Correct Option: (C)
Assertion (A) is true (\(AD/AB = AE/AC \implies 3/9 = 4/AC \implies AC = 12\text{ cm}\)). Reason (R) is false because formula is \(AD/AB = AE/AC\) or \(AD/DB = AE/EC\).
Assertion (A) is true (\(AD/AB = AE/AC \implies 3/9 = 4/AC \implies AC = 12\text{ cm}\)). Reason (R) is false because formula is \(AD/AB = AE/AC\) or \(AD/DB = AE/EC\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
Two isosceles triangles with equal vertical angles are similar.
Reason (R)
All isosceles triangles are always similar.
Correct Option: (C)
Assertion (A) is true (if vertical angles are equal, base angles are equal, so AA similar). Reason (R) is false because two isosceles triangles with different vertical angles are not similar.
Assertion (A) is true (if vertical angles are equal, base angles are equal, so AA similar). Reason (R) is false because two isosceles triangles with different vertical angles are not similar.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
In \(\triangle ABC\), \(DE \parallel BC\) with \(AD = 2\text{ cm}, DB = 3\text{ cm}, AE = 4\text{ cm}\). Then \(EC = 6\text{ cm}\).
Reason (R)
By Basic Proportionality Theorem, \(\frac{AD}{DB} = \frac{AE}{EC} \implies \frac{2}{3} = \frac{4}{EC} \implies 2 EC = 12 \implies EC = 6\text{ cm}\).
Correct Option: (A)
Applying BPT: \(2/3 = 4/EC \implies EC = 6\text{ cm}\). Reason explains Assertion.
Applying BPT: \(2/3 = 4/EC \implies EC = 6\text{ cm}\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(\triangle ABC \sim \triangle DEF\), then \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\).
Reason (R)
If two triangles are similar, their corresponding angles are equal.
Correct Option: (B)
Both A and R are true basic properties of similar triangles.
Both A and R are true basic properties of similar triangles.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\triangle ABC \sim \triangle PQR\) with \(\angle A = 80^\circ, \angle B = 60^\circ\), then \(\angle P = 60^\circ\).
Reason (R)
In similar triangles \(\triangle ABC \sim \triangle PQR\), corresponding angles are \(\angle A = \angle P = 80^\circ\) and \(\angle B = \angle Q = 60^\circ\).
Correct Option: (D)
Assertion (A) is false (\(\angle P = \angle A = 80^\circ\)). Reason (R) is true.
Assertion (A) is false (\(\angle P = \angle A = 80^\circ\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The line joining the mid-points of two sides of a triangle is parallel to the third side.
Reason (R)
If a line divides two sides of a triangle in the same ratio \(1:1\), it is parallel to the third side by Converse of Basic Proportionality Theorem.
Correct Option: (A)
By converse of BPT, equal ratio \(1:1\) implies parallelism. Reason explains Assertion.
By converse of BPT, equal ratio \(1:1\) implies parallelism. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
In \(\triangle ABC\), if \(D\) is a point on \(BC\) such that \(AD\) bisects \(\angle A\), then \(\frac{BD}{DC} = \frac{AB}{AC}\).
Reason (R)
The angle bisector of a triangle divides the opposite side in the ratio of perimeter.
Correct Option: (C)
Assertion (A) is Angle Bisector Theorem (true). Reason (R) is false because it divides in ratio of adjacent sides, not perimeter.
Assertion (A) is Angle Bisector Theorem (true). Reason (R) is false because it divides in ratio of adjacent sides, not perimeter.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If in two triangles, corresponding angles are equal, then corresponding sides are equal.
Reason (R)
If corresponding angles are equal, the triangles are similar, but their sides are in proportion, not necessarily equal.
Correct Option: (D)
Assertion (A) is false (sides are proportional, equal only if congruent). Reason (R) is true.
Assertion (A) is false (sides are proportional, equal only if congruent). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(DE \parallel BC\) in \(\triangle ABC\) with \(AD = x, DB = x-2, AE = x+2, EC = x-1\), then \(x = 4\).
Reason (R)
By BPT, \(AD \times AE = DB \times EC\).
Correct Option: (C)
Assertion (A) is true (\(x/(x-2) = (x+2)/(x-1) \implies x(x-1) = (x-2)(x+2) \implies x^2 - x = x^2 - 4 \implies x = 4\)). Reason (R) is false because BPT states \(AD/DB = AE/EC \implies AD \times EC = DB \times AE\).
Assertion (A) is true (\(x/(x-2) = (x+2)/(x-1) \implies x(x-1) = (x-2)(x+2) \implies x^2 - x = x^2 - 4 \implies x = 4\)). Reason (R) is false because BPT states \(AD/DB = AE/EC \implies AD \times EC = DB \times AE\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If \(\triangle ABC \sim \triangle PQR\) with \(\angle A = 50^\circ\) and \(\angle B = 70^\circ\), then \(\angle R = 60^\circ\).
Reason (R)
The sum of interior angles of a triangle is \(180^\circ\).
Correct Option: (B)
Both A and R are true. \(\angle C = 180 - (50+70) = 60^\circ\). Since \(\triangle ABC \sim \triangle PQR\), \(\angle R = \angle C = 60^\circ\).
Both A and R are true. \(\angle C = 180 - (50+70) = 60^\circ\). Since \(\triangle ABC \sim \triangle PQR\), \(\angle R = \angle C = 60^\circ\).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
In \(\triangle ABC\), \(D\) and \(E\) are points on \(AB\) and \(AC\) such that \(DE \parallel BC\). If \(AD/DB = 3/5\) and \(AC = 5.6\text{ cm}\), then \(AE = 2.1\text{ cm}\).
Reason (R)
By BPT, \(\frac{AD}{AB} = \frac{AE}{AC}\).
Correct Option: (B)
Both A and R are true. \(AD/AB = 3/(3+5) = 3/8\). \(AE = (3/8) \times 5.6 = 2.1\text{ cm}\).
Both A and R are true. \(AD/AB = 3/(3+5) = 3/8\). \(AE = (3/8) \times 5.6 = 2.1\text{ cm}\).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\triangle ABC \sim \triangle DEF\) such that \(AB = 3\text{ cm}, DE = 6\text{ cm}\) and area of \(\triangle ABC = 15\text{ cm}^2\), then area of \(\triangle DEF = 60\text{ cm}^2\).
Reason (R)
The ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides: \(\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{3}{6}\right)^2 = \frac{1}{4}\).
Correct Option: (A)
Ratio of areas \(= (3/6)^2 = (1/2)^2 = 1/4\). Area(\(\triangle DEF\)) \(= 4 \times 15 = 60\text{ cm}^2\). Reason explains Assertion.
Ratio of areas \(= (3/6)^2 = (1/2)^2 = 1/4\). Area(\(\triangle DEF\)) \(= 4 \times 15 = 60\text{ cm}^2\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If two triangles have equal areas, they must be similar.
Reason (R)
Two triangles with equal areas need not be similar unless their corresponding sides are proportional.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
In \(\triangle ABC\), \(DE \parallel BC\). If \(AD = 2, DB = 3\), then \(DE/BC = 2/3\).
Reason (R)
Since \(\triangle ADE \sim \triangle ABC\), \(\frac{DE}{BC} = \frac{AD}{AB} = \frac{2}{2+3} = \frac{2}{5}\).
Correct Option: (D)
Assertion (A) is false (\(DE/BC = 2/5\), not \(2/3\)). Reason (R) is true.
Assertion (A) is false (\(DE/BC = 2/5\), not \(2/3\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
All congruent triangles are similar, but similar triangles need not be congruent.
Reason (R)
Congruent triangles have equal corresponding sides (ratio \(1:1\)) and equal angles, satisfying both ratio equality and angle equality required for similarity.
Correct Option: (A)
Congruent triangles have ratio of corresponding sides = 1, so they are similar. Reason explains Assertion.
Congruent triangles have ratio of corresponding sides = 1, so they are similar. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
All circles are similar.
Reason (R)
All squares are similar.
Correct Option: (B)
Both are true mathematical facts regarding geometric similarity of regular shapes.
Both are true mathematical facts regarding geometric similarity of regular shapes.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\triangle ABC \sim \triangle DEF\), then \(\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)} = \frac{AB}{DE}\).
Reason (R)
The ratio of areas of two similar triangles is equal to the square of ratio of their corresponding sides.
Correct Option: (D)
Assertion (A) is false (misses the square). Reason (R) is true.
Assertion (A) is false (misses the square). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Two polygons of the same number of sides are similar if their corresponding angles are equal and corresponding sides are in the same ratio.
Reason (R)
Two triangles are similar if their corresponding angles are equal.
Correct Option: (B)
Both A and R are true definitions of similarity.
Both A and R are true definitions of similarity.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If in two triangles \(\triangle ABC\) and \(\triangle PQR\), \(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\), then \(\triangle ABC \sim \triangle PQR\).
Reason (R)
If corresponding sides of two triangles are proportional, their corresponding angles are equal, and the triangles are similar by SSS similarity criterion.
Correct Option: (A)
SSS similarity criterion states that if three sides are in proportion, the triangles are similar. Reason explains Assertion.
SSS similarity criterion states that if three sides are in proportion, the triangles are similar. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
All equilateral triangles are similar.
Reason (R)
Equilateral triangles have all interior angles equal to \(60^\circ\), satisfying AA similarity criterion.
Correct Option: (A)
Since every equilateral triangle has angles \(60^\circ, 60^\circ, 60^\circ\), any two equilateral triangles have equal corresponding angles and are similar by AAA/AA similarity. Reason explains Assertion.
Since every equilateral triangle has angles \(60^\circ, 60^\circ, 60^\circ\), any two equilateral triangles have equal corresponding angles and are similar by AAA/AA similarity. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.