SSC CGL (Prelims) General: Triangles questions with solutions

Q1. Two right-angled triangular panels, LMN and PQR, have \(\angle M = \angle Q = 90^\circ\). If the hypotenuse \(LN\) equals \(PR\) and side \(LM\) equals \(PQ\), are the triangles congruent? If so, by what rule?

1. Yes, by SSS
2. Yes, by SAS
3. Yes, by RHS
4. Yes, by ASA

Answer: Yes, by RHS

Since both triangles are right-angled, the equal hypotenuse and one corresponding side are sufficient to establish congruence. This is exactly the RHS (Right angle-Hypotenuse-Side) criterion.

Q2. In triangle PQR, medians PM and QN intersect at G. If the length of median PM is 15 cm, what is the length of segment PG?

1. 4 cm
2. 5 cm
3. 7.5 cm
4. 10 cm

Answer: 10 cm

The centroid divides every median in the ratio 2:1 from the vertex. Therefore, \(PG = \frac{2}{3} \times PM = \frac{2}{3} \times 15 = 10\) cm.

Q3. A triangle can have:

1. Two obtuse angles
2. One acute angle and two right angles
3. Only one obtuse angle
4. Three angles each equal to 70°

Answer: Only one obtuse angle

In a triangle, the sum of interior angles is \(180^\circ\). If one angle is obtuse, it is already greater than \(90^\circ\), leaving less than \(90^\circ\) for the other two angles combined. Therefore, a triangle can have only one obtuse angle.

Q4. If in $\triangle ABC$ and $\triangle DEF$, $AB = DE$, $\angle B = \angle E$, and $BC = EF$, which rule proves $\triangle ABC \cong \triangle DEF$?

1. SSS
2. SAS
3. ASA
4. RHS

Answer: SAS

The triangles have two corresponding sides equal, $AB=DE$ and $BC=EF$, and the included angle between them is also equal, $\angle B=\angle E$. Therefore, the congruence criterion is SAS.

Q5. In triangle $XYZ$, if the angle bisectors of $\angle Y$ and $\angle Z$ intersect at point $O$, and $\angle X = 70^\circ$, what is the measure of $\angle YOZ$?

1. 115°
2. 120°
3. 130°
4. 125°

Answer: 125°

Since $O$ is the incenter, the angle between the bisectors of $\angle Y$ and $\angle Z$ is $\angle YOZ = 90^\circ + \frac{\angle X}{2}$. With $\angle X = 70^\circ$, we get $90^\circ + 35^\circ = 125^\circ$.

Q6. Two triangles, \(\triangle ABC\) and \(\triangle DEF\), have \(AB = DE\), \(\angle ABC = \angle DEF\), and \(BC = EF\). By which rule are they congruent?

1. SSS
2. ASA
3. SAS
4. RHS

Answer: SAS

The triangles have two corresponding sides equal and the included angle between them equal. This matches the Side-Angle-Side (SAS) congruence criterion. Therefore, the triangles are congruent by SAS.