Test:Triangles- 1 - Class 9 MCQ

- The problem involves two triangles with corresponding side lengths: ( AB = QR ), ( BC = RP ), and ( CA = QP ).
- According to the Side-Side-Side (SSS) Congruence Theorem, two triangles are congruent if all three pairs of corresponding sides are equal.
- To match the sides correctly with the given order, note:
- ( CA = QP )
- ( AB = QR )
- ( BC = RP )
- Therefore, the correct correspondence is △CAB ≅ △PQR

In △ABC,
∠ACD+∠ACB = 180^∘ (Linear pair)
115^∘+∠ACB =180^∘
∠ACB = 180^∘−115^∘=65^∘

since AC = BC then ∠ABC = ∠BAC = X
x + x + 65^∘ = 180^∘
2x = 180^∘- 65^∘
2x = 115^∘
x = 57.5^∘

If the bisector of angle A of a triangle is perpendicular to the base BC of the triangle then the triangle ABC is:

B: Isosceles

Solution:

- The angle bisector of angle A divides the angle into two equal parts.
- For this bisector to be perpendicular to base BC, angles B and C must be equal.
- This means that triangle ABC has two equal sides opposite these equal angles.
- Therefore, triangle ABC is isosceles.

Given:

BE = CD

Concept Used:

When 2 sides of a triangle are equal, then it is isosceles.

When 2 angles and 1 side of 2 triangles is equal, then both the triangles are similar.

Calculations:

In △ABE and △ACD,

BE = CD (Given)

∠BEA = ∠CDA (90° each)

∠BAE = ∠CAD (Common Angle)

∠ABE = ∠ACE (By Sum angle property)

⇒ △ABE is similar to △ACD

⇒ AB = AC

∴ If the altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is isosceles.

To determine the congruence rule for ΔAOC and ΔXYZ:

• Given: ∠A = ∠X


• AO = YZ (one pair of corresponding sides)


• AC = XY (another pair of corresponding sides)


• Two pairs of sides and the included angle are equal


•  

Therefore, by the SAS (Side-Angle-Side) congruence rule, ΔAOC ≅ ΔXYZ.

The criteria for congruence of triangles are:

• RHS: Right Angle-Hypotenuse-Side, which is only used for right triangles 

• SSS: Side-Side-Side, where all three sides of two triangles are equal
• SAS: Side-Angle-Side, where two sides and the angle between them are equal
• ASA: Angle-Side-Angle, where two angles and the included side of one triangle are equal to the corresponding angles and sides of another triangle
• AAS: Angle-Angle-Side, where two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle

△ABD ≅ △ADC can be done on all of these ways by using the properties of isoceles triangle and ∠B = ∠C and AD⊥BC

Explanation: This is the SSS similarity criterion, which states that if the sides of two triangles are in proportion, the triangles are similar