Short & Long Answer Questions: Geometric Twins

Q1. The angles of a triangle are 30°, 70°, and 80°. Explain why these measurements alone are not sufficient to guarantee congruence between two triangles.

Sol: The three angles of a triangle being 30°, 70°, and 80° determine only the shape of the triangle, not its size.

Many triangles of different sizes can be drawn having the same three angles. Such triangles are similar but not necessarily congruent.

Since congruent triangles must have both the same shape and the same size, angle measurements alone are not sufficient to guarantee congruence.

Q2. In ΔABC, AB = AC, and altitude AD is drawn to BC. Prove that ∠B = ∠C using congruence.

Sol: Given: AB = AC (isosceles triangle).
Draw AD ⟂ BC.

In triangles ΔADB and ΔADC:

• AD = AD (common)
• ∠ADB = ∠ADC = 90° (altitude)
• AB = AC (given)

Thus, by RHS congruence condition,
ΔADB ≅ ΔADC.
Corresponding angles are equal, so ∠B = ∠C.

Q3. In a rectangle ABCD, show that ΔABD ≅ ΔCDB. Give the correct correspondence of vertices.

Sol: Given: ABCD is a rectangle.

In rectangle ABCD:

• AB = CD (opposite sides of a rectangle)

• AD = BC (opposite sides of a rectangle)

• BD = BD (common side)

Thus, in triangles ΔABD and ΔCDB, all three corresponding sides are equal.

∴ ΔABD ≅ ΔCDB (by SSS congruence condition)

Correct correspondence of vertices:
A ↔ C, B ↔ D, D ↔ B

Hence, ΔABD ≅ ΔCDB.

Q4. Given, OB = OC and OA = OD. Prove that AB ∥ CD using triangle congruence.

Sol: Given:
OB = OC
OA = OD

To prove: AB ∥ CD

Consider triangles ΔAOB and ΔDOC.

• OA = OD (given)

• OB = OC (given)

• ∠AOB = ∠DOC (vertically opposite angles)

∴ ΔAOB ≅ ΔDOC (by SAS congruence condition)

Therefore, corresponding angles are equal:
∠ABO = ∠OCD

These angles are alternate interior angles.

∴ AB ∥ CD.

Hence proved.

Q5.  AB = AD and BC = CD. Prove that AC bisects both ∠BAD and ∠BCD.

Sol: Given:
AB = AD
BC = CD

To prove: AC bisects ∠BAD and ∠BCD

Consider triangles ΔBAC and ΔCAD.

• AB = AD (given)
• BC = CD (given)
• AC = AC (common side)

∴ ΔBAC ≅ ΔCAD (by SSS congruence condition)

Since corresponding angles of congruent triangles are equal:

• ∠BAC = ∠CAD
• ∠BCA = ∠ACD

Therefore, AC divides both ∠BAD and ∠BCD into two equal parts.

Hence, AC bisects both angles.

Q6. ΔABC is equilateral. A point D lies on BC such that BD = DC. Prove that ΔABD ≅ ΔACD, and hence prove ∠BAD = ∠CAD.​Sol: Given:
ΔABC is equilateral ⇒ AB = AC
BD = DC

To prove:
(i) ΔABD ≅ ΔACD
(ii) ∠BAD = ∠CAD

Consider triangles ΔABD and ΔACD.

• AB = AC (definition of equilateral triangle)

• BD = DC (given)

• AD = AD (common side)

∴ ΔABD ≅ ΔACD (by SSS congruence condition)

Hence, corresponding angles are equal:
∠BAD = ∠CAD

Therefore, AD bisects ∠A.