Question

In ΔABC, ∠B = 50°, ∠ADB = 80°, AD = DC.

Which of the following is true?

Options

• AD > AB

• AB > AD

• AD > AC

• AB > AC

Answer 1

AB > AD

Explanation:

Given:

• In ΔABC
• ∠B = 50°
• ∠ADB = 80°
• AD = DC ⇒ Triangle ΔDBC is isosceles

From previous reasoning:

• ∠DBC = ∠DCB = x
• x + x + 80° = 180°
  ⇒ 2x = 100°
  ⇒ x = 50°

So:

• ∠C = 50°
• ∠A = 180° – 50° – 50° = 80°

Now compare sides opposite these angles in triangle ΔABC:

• ∠A = 80° → Opposite side BC
• ∠B = 50° → Opposite side AC
• ∠C = 50° → Opposite side AB

So:

• BC > AB = AC

Thus:

• AB = AC and both < BC
• Since AD = DC, point D is on segment BC and AB is opposite angle C = 50°, AC is opposite angle B = 50°, so AB = AC

Now examine options:

1. AD > AB → Not supported
2. AB > AD → Since AB = AC > AD (AD is only a part of AC)
3. AD > AC → False
4. AB > AC  → They are equal