The perpendicular bisector of side AB of ∆ABC intersects the extension of side AC at D.

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The perpendicular bisector of side AB of ∆ABC intersects the extension of side AC at D. Find the measure of ∠ABC if m∠CBD = 16° and m∠ACB = 118°.

The Correct Answer and Explanation is:

To find the measure of angle ∠ABC, let’s analyze the given information step by step:

Given:

  • ∠CBD = 16°
  • ∠ACB = 118°
  • D is the point where the perpendicular bisector of side AB intersects the extension of side AC.

Step-by-step Solution:

In triangle ABC:

  • Point D lies on the extension of AC, and line CD bisects angle ABC, since ∠CBD = 16°.
  • Thus, ∠ABC = ∠CBD + ∠ABD = 2 × ∠CBD = 2 × 16° = 32°

So, m∠ABC = 32°

We can also double-check with the triangle angle sum rule:

  • The sum of interior angles of triangle ABC is always 180°.
  • Given:
    ∠ACB = 118°,
    ∠ABC = 32°
    Then:
    ∠BAC = 180° − (∠ABC + ∠ACB) = 180° − (32° + 118°) = 30°

Thus, all angles in triangle ABC are:

  • ∠BAC = 30°
  • ∠ABC = 32°
  • ∠ACB = 118°

Final Answer:

The measure of ∠ABC is 32°.

Book-Style Explanation (300 Words):

In any triangle, the sum of the interior angles is always 180°. In triangle ABC, we are given the following information: angle CBD is 16°, and angle ACB is 118°. Furthermore, the perpendicular bisector of side AB intersects the extension of side AC at point D.

Because CD is a perpendicular bisector and intersects the extension of AC, and because angle CBD is given as 16°, we can deduce that CD is also an angle bisector of ∠ABC. Since ∠CBD is one half of angle ∠ABC, this means that:∠ABC=2×∠CBD=2×16∘=32∘\angle ABC = 2 \times \angle CBD = 2 \times 16^\circ = 32^\circ∠ABC=2×∠CBD=2×16∘=32∘

Now, using the triangle angle sum property:∠ABC+∠BAC+∠ACB=180∘\angle ABC + \angle BAC + \angle ACB = 180^\circ∠ABC+∠BAC+∠ACB=180∘

We are given that:

  • ∠ABC = 32°
  • ∠ACB = 118°

Then,∠BAC=180∘−(32∘+118∘)=180∘−150∘=30∘\angle BAC = 180^\circ – (32^\circ + 118^\circ) = 180^\circ – 150^\circ = 30^\circ∠BAC=180∘−(32∘+118∘)=180∘−150∘=30∘

All three interior angles of triangle ABC now make sense and follow the triangle sum rule:

  • ∠ABC = 32°
  • ∠ACB = 118°
  • ∠BAC = 30°

Therefore, the correct and final measure of angle ∠ABC is 32 degrees.

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