Question

\(\triangle ABC\) and \(\triangle DEF\) are such that \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}\) and \(\angle A = 40^\circ\), \(\angle B = 80^\circ\); then the measure of \(\angle F\) is

• A. 30°
• B. 45°
• C. 60°
• D. 40°

Hint:

First, establish the similarity and identify the corresponding angles correctly. Then, use the angle sum property in the triangle for which you have two angles to find the third one.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The condition \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}\) means that the sides of \(\triangle ABC\) are proportional to the corresponding sides of \(\triangle DEF\). By the Side-Side-Side (SSS) similarity criterion, this implies that \(\triangle ABC \sim \triangle DEF\).

Step 2: Key Formula or Approach:
In similar triangles, corresponding angles are equal. The correspondence is given by the order of vertices in the ratio.
A corresponds to D, B corresponds to E, and C corresponds to F.
Therefore, \(\angle A = \angle D\), \(\angle B = \angle E\), and \(\angle C = \angle F\).
The sum of angles in a triangle is 180°.

Step 3: Detailed Explanation:
We need to find the measure of \(\angle F\). Since \(\triangle ABC \sim \triangle DEF\), we have \(\angle F = \angle C\).
We can find \(\angle C\) using the angle sum property in \(\triangle ABC\).
We are given \(\angle A = 40^\circ\) and \(\angle B = 80^\circ\).
\[ \angle A + \angle B + \angle C = 180^\circ \] \[ 40^\circ + 80^\circ + \angle C = 180^\circ \] \[ 120^\circ + \angle C = 180^\circ \] \[ \angle C = 180^\circ - 120^\circ = 60^\circ \] Since \(\angle F = \angle C\), the measure of \(\angle F\) is 60°.

Step 4: Final Answer:
The measure of \(\angle F\) is 60°.