Question

In \( \triangle ABC \) and \( \triangle DEF \), \( \angle A = \angle E = 40^\circ \) and AB : ED = AC : EF and \( \angle F = 65^\circ \) then \( \angle B = ? \)

• A. \(35^\circ\)
• B. \(65^\circ\)
• C. \(75^\circ\)
• D. \(85^\circ\)

Hint:

1. Identify similarity: Given \(\angle A = \angle E\) and \(\frac{AB}{ED} = \frac{AC}{EF}\). This is SAS similarity for \(\triangle ABC \sim \triangle EDF\). (Careful with the order of vertices in \(\triangle DEF\)). 2. Corresponding angles are equal: \(\angle A \leftrightarrow \angle E\) \(\angle B \leftrightarrow \angle D\) \(\angle C \leftrightarrow \angle F\) 3. Use given \(\angle F = 65^\circ\). So, \(\angle C = 65^\circ\). 4. In \(\triangle ABC\), \(\angle A + \angle B + \angle C = 180^\circ\). \(40^\circ + \angle B + 65^\circ = 180^\circ\). \(\angle B + 105^\circ = 180^\circ\). \(\angle B = 75^\circ\).

Answer 1

The Correct Option is C

Concept: This problem involves similarity of triangles using the SAS (Side-Angle-Side) similarity criterion. If two triangles are similar, their corresponding angles are equal. Step 1: Analyze the given information for similarity We are given: % Option (A) \( \angle A = \angle E = 40^\circ \) (One pair of corresponding angles is equal). % Option (B) AB : ED = AC : EF. This can be rewritten as \( \frac{AB}{ED} = \frac{AC}{EF} \). This means the ratio of sides including the angle A in \(\triangle ABC\) (AB, AC) is equal to the ratio of sides including the angle E in \(\triangle DEF\) (ED, EF). The conditions \( \angle A = \angle E \) and \( \frac{AB}{ED} = \frac{AC}{EF} \) satisfy the SAS similarity criterion. The angle (\(\angle A\) or \(\angle E\)) is included between the sides whose ratios are given. So, \(\triangle ABC \sim \triangle EDF\). Note the order of vertices for similarity: A corresponds to E. Side AB (adjacent to A) corresponds to side ED (adjacent to E). Side AC (adjacent to A) corresponds to side EF (adjacent to E). Thus, the similarity is \(\triangle ABC \sim \triangle EDF\). Step 2: Corresponding angles in similar triangles Since \(\triangle ABC \sim \triangle EDF\), their corresponding angles are equal:
\(\angle A = \angle E\) (Given as \(40^\circ\))
\(\angle B = \angle D\) (This is what we need to find, or related to it)
\(\angle C = \angle F\) Step 3: Use the given angle \(\angle F\) We are given \(\angle F = 65^\circ\). From the similarity \(\triangle ABC \sim \triangle EDF\), we have \(\angle C = \angle F\). So, \(\angle C = 65^\circ\). Step 4: Find \(\angle B\) using the sum of angles in \(\triangle ABC\) The sum of angles in any triangle is \(180^\circ\). In \(\triangle ABC\): \(\angle A + \angle B + \angle C = 180^\circ\). We know \(\angle A = 40^\circ\) and we just found \(\angle C = 65^\circ\). Substitute these values: \[ 40^\circ + \angle B + 65^\circ = 180^\circ \] \[ \angle B + 105^\circ = 180^\circ \] \[ \angle B = 180^\circ - 105^\circ \] \[ \angle B = 75^\circ \] Therefore, \(\angle B = 75^\circ\).