Question

In \(\triangle DEF\) and \(\triangle PQR\) it is given that \(\angle D = \angle Q\) and \(\angle R = \angle E\), then which of the following is correct?

• A. \(\angle F = \angle P\)
• B. \(\angle F = \angle Q\)
• C. \(\angle D = \angle P\)
• D. \(\angle E = \angle P\)

Hint:

When two pairs of angles are given as equal, the third pair of angles must also be equal. Simply match the remaining angle from the first triangle (\(\angle F\)) with the remaining angle from the second triangle (\(\angle P\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar by the Angle-Angle (AA) similarity criterion. A consequence of this is that their third angles must also be equal.

Step 2: Key Formula or Approach:
The sum of angles in any triangle is 180°.
In \(\triangle DEF\), \(\angle D + \angle E + \angle F = 180^\circ\).
In \(\triangle PQR\), \(\angle P + \angle Q + \angle R = 180^\circ\).

Step 3: Detailed Explanation:
We are given:
\(\angle D = \angle Q\) (Equation 1)
\(\angle E = \angle R\) (Equation 2)
From the sum of angles in \(\triangle DEF\), we can write \(\angle F = 180^\circ - \angle D - \angle E\).
From the sum of angles in \(\triangle PQR\), we can write \(\angle P = 180^\circ - \angle Q - \angle R\).
Now substitute the given equalities (Equations 1 and 2) into the equation for \(\angle P\):
\[ \angle P = 180^\circ - (\angle D) - (\angle E) \] Comparing this with the equation for \(\angle F\), we see that:
\[ \angle F = \angle P \]

Step 4: Final Answer:
The correct statement is \(\angle F = \angle P\).