In isosceles triangle ABC, AB = AC and angle A = 46°. Point D lies on side AB, and point...

1. VISUALIZE the problem setup

• Given information:
  • Triangle ABC is isosceles with \(\mathrm{AB = AC}\)
  • Angle \(\mathrm{A = 46°}\)
  • \(\mathrm{DE}\) is parallel to \(\mathrm{BC}\)
  • \(\mathrm{D}\) is on \(\mathrm{AB}\), \(\mathrm{E}\) is on \(\mathrm{AC}\)
• Drawing a clear diagram helps you see which angles relate to each other

2. INFER the solution strategy

• Since we need angle BDE, we should work systematically:
  • First find the base angles of the isosceles triangle
  • Then use parallel line properties to connect to our target angle
  • Finally use angle relationships on the straight line

3. APPLY isosceles triangle properties

• In isosceles triangle ABC with \(\mathrm{AB = AC}\):
  • Base angles are equal: \(\mathrm{angle\ ABC = angle\ ACB}\)
  • All angles sum to 180°:

\(\mathrm{angle\ ABC + angle\ ACB + 46° = 180°}\)

• So:

\(\mathrm{2 × angle\ ABC = 134°}\)

which gives

\(\mathrm{angle\ ABC = 67°}\)

4. INFER the parallel line relationship

• Since \(\mathrm{DE \parallel BC}\) and \(\mathrm{AB}\) cuts through both lines:
  • Angle ADE and angle ABC are corresponding angles
  • Therefore: \(\mathrm{angle\ ADE = 67°}\)

5. APPLY supplementary angle properties

• Points B, D, and A all lie on the same straight line AB
• This means angles BDE and ADE form a linear pair
• So:

\(\mathrm{angle\ BDE + angle\ ADE = 180°}\)

• Therefore:

\(\mathrm{angle\ BDE = 180° - 67° = 113°}\)

Answer: 113

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find the base angles correctly \(\mathrm{(67°)}\) but then get confused about which angle relationships to use next. They might try to directly relate angle BDE to the base angles without recognizing the parallel line connection first.

This leads to confusion and guessing since they can't connect their known angles to the target angle.

Second Most Common Error:

Poor VISUALIZE execution: Students attempt the problem without drawing a diagram or with an unclear diagram. Without seeing the geometric relationships visually, they confuse which angles are corresponding, which are supplementary, and how the parallel lines create angle relationships.

This causes them to get stuck early in the problem and resort to random guessing.

The Bottom Line:

This problem requires students to INFER a multi-step strategy connecting three different geometric concepts (isosceles triangles, parallel lines, and supplementary angles). Success depends on recognizing that each concept builds toward the next in a logical sequence.