https://file.aiquickdraw.com/mathbot/file/1727433699578brv5vt6a.jpg

Let's break down the information from the diagram and solve the problems step by step.

### Given:
1. \( BD \) bisects \( \angle ABC \).
2. \( DE \parallel AB \).
3. \( AB \perp BC \).

This forms a triangle \( \triangle ABC \) with several important geometric properties due to the bisector and parallel lines. We'll use these facts to find the angles as requested in questions 26 to 30.

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### Question 26: Find \( m \angle DEB \)
- Since \( DE \parallel AB \) and \( AB \perp BC \), \( \angle DEB \) is a right angle.
- Therefore, \( m \angle DEB = 90^\circ \).

### Question 27: Find \( m \angle DBE \)
- \( BD \) bisects \( \angle ABC \), meaning it divides \( \angle ABC \) into two equal angles.
- Since \( AB \perp BC \), we know \( \angle ABC = 90^\circ \).
- Thus, \( \angle ABD = \angle DBC = 45^\circ \).
- Therefore, \( m \angle DBE = 45^\circ \).

### Question 28: Find \( m \angle EDB \)
- Triangle \( \triangle DEB \) is a right triangle (from question 26, \( m \angle DEB = 90^\circ \)).
- Using the triangle sum theorem:
  \[
  m \angle DEB + m \angle DBE + m \angle EDB = 180^\circ
  \]
  Substituting known values:
  \[
  90^\circ + 45^\circ + m \angle EDB = 180^\circ
  \]
  Solving for \( m \angle EDB \):
  \[
  m \angle EDB = 45^\circ
  \]

### Question 29: Find \( m \angle ABD \)
- As found earlier, \( BD \) bisects \( \angle ABC \), so \( m \angle ABD = 45^\circ \).

### Question 30: Find an angle congruent to \( \angle BAD \)
- By the parallel line property, \( \angle BAD \) corresponds to \( \angle ADE \) because of alternate interior angles (since \( AB \parallel DE \)).
- Therefore, \( \angle ADE \cong \angle BAD \).

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Would you like further details or explanations on any step?

Here are 5 related questions to explore further:
1. How do you calculate the length of a segment if one of the angles is known in a right triangle?
2. Can you prove why alternate interior angles are congruent when lines are parallel?
3. What is the relationship between angle bisectors and side ratios in a triangle?
4. How would you find the area of triangle \( \triangle ABC \) using the given angles?
5. How would this problem change if \( BD \) were not the bisector?

**Tip:** The sum of angles in any triangle is always \( 180^\circ \), a fact you can apply in many geometric problems!

Use the figure and the given conditions to find the measures of angles DEB, DBE, EDB, ABD, and an angle congruent to BAD.

This problem involves using the triangle sum theorem and angle bisector theorem to find the measures of various angles in a triangle, given that one line bisects an angle and two lines are parallel. Suitable for high school students in Grades 9-10.

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