https://file.aiquickdraw.com/mathbot/file/1733048037910tuv3g195.jpg

From the image, the problem involves solving for the angles of a trapezoid \(ABCD\), which is divided by diagonal \(AC\) into two triangles: an isosceles triangle \(ADC\) and a right triangle \(ACB\). The known conditions are:

1. The isosceles triangle \(ADC\) has one angle \( \angle ADC = 128^\circ \).
2. Triangle \(ACB\) is a right triangle, implying \( \angle ACB = 90^\circ \).

### Steps to Solve:

1. **Analyze Triangle \(ADC\):**
   - In triangle \(ADC\), since it is isosceles, the base angles \( \angle DAC \) and \( \angle DCA \) are equal.
   - The sum of the angles in a triangle is \(180^\circ\). Therefore:
     \[
     \angle DAC + \angle DCA + \angle ADC = 180^\circ
     \]
     Substituting \( \angle ADC = 128^\circ \):
     \[
     \angle DAC + \angle DCA = 52^\circ
     \]
     Since \( \angle DAC = \angle DCA \), each of these angles is:
     \[
     \angle DAC = \angle DCA = \frac{52^\circ}{2} = 26^\circ
     \]

2. **Analyze Triangle \(ACB\):**
   - In triangle \(ACB\), one angle is \( \angle ACB = 90^\circ \), and the sum of all angles is \(180^\circ\). Thus:
     \[
     \angle CAB + \angle ABC + \angle ACB = 180^\circ
     \]
     Substituting \( \angle ACB = 90^\circ \):
     \[
     \angle CAB + \angle ABC = 90^\circ
     \]
     From the trapezoid geometry, \( \angle CAB = \angle DAC = 26^\circ \) (shared angle with triangle \(ADC\)).
     Therefore:
     \[
     \angle ABC = 90^\circ - 26^\circ = 64^\circ
     \]

3. **Trapezoid Angles:**
   - The angles of the trapezoid \(ABCD\) are:
     - \( \angle DAB = \angle DAC + \angle CAB = 26^\circ + 26^\circ = 52^\circ \),
     - \( \angle ABC = 64^\circ \),
     - \( \angle BCD = \angle DCA = 26^\circ \),
     - \( \angle ADC = 128^\circ \).

### Final Answer:
The angles of trapezoid \(ABCD\) are:
- \( \angle DAB = 52^\circ \),
- \( \angle ABC = 64^\circ \),
- \( \angle BCD = 26^\circ \),
- \( \angle ADC = 128^\circ \).

Let me know if you would like a detailed geometric diagram or further explanation!

---

#### Related Questions:
1. What is the sum of the interior angles of a trapezoid?
2. How can you identify whether a trapezoid is isosceles based on its angles?
3. What properties define the diagonal of a trapezoid in geometry?
4. How does the angle-sum property of triangles help in solving problems in trapezoids?
5. Can the given trapezoid be inscribed in a circle? Why or why not?

#### Tip:
Always start by identifying shared angles and using symmetry in isosceles triangles to simplify problems.

The diagonal AC divides trapezoid ABCD into an isosceles triangle ADC and a right triangle ACB. Given ∠ADC = 128°, calculate the angles of the trapezoid.

Learn how to calculate the angles of a trapezoid ABCD divided by diagonal AC into an isosceles triangle and a right triangle. This problem involves angle sum properties and symmetry in triangles. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation