## Step-by-Step Solution Let's analyze the diagram: - \( \triangle BCE \) is the large triangle. - \( AD \) is a transversal, with \( AB \parallel CD \) (indicated by matching arrows). - Angles are given as: \( \angle BAD = 105^\circ \) \( \angle ADC = 124^\circ \) We need to find all the unknown angles in the triangle. --- ### **Step 1: Identify Corresponding Angles** Since \( AB \parallel CD \), the angle at \( A \) (i.e., \( \angle BAD \)) and the angle at \( D \) (i.e., \( \angle DCB \)) are **corresponding angles**. So: \[ \angle EBC = \angle BAD = 105^\circ \] --- ### **Step 2: Find the Angle at \( E \)** Notice that the angles \( \angle BAD \) and \( \angle ADC \) are **interior angles on the same side of the transversal** \( AE \). The sum of the interior angles on the same side of a transversal is \( 180^\circ \): \[ \angle BAD + \angle ADC = 105^\circ + 124^\circ = 229^\circ \] But this sum is greater than \( 180^\circ \), which means our assumption is incorrect. Let's look at triangle \( BCE \). --- ### **Step 3: Sum of Angles in Triangle \( \triangle BCE \)** In triangle \( \triangle BCE \): \[ \angle EBC + \angle BCE + \angle BE = 180^\circ \] We know: - \( \angle EBC = 105^\circ \) (from above). Now, \( \angle BCE \) is the angle at \( C \), which is **corresponding** to \( \angle ADC = 124^\circ \). So, \[ \angle BCE = 124^\circ \] --- ### **Step 4: Find the Remaining Angle at \( E \)** \[ \angle E = 180^\circ - (\angle EBC + \angle BCE) \] \[ = 180^\circ - (105^\circ + 124^\circ) \] \[ = 180^\circ - 229^\circ \] \[ = -49^\circ \] This is not possible (angles can't be negative). Let's reconsider the geometry. --- ### **Step 5: Correct Interpretation** Since the parallel lines are \( AB \parallel DC \), and the transversal is \( AE \) and \( DE \): - \( \angle BAD = 105^\circ \) and \( \angle DCE = 124^\circ \). - \( \triangle BCE \) is the triangle formed. Let \( x = \angle EBC \), \( y = \angle BCE \), and \( z = \angle BE \). Given the parallel lines, \( \angle BAD \) and \( \angle BCE \) are **alternate interior angles**: \[ \angle BCE = \angle BAD = 105^\circ \] Similarly, \( \angle DCE \) and \( \angle EBC \) are **alternate interior angles**: \[ \angle EBC = \angle DCE = 124^\circ \] Now, for the triangle: \[ x = 124^\circ, \quad y = 105^\circ \] So, \[ z = 180^\circ - (124^\circ + 105^\circ) \] \[ z = 180^\circ - 229^\circ \] \[ z = -49^\circ \] Again, negative angle, which is not possible. --- ### **Step 6: Actual Triangle and Angles** Let's check the triangle carefully: Is the triangle \( \triangle ABE \) or \( \triangle CDE \)? Given the parallel lines and transversals, the correct triangle is \( \triangle ABE \). Let's denote: - \( \angle BAE = 105^\circ \) - \( \angle AED \) (unknown) - \( \angle EAB \) (unknown) If the parallel lines are \( AB \parallel CD \), and the transversal is \( AE \), then: - \( \angle BAD = 105^\circ \) is **corresponding** to \( \angle AED \) (at \( E \)). - \( \angle ADC = 124^\circ \) is **corresponding** to \( \angle ABE \) (at \( B \)). So, \[ \angle ABE = 124^\circ \] \[ \angle BAE = 105^\circ \] Sum of angles in triangle \( ABE \): \[ \angle ABE + \angle BAE + \angle AEB = 180^\circ \] \[ 124^\circ + 105^\circ + \angle AEB = 180^\circ \] \[ 229^\circ + \angle AEB = 180^\circ \] \[ \angle AEB = 180^\circ - 229^\circ = -49^\circ \] --- ### **Step 7: Final Check** Given the repeated negative result, **the diagram may have an error, or the parallel lines are not as indicated**. However, based on the standard interpretation: #### **Final Answers (Based on Corresponding Angles):** - **Angle at \( B \):** \( 124^\circ \) - **Angle at \( A \):** \( 105^\circ \) - **Angle at \( E \):** \( -49^\circ \) (not possible for a triangle) --- ## **Conclusion** The sum of the given angles exceeds \( 180^\circ \), which is not possible for a triangle. There may be a mislabeling in the diagram or the parallel lines. **Please check the diagram or provide additional information to clarify the question.** --- ### **Summary Table** | Angle Location | Value | |:--------------:|:----------| | At \( B \) | \( 124^\circ \) | | At \( A \) | \( 105^\circ \) | | At \( E \) | Not possible (negative angle) | --- If you have further clarification or an updated diagram, I can assist further!