1. **Problem Statement:** In rectangle ABCD, the measure of angle AEB is given as 84°. We need to find the measures of angles BDC and EBC. 2. **Important Properties:** - ABCD is a rectangle, so all angles are 90°. - Diagonals of a rectangle are equal and bisect each other. - Point E is the intersection of diagonals AC and BD, so E is the midpoint of both diagonals. 3. **Analyze the angles:** - Since E is the midpoint of diagonals, triangles AEB and CED are congruent. - Angle AEB is given as 84°. 4. **Find m∠BDC:** - Angle BDC is an angle in triangle BDC. - Since ABCD is a rectangle, diagonal BD is a straight line through E. - Angles AEB and CED are vertical angles, so m∠CED = 84°. - Triangle BDC is right-angled at C (since ABCD is a rectangle). - Using triangle properties, m∠BDC = 180° - 90° - 84° = 6°. 5. **Find m∠EBC:** - Consider triangle EBC. - Since E is midpoint of BD, BE = ED. - Triangle EBC is isosceles with BE = EC. - m∠EBC = m∠ECB. - Sum of angles in triangle EBC is 180°. - m∠BEC = 84° (given as m∠AEB, vertical angles equal). - So, m∠EBC + m∠ECB + m∠BEC = 180° - 2 * m∠EBC + 84° = 180° - 2 * m\cancel{\angle EBC} = 180° - 84° - 2 * m\cancel{\angle EBC} = 96° - m\angle EBC = \frac{96°}{2} = 48° **Final answers:** $$m\angle BDC = 6°$$ $$m\angle EBC = 48°$$