The figure shows a rectangle ABCD where the diagonals AC and BD intersect at E. Given that BEC = 52°, find

(I) ADB
(ii) ACD

(i) 64°

(ii) 26°

Step-by-step explanation:

(i) m<ADB = 64°

(ii) m<ACD = 26°

Step-by-step explanation:

I use this to mean the measure of angle ADC: m<ADC

(i)

m<BEC = 52°

<DEA and <BEC are vertical angles, so they are congruent.

m<DEA = m<BEC = 52°

Polygon ABCD is a rectangle. When the diagonals, AC and BD, of a rectangle are drawn, 4 triangles are formed. Each two opposite triangles are congruent, and all 4 triangles are isosceles.

Triangle ADE is an isosceles triangle with congruent sides ED and EA. That makes opposite angles ADB and DAC congruent. Also the sum of the measures of the angles of a triangle is 180°.

m<ADB + m<DAC + m<DEA = 180°

m<ADB = m<DAC; m<DEA = 52°

2m<ADB + 52° = 180°

2m<ADB = 128°

m<ADB = 64°

(ii)

Angles DEC and BEC form a linear pair. That means the angles are supplementary, and the sum of their measures is 180°.

m<DEC + m<BEC = 180°

m<DEC + 52° = 180°

m<DEC = 128°

Triangle DEC is an isosceles triangle with congruent sides ED and EC. That makes opposite angles ACD and BDC congruent.

m<ACD + m<BDC + m<DEC = 180°

2m<ACD + 128° = 180°

2m<ACD = 52°

m<ACD = 26°