In Exercises 31-36, the diagonals of rhombus ABCD intersect at E . Given that m/_BAC=53°,DE=8 . and EC=6 , find the indicated measure. 31. m/_DAC 32. m/_AED 33. m/_ADC 34. DB 35. AE 36. AC

<p> <br />1. The measure of angle DAC: The diagonals of a rhombus bisect the vertex angles which means ∠DAC will be equal to ∠BAC. Since ∠BAC=53°, hence m∠DAC=53° .<br />2. The measure of angle CONET_STEP_TO_USE_HTMLFormatCharactersAED : All sides of a rhombus are equal, thus triangle AEC (triangles formed by the diagonals) are also congruent. That makes AEC isosceles, hence the base angles are equal. ∠AED = ∠AEC. Since we know that the diagonals will add up to 180° which means : ∠AED + ∠AED + ∠ADC = 180, We found that ∠ADC is 180 - 106 because the angle ∠ADB (which is a straight line) equals 180. Which gets us to the equation ∠AED+∠AED = 180 - 106 =74 Therefore, ∠AED =74/2 Hence, m∠AED =37°.<br />3. To find the measure of ∠ADC: Any quad made from a rhombus is bisected into two equal right triangles ADE and CDE, hence m∠ADC is supplementary to m∠ADE as they are on the same line and the sum up to m∠ADC=180° -37°, so m∠ADC=143°.<br />4. Measuring DB ( Diagonal BD): Right triangles ADE and CDE would have a hypotenuse at 𝐷𝐵=A√(𝑎^2+(2b)^2) where DB= √〖8^2+(2*6)〗^2. Afterwards calculate this giving us around 14.1421 that equates to approx. 14.<br />5. AEC (length of AE): All sides of a rhombus are equal, thus will be equal to DE or EC . Thus, AE=DE= EC =6 or 8.<br />6. Calculating AC ( Diagonal AC,in terms of DE, EC): It's worth reminding that the diagonals of a rhombus bisect into two parts . Thus AC would equate to 2 * EC =2 *6 = 12.-(exploration stage COMPLETED)</p>