The sum of the measures of interior angles in a quadrilateral is 360 degrees. A quadrilateral is any shape that has <u>4 sides</u>. ABCD is a square, and squares have 4 sides, so the sum of the angles inside of the square will be equal to 360. A square has 4 angles, each with equal measure, so to find the measure of one angle, divide 360 by 4: 360/4=90. Each angle in a square measures 90 degrees.
Triangle DEF is an equilateral. This means that each side is equal, but it also means that each angle will be equal too. The sum of the measure of interior angles in a triangle is 180 degrees, and a triangle has 3 angles, therefore you can find the measure of each angle in an equilateral by dividing 180 by 3: 180/3=60. Each angle in an equilateral triangle measures 60 degrees.
The measure of a straight line is always 180 degrees. CDF is a straight line, and it is made up of 3 angles:
-Angle CDA
-Angle ADE
-Angle EDF
Since the measure of CDF has to be 180 degrees, the measures of angles CDA, ADE, and EDF must add up to 180.
Angle CDA is an angle of a square, so it is equal to 90 degrees
Angle EDF is an angle of an equilateral triangle, so it is equal to 60 degrees.
The angles must add up to 180, so 90+60+ADE=180
You can subtract 90 and 60 from 180 to find the measure of ADE: 180-90-60=30
The question tells us that triangle ADE is isosceles. There is a theorem called base angles theorem, and it states that if two sides of a triangle are equal in length, then the two angles opposite from them will be equal in measure. Using this theorem, we can find that the measure of angle ADE is equal to the measure of AED. We already found out that ADE is 30 degrees, so AED is also 30 degrees. But the angle we are looking for is AEF:
AEF is made up of angles AED and DEF, so the measures of angles AED+DEF=AEF. DEF is part of the equilateral triangle, so we know it is 60 degrees. We already found that AED is 30 degrees, so add 30 and 60 degrees: 30+60=90. AEF is 90 degrees.
