1) Five of the six interior angles of a hexagon have measures 111◦ , 122◦ , 133◦ , 144◦ , and 155◦ . What is the measure of the unknown interior angle?
2) The length of a diagonal of a square is 3 √ 2. What is the area of the square?
3) Quadrilateral ABCD is inscribed in a circle. We know ∠BAD = 51◦ and ∠ABC = 66◦ . Find ∠ADC.
4) DC is tangent to circle O at C. If ∠ABC = 57◦ , ∠BAC = 60◦ , and ∠BCA = 63◦ , then what is the measure of ∠ACD?
5) Triange ABC is inscribed in circle O. We know ∠AOB = 106◦ , ∠AOC = 124◦ . Find the measure of ∠BAC.
#1 Answer: 55 degrees
The angles added up are 665 degrees.
The angles of a hexagon add to 720 degrees.
The missing angle is therefore 55 degrees.
#2 Answer: 9 (square units)
The diagonal of a square forms a right triangle, with the diagonal as hypotenuse and two sides as legs.
The two sides are equal.
If we let one side have length x, then
by Pythagorean Theorem.
The area of the square is x^2, and we found it to be 9.
A useful formula is that the area of the square when you have the diagonal is (diagonal^2)/2
You are very welcome!
Opposite angles in a cyclic quadrilateral are supplementary
angle ABC + angle ADC = 180
66 + angle ADC = 180 subtract 66 from both sides
angle ADC = 114°
Angle ACD = (1/2) minor arc AC
Note that angle ABC is an inscribed angle.....so......the arc that it intercepts ( minor arc AC) = twice this = 114°
Angle ACD = (1/2) minor arc AC = (1/2)(114°) = 57°