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# Geometry Questions

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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. Thanks!

Mar 22, 2020

#1
+2

#1 Explanation:

The angles added up are 665 degrees.

The angles of a hexagon add to 720 degrees.

The missing angle is therefore 55 degrees.

#2 Explanation:

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

x^2+x^2=(3sqrt2)^2

by Pythagorean Theorem.

2x^2=18

x^2=9

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!

:P

Mar 22, 2020
#2
+1

(3)

Opposite angles in a  cyclic quadrilateral are supplementary

Therefore......

angle ABC  +  angle ADC    =   180

66       +   angle ADC   =  180             subtract  66 from both sides   Mar 22, 2020
#3
+2

(4)

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°

So

Angle ACD  = (1/2)  minor arc AC  =  (1/2)(114°)  = 57°   Mar 22, 2020
#4
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(5)

Central angle BOC  =  360°  - 106° - 124°  =  130°

Since  angle BAC   is an inscribed angle  intercepting the same arc as central angle BOC, it has (1/2) the measure of angle BOC  =   (1/2) (130°)  =  65°   Mar 22, 2020