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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
avatar+1116 
+2

#1 Answer: 55 degrees

 

#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 Answer: 9 (square units)

 

#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
avatar+111330 
+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

 

                       angle ADC   =  114°

 

 

cool cool cool

 Mar 22, 2020
 #3
avatar+111330 
+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°

 

 

 

cool cool cool

 Mar 22, 2020
 #4
avatar+111330 
+1

(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°

 

 

cool cool cool

 Mar 22, 2020

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