9th Grade Geometry Homework

Question

Suppose <A and <B are complementary angles, and <C and <D are also complementary angles. If m<A = 2x+3, m<B = y-2, m<C = 2x-y and m<D = x-1, find the values of all four angles.

Answer 1

Complementary angles add up to 90 degrees.  We have 2 sets. of complementary angles.

 

Set 1:  angle A and angle B

Set 2: angle C and angle D

 

 

∠A + ∠B = 90

2x + 3 + y - 2 = 90

2x + y + 1 = 90

2x + y = 89

 

 

∠C + ∠D = 90

2x - y + x - 1 = 90

3x - y = 91

 

 

We have 2 equations to work with.  A system of equations.

 

2x + y = 89            eq1

3x - y = 91              eq2

 

 

If we add the equations, we can eliminate the y terms.

 

5x = 180

 x = 36

 

 

Now we substitute this value of x into eq1 to solve for y.

 

2(36) + y = 89

72 + y = 89

        y = 17

 

x = 36

y = 17

 

 

Substitute these x and y values back into the 4 original expressions to get your 4 angles.

 

 

 

 

Answer 2

m∠A + m∠B = 90

(2x + 3) + (y - 2) = 90

2x + y + 1 = 90

 

m∠C + m∠D = 90

(2x - y) + (x - 1) = 90

3x - y - 1 = 90

 

2x + y + 1 = 90

3x - y - 1 = 90              add bottom equation to top

 

5x = 180

x = 36

 

2(36) + y + 1 =90

72 + y + 1 =90           combine

73 + y = 90                 subtract 73 from both sides

y = 17

 

Now substitute the values for x and y into the original four measures.