Find the measure of each angle

Question

In triangle ABC the measure of angle B is 6 degrees more than three times the measure of angle A. The measure of angle C is 39 degrees more than the measure of angle A. Find the measure of each angle.

Answer 1

High, Angel

 

It is tricky getting all these "round-about" clues about the measures of the angles of this mysterious triangle!  

 

I tell my students the cavemen were really busy the day they sat around in the cave and started drawing math problems on the walls instead of their usual cave art.

 

While the explanation looks hideously long, it is rather straightforward.....I just use a LOT of words to break it down to the "nuts and bolts," so I don't lose or confuse you. (Please let me know how I did!!!)

 

Sure hope this comes across all lined up.  :)

 

 

 

The text give us all the clues we need to solve this.

 

1.  The measure of angle A will just be called A (until we figure the rest out)

 

2.  For angle B, we are told it is 6 more than 3 times A, so we say it is   6 + 3A

 

3.  For angle C, it is 39 more than A, so we say it is 39 + A

 

4.  You also have to remember this VITALl fact about triangles:  The measures of all 3 angles add up to 180 degrees. 

 

Soooo.... HERE IS OUR MATH PROBLEM:  A + B + C = 180

 

Follow these steps, and you will have beaten the cavemen at their own game!

 

1.  Let's substitute all this gobbledygook for the A, B,  and C into the equation.

 

 

                                                                    A    +     B       +      C      = 180

                                                                     |           |                 |

 NOW OUR MATH PROBLEM LOOKS LIKE THIS:  A  +  (6 + 3A) +  (39 + A) = 180.

 

 

2.  Now combine like terms, which just means add all the A's together:

 

A + 39 + 3A + 6 + A

|              |             |

A        + 3A         + A = 5A (since each A that stands alone is understood to be 1A)

 

3.  Next, add all the real (non-gobbledygook) numbers together:

A + 39 + 3A + 6 + A
       |              |

       39         + 6 = 45

 

OUR PROBLEM NOW LOOKS LIKE THIS:  5A + 45 = 180

 

OUR GOAL is to get:   A = (answer)

 

You will do what I call "opposite on both sides" to achieve that goal. 

 

Keep ONE thing in mind:

 

[Our equation MUST remain balanced (like a seesaw), so you must do exactly the same on both sides of the equal sign.]  

 

 

1.  It's too early to work with the gobbledygook number (5A), so let's get rid  of the real number first.  To make the "+45" magically disappear, we need to turn it into a zero--since it is being added--by doing the opposite of addition:  subtraction

 

 

Thus:

 

                                                                     5A + 45 = 180

                                                                      |     -45    -45

                                                                      |       0     135

                                                                      |                |

NOW OUR MATH PROBLEM LOOKS LIKE THIS:   5A      =     135

 

2.  Now, let's get rid of the 5 in front of the A.  We know that we added all the A's up to get 5, and that 5A really means 5 X A--multiplication--so we'll do the opposite of multiplication (division) on both sides of the = to get rid of the 5 and come up with an answer.  [Notice division looks like we turn things into fractions--just works easier that way in Algebra.]

 

 

 

                                                                5A = 135     [The A has to stay, so we only divide by 5]

                                                                5        5

 

NOW WE HAVE THE MEASURE OF ANGLE A:    27 degrees

 

....But we are not done yet.  We need the measure of all 3 angles.

 

 

To get B and C, we simply go back to our original clues:

 

B    6 + 3A  = B

      6 + 3(27) = B

      6 + 81 = 87

 

So B is 87 degrees

 

 

C    39 + A = C

      39 + 27 = 66

 

So C is 66 degrees

 

 

 

Just to be sure, we didn't goof, let's add up all our angles to make sure they make 180 degrees!

 

27 + 87 + 66 = 180 degrees!!  So yep, we did it correctly.

 

 

Math can be fun once you know the trick!!!

 

 

Keep smiling,

 

Miss Peggy (Margaret S)

 

 

 

 

 

Answer 2

hi Angel--

 

remember that all 3 angles in ANY triangle ALWAYS add up to 180 degrees.

 

so you have angle A, 

 

then you have angle B, but let's call it "angle 3a + 6,"

 

 

and angle C can be called "angle A + 39."

 

so you have A + 3A + 6 + A + 39 = 180, otherwise known as:

 

5A + 45 = 180.

 

Combine terms to get 5A = 135, then divide by 5, which gives you:

 

A = 27. You should be able to figure out the other 2 from there!

 

Matt