In a right triangle, one of the two small angles is 10° less than 3… Step by Step

Step by step

The question

Question: In a right triangle, one of the two small angles is 10° less than 3 times the other.

Find the two small angles.

1 Find what we know

What do we know?

A right triangle has one angle of 90°.

One small angle is 10° less than 3 times the other.

What do we want?

The two small angles.

2 Name the unknown

We know the other angles from this one.

So we call one small angle x.

x = small angle

The other small angle is 10° less than 3 times the other: 3x − 10.

3x − 10 = other small angle

Let's draw it.

The angles are x, 3x − 10 and 90°.

The three angles of a triangle always add up to 180°.

small + small + 90° = 180°

3 Write the equation

In words: the two small angles and the right angle make 180.

x + (3x − 10) + 90 = 180

So the equation is x + (3x − 10) + 90 = 180.

Now we solve it.

x + (3x − 10) + 90 = 180

Careful!

The angles of a triangle make 180°, not 360°.

360° is all the way round a point.

Careful: not x + (3x − 10) + 90 = 360 ✗

4 Open the brackets

A bracket with a plus or nothing in front can just be removed.

x + 3x − 10 + 90 = 180

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

Plain numbers go together.

x + 3x = 4x.

−10 + 90 = 80.

x + 3x = 4x

−10 + 90 = 80

Now we have 4x + 80 = 180.

4x + 80 = 180

We want x by itself.

Two things are stuck to x: times 4 and + 80.

The + 80 is on the outside, so it comes off first.

Then the times 4.

6 Remove the + 80

To remove + 80, we take away 80.

We do it on both sides, so they stay equal.

4x ~~+ 80 − 80~~ = 180 − 80

+ 80 and − 80 make zero.

Cross them out.

180 − 80 = 100.

180 − 80 = 100

Now we have 4x = 100.

4x = 100

7 Remove the 4

4x means 4 times x.

We want just x.

To remove times 4, we divide by 4.

We do it on both sides, so they stay equal.

4x4 = 1004

4 divided by 4 is 1.

So just x is left.

100 divided by 4 is 25.

1004 = 25

So x = 25!

x = 25

8 Answer the question

The other small angle: 3 × 25 − 10 = 65.

3x − 10 = 3 × 25 − 10 = 65

So the angles are 25° and 65°.

9 Check

Let's check in the story.

All the angles: 25 + 65 + 90 = 180, like a triangle must.

25 + 65 + 90 = 180

And the other small angle is 10° less than 3 times the other, like the story says.

So the answer is right!

65 = 3 × 25 − 10

Let's look back.

First: find what we know.

Then: name the unknown.

Then: write the equation.

Then: solve it.

Then: answer the question.

Last: check.

Answer