## Set the Height of a Triangle

The height of a triangle is the segment that connects a vertex to the opposite side such that it creates a 90-degree angle.

In every triangle, there are three heights, as there are three vertices from which the height can be calculated relative to the side that is opposite to each of them.

The height can be found either inside or outside of the triangle. If it does not run through the interior of the triangle, it is called an external height.

Below, we provide you with some examples of triangle heights:

## Examples with solutions for Triangle Height

### Exercise #1

Can a triangle have two right angles?

### Step-by-Step Solution

The sum of angles in a triangle is 180 degrees. Since two angles of 90 degrees equal 180, a triangle can never have two right angles.

No

### Exercise #2

ABC is an isosceles triangle.

What is the size of angle $∢\text{ADC}$?

### Step-by-Step Solution

In an isosceles triangle, the median to the base is also the height to the base.

That is, side AD forms a 90° angle with side BC.

That is, two right triangles are created.

Therefore, angle ADC is equal to 90 degrees.

90

### Exercise #3

Which of the following is the height in triangle ABC?

### Step-by-Step Solution

Let's remember the definition of height of a triangle:

A height is a straight line that descends from the vertex of a triangle and forms a 90-degree angle with the opposite side.

The sides that form a 90-degree angle are sides AB and BC. Therefore, the height is AB.

AB

### Exercise #4

Given the following triangle:

Write down the height of the triangle ABC.

### Step-by-Step Solution

An altitude in a triangle is the segment that connects the vertex and the opposite side, in such a way that the segment forms a 90-degree angle with the side.

If we look at the image it is clear that the above theorem is true for the line AE. AE not only connects the A vertex with the opposite side. It also crosses BC forming a 90-degree angle. Undoubtedly making AE the altitude.

AE

### Exercise #5

Tree angles have the sizes:

50°, 41°, and 81.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We'll add the three angles to see if their sum equals 180:

$50+41+81=172$

Therefore, these cannot be the values of angles in any triangle.

Impossible.

### Exercise #6

Tree angles have the sizes 56°, 89°, and 17°.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's calculate the sum of the angles to see what total we get in this triangle:

$56+89+17=162$

The sum of angles in a triangle is 180 degrees, so this sum is not possible.

Impossible.

### Exercise #7

Tree angles have the sizes:

31°, 122°, and 85.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We'll add the three angles to see if their sum equals 180:

$31+122+85=238$

Therefore, these cannot be the values of angles in any triangle.

Impossible.

### Exercise #8

Three angles measure as follows: 60°, 50°, and 70°.

Is it possible that these are angles in a triangle?

### Step-by-Step Solution

Recall that the sum of angles in a triangle equals 180 degrees.

Let's add the three angles to see if their sum equals 180:

$60+50+70=180$

Therefore, it is possible that these are the values of angles in some triangle.

Possible.

### Exercise #9

Tree angles have the sizes 94°, 36.5°, and 49.5. Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We'll add the three angles to see if their sum equals 180:

$94+36.5+49.5=180$

Therefore, these could be the values of angles in some triangle.

Possible.

### Exercise #10

True or false?

$\alpha+\beta=180$

### Step-by-Step Solution

Given that the angles alpha and beta are on the same straight line and given that they are adjacent angles. Together they are equal to 180 degrees and the statement is true.

True

### Exercise #11

Tree angles have the sizes:

69°, 93°, and 81.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We'll add the three angles to see if their sum equals 180:

$69+81+93=243$

Therefore, these cannot be the values of angles in any triangle.

No.

### Exercise #12

Tree angles have the sizes:

90°, 60°, and 30.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We'll add the three angles to see if their sum equals 180:

$90+60+30=180$

Therefore, these could be the values of angles in some triangle.

No.

### Exercise #13

Tree angles have the sizes:

90°, 60°, and 40.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We'll add the three angles to see if their sum equals 180:

$90+60+40=190$

Therefore, these cannot be the values of angles in any triangle.

Yes.

### Exercise #14

Tree angles have the sizes:

76°, 52°, and 52°.

Is it possible that these angles are in a triangle?

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

We will add the three angles to find out if their sum equals 180:

$76+52+52=180$

Therefore, these could be the values of angles in some triangle.

Yes.

### Exercise #15

Find the measure of the angle $\alpha$

### Step-by-Step Solution

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Now let's input the known data:

$120+27+\alpha=180$

$147+\alpha=180$

We'll move the term to the other side and keep the appropriate sign:

$\alpha=180-147$

$\alpha=33$