Exterior angles of a triangle - Examples, Exercises and Solutions

The exterior angle of a triangle is the one that is found between the original side and the extension of the side.
The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.

It is defined as follows:

A1 - Exterior angle of a triangle

α=A+Bα=∢A+∢B

Practice Exterior angles of a triangle

examples with solutions for exterior angles of a triangle

Exercise #1

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCEEEDDD

Video Solution

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 drawing, we can notice that the previous theorem is true for the line AE that crosses BC and forms a 90-degree angle, comes out of vertex A and therefore is the altitude of the triangle.

Answer

AE

Exercise #2

ABC is an isosceles triangle.

AD is the median.

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

AAABBBCCCDDD

Video Solution

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.

Answer

90

Exercise #3

Which of the following is the height in triangle ABC?

AAABBBCCCDDD

Video Solution

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.

Answer

AB

Exercise #4

Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.

Can these angles make a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

56+89+17=162 56+89+17=162

The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

Exercise #5

Angle A equals 90°.
Angle B equals 115°.
Angle C equals 35°.

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they are equal to 180 degrees:

90+115+35=240 90+115+35=240
The sum of the given angles is not equal to 180, so they cannot form a triangle.

Answer

No.

examples with solutions for exterior angles of a triangle

Exercise #1

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

Video Solution

Step-by-Step Solution

We add the three angles to see if they equal 180 degrees:

30+60+90=180 30+60+90=180
The sum of the angles equals 180, so they can form a triangle.

Answer

Yes

Exercise #2

Triangle ADE is similar to isosceles triangle ABC.

Angle A is equal to 50°.

Calculate angle D.

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Triangle ABC is isosceles, therefore angle B is equal to angle C. We can calculate them since the sum of the angles of a triangle is 180:

18050=130 180-50=130

130:2=65 130:2=65

As the triangles are similar, DE is parallel to BC

Angles B and D are corresponding and, therefore, are equal.

B=D=65

Answer

65 65 °

Exercise #3

True or false?

α+β=180 \alpha+\beta=180

αβ

Video Solution

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.

Answer

True

Exercise #4

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

Is it possible that these are angles in a triangle?

Video Solution

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 60+50+70=180

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

Answer

Possible.

Exercise #5

Find the measure of the angle α \alpha

505050AAABBBCCC50

Video Solution

Step-by-Step Solution

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

Therefore, we will use the following formula:

A+B+C=180 A+B+C=180

Now let's insert the known data:

α+50+50=180 \alpha+50+50=180

α+100=180 \alpha+100=180

We will simplify the expression and keep the appropriate sign:

α=180100 \alpha=180-100

α=80 \alpha=80

Answer

80

examples with solutions for exterior angles of a triangle

Exercise #1

What kind of triangle is shown in the diagram below?

535353117117117212121AAABBBCCC

Video Solution

Step-by-Step Solution

We calculate the sum of the angles of the triangle:

117+53+21=191 117+53+21=191

It seems that the sum of the angles of the triangle is not equal to 180°,

Therefore, the figure can not be a triangle and the drawing is incorrect.

Answer

The triangle is incorrect.

Exercise #2

ABC is an isosceles triangle.

A=4x ∢A=4x

B=2x ∢B=2x

Calculate the value of x.

AAABBBCCC4x2x

Video Solution

Step-by-Step Solution

As we know that triangle ABC is isosceles.

B=C=2X B=C=2X

It is known that in a triangle the sum of the angles is 180.

Therefore, we can calculate in the following way:

2X+2X+4X=180 2X+2X+4X=180

4X+4X=180 4X+4X=180

8X=180 8X=180

We divide the two sections by 8:

8X8=1808 \frac{8X}{8}=\frac{180}{8}

X=22.5 X=22.5

Answer

22.5

Exercise #3

Triangle ABC isosceles.

AB = BC

Calculate angle ABC and indicate its type.

45°45°45°AAABBBCCC

Video Solution

Step-by-Step Solution

Given that it is an isosceles triangle:AB=BC AB=BC

It is possible to argue that:BAC=ACB=45 BAC=ACB=45

Since the sum of the angles of a triangle is 180, the angle ABC will be equal to:

1804545=90 180-45-45=90

Since the angle ABC measures 90 degrees, it is a right triangle.

Answer

90°, right angle.

Exercise #4

It is known that angles A and D are equal to 90 degrees

Angle DEB is equal to 95 degrees

Complete the value of angle GDC based on the data from the figure.

505050404040707070AAABBBCCCDDDEEEFFFGGG3025

Video Solution

Step-by-Step Solution

Note that the GDC angle is part of the EDC angle.

Therefore, we can write the following expression:

GDC+EDG=EDC GDC+EDG=EDC

Since we know that angle D equals 90 degrees, we will substitute the values in the formula:

GDC+40=90 GDC+40=90

We will simplify the expression and keep the appropriate sign:

GDC=9040 GDC=90-40

GDC=50 GDC=50

Answer

50

Exercise #5

ABC Right triangle

Since BD is the median

Given AC=10.

Find the length of the side BD.

AAABBBCCCDDD10

Video Solution

Step-by-Step Solution

We calculate BD according to the rule:

In a right triangle, the midpoint of the hypotenuse is equal to half of the hypotenuse.

That is:

BD is equal to half of AC:

Given that: AC=10 AC=10

BD=10:2=5 BD=10:2=5

Answer

5

Topics learned in later sections

  1. Area
  2. The Sum of the Interior Angles of a Triangle
  3. The sides or edges of a triangle
  4. Triangle Height
  5. Types of Triangles
  6. Obtuse Triangle
  7. Equilateral triangle
  8. Identification of an Isosceles Triangle
  9. Scalene triangle
  10. Acute triangle
  11. Isosceles triangle
  12. The Area of a Triangle
  13. Area of a right triangle
  14. Area of Isosceles Triangles
  15. Area of a Scalene Triangle
  16. Area of Equilateral Triangles
  17. Perimeter
  18. Triangle
  19. Perimeter of a triangle