Exterior angles of a triangle - Examples, Exercises and Solutions

Understanding Exterior angles of a triangle

Complete explanation with examples

Exterior angles of a triangle

The exterior angle of a triangle is the one that is found between the original side and the extension of the side.

Key Properties:

  • The exterior angle is equal to the sum of the two interior angles of the triangle that are not adjacent to it.
  • Each exterior angle is supplementary to its adjacent interior angle, meaning their sum is 180180^\circ.
  • The sum of all exterior angles of a triangle is always 360360^\circ, no matter the shape of the triangle.

It is defined as follows:

α=A+Bα=∢A+∢B

A1 - Exterior angle of a triangle

Detailed explanation

Practice Exterior angles of a triangle

Test your knowledge with 63 quizzes

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

Examples with solutions for Exterior angles of a triangle

Step-by-step solutions included
Exercise #1

Look at the triangle ABC below.

AD=12AB AD=\frac{1}{2}AB

BE=12EC BE=\frac{1}{2}EC

What is the median in the triangle?

AAABBBCCCEEEDDD

Step-by-Step Solution

A median in a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Here, we need to find such a segment in triangle ABC \triangle ABC .

Let's analyze the given conditions:

  • AD=12AB AD = \frac{1}{2}AB : Point D D is the midpoint of AB AB .
  • BE=12EC BE = \frac{1}{2}EC : Point E E is the midpoint of EC EC .

Given that D D is the midpoint of AB AB , if we consider the line segment DC DC , it starts from vertex D D and ends at C C , passing through the midpoint of AB AB (which is D D ), fulfilling the condition for a median.

Therefore, the line segment DC DC is the median from vertex A A to side BC BC .

In summary, the correct answer is the segment DC DC .

Answer:

DC

Exercise #2

Look at triangle ABC below.

What is the median of the triangle and to which side is it drawn?

AAABBBCCCDDDEEE

Step-by-Step Solution

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In triangle ABC \triangle ABC , we need to identify such a median from the diagram provided.

Step 1: Observe the diagram to identify the midpoint of each side.

Step 2: It is given that point E E is located on side AC AC . If E E is the midpoint of AC AC , then any line from a vertex to point E E would be a median.

Step 3: Check line segment BE BE . This line runs from vertex B B to point E E .

Step 4: Since E E is labeled as the midpoint of AC AC , line BE BE is the median of ABC \triangle ABC drawn to side AC AC .

Therefore, the median of the triangle is BE BE for AC AC .

Answer:

BE for AC

Exercise #3

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCEEEDDD

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.

Answer:

AE

Video Solution
Exercise #4

In an isosceles triangle, the angle between ? and ? is the "base angle".

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer:

Side, base.

Exercise #5

Indicates which angle is greater

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Answer:

Video Solution

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