Sum and Difference of Angles - Examples, Exercises and Solutions

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

$30+60+90=180$

The sum of the angles equals 180, therefore they can form a triangle.

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

$56+89+17=162$

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

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

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

Remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since the lines are perpendicular to each other, the marked angles are right angles each equal to 90 degrees.

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

To determine if the statement that "the sum of the adjacent angles is 180" is true, follow these steps:

• Step 1: Define Adjacent Angles
  

  Adjacent angles are two angles that have a common vertex and a common side but do not overlap. In geometry, when these angles form a straight line, they are known as a linear pair.

• Step 2: Apply the Linear Pair Theorem
  

  The Linear Pair Theorem states that if two angles are adjacent and form a linear pair (i.e., the non-common sides form a straight line), then these angles are supplementary. This means that their sum is $180^\circ$.

• Step 3: Conclusion
  

  Therefore, when adjacent angles form a linear pair on a straight line, their sum is indeed $180^\circ$.

This validates the statement that "the sum of the adjacent angles is 180" for linear pairs, making the statement True.

This corresponds to the answer choice stating: True.

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

The unmarked angle is adjacent to an angle of 160 degrees.

Remember: the sum of adjacent angles is 180 degrees.

Therefore, the size of the unknown angle is:

$180-160=20$

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:

$180-50=130$

$130:2=65$

As the triangles are similar, DE is parallel to BC

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

B=D=65

Since in an equilateral triangle all sides are equal and all angles are equal. It is also known that in a triangle the sum of angles is 180°, we can calculate X in the following way:

$X+5+X+5+X+5=180$

$3X+15=180$

$3X=180-15$

$3X=165$

Let's divide both sides by 3:

$\frac{3X}{3}=\frac{165}{3}$

$X=55$

We can see from the diagram that angle DBC equals 100 degrees.

We can also see that the size of angle ABC is shown and equals 40 degrees.

Therefore, the answer is 40.

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.

Since this is an equilateral triangle, all angles are also equal.

As the sum of angles in a triangle is 180 degrees, each angle is equal to 60 degrees. (180:3=60)

From this, we can conclude that: $60=8x$

Let's divide both sides by 8:

$\frac{60}{8}=\frac{8x}{8}$

$7.5=x$

If we try to draw two obtuse angles and connect them to form a triangle (i.e: only 3 sides), we will see that it is not possible.

Therefore, the answer is no.

In order to calculate the value of angle ABC, we must calculate the sum of all the given angles.

That is:

$ABC=60+50$

$ABC=110$