Finding Missing Angles

Finding missing angles typically involves applying one or more of the following angle relationships

Table of Contents:

• Finding Missing Angles
• Solved Examples
• FAQs

Finding missing angles typically involves applying one or more of the following angle relationships:

Vertical Angles: When two lines intersect, the angles opposite each other are called vertical angles and are congruent.

Complementary Angles: Two angles are complementary if their sum is 90 degrees. This means that one angle can be found by subtracting the other from 90 degrees.

Supplementary Angles: Two angles are supplementary if their sum is 180 degrees. This means that one angle can be found by subtracting the other from 180 degrees.

Exterior Angles: An exterior angle of a triangle is formed by extending one of its sides. The measure of an exterior angle is equal to the sum of the two remote interior angles.

Here are some steps to follow when finding missing angles:

Identify the angle relationships that apply to the given angles.

Use the given information to set up an equation or system of equations that relates the missing angle(s) to the known angles.

Solve the equation(s) to find the value(s) of the missing angle(s).

Check your answer(s) by verifying that the sum of the known angles and the missing angles is equal to the total angle measure of the figure.

Note that finding missing angles can be more complex in some cases, such as when dealing with non-standard shapes or when multiple angle relationships apply. In those cases, it may be helpful to draw a diagram and use properties of polygons or trigonometric functions to solve the problem.

Finding missing angles in Triangles 

To find missing angles in triangles, we can use a variety of angle relationships, including:

The sum of angles in a triangle is 180 degrees.

Exterior angles of a triangle are equal to the sum of the two opposite interior angles.

The angles opposite two congruent sides of a triangle are congruent.

The larger angle of a triangle is opposite the longer side, and the smaller angle is opposite the shorter side.

Here are some steps to follow when finding missing angles in triangles:

Identify the angle relationships that apply to the given angles and the missing angles.

Use the given information to set up an equation or system of equations that relates the missing angle(s) to the known angles.

Solve the equation(s) to find the value(s) of the missing angle(s).

Check your answer(s) by verifying that the sum of the known angles and the missing angles is equal to 180 degrees.

Note that in some cases, finding the missing angle(s) may require multiple steps or involve applying multiple angle relationships. It can be helpful to draw a diagram and label the angles to make it easier to see which relationships apply.

Finding missing angles in Trapezoid 

To find missing angles in a trapezoid, we can use several angle relationships, including:

• The sum of angles in a quadrilateral is 360 degrees.
• Adjacent angles in a trapezoid are supplementary.
• One opposite side of a trapezium is congruent.

Here are some steps to follow when finding missing angles in a trapezoid:

Identify the angle relationships that apply to the given angles and the missing angles.

Use the given information to set up an equation or system of equations that relates the missing angle(s) to the known angles.

Solve the equation(s) to find the value(s) of the missing angle(s).

Check your answer(s) by verifying that the sum of the known angles and the missing angles is equal to 360 degrees.

Note that in some cases, finding the missing angle(s) may require multiple steps or involve applying multiple angle relationships. It can be helpful to draw a diagram and label the angles to make it easier to see which relationships apply. Additionally, it is important to note that the angles in a trapezoid can vary widely depending on the shape and size of the trapezoid.

Finding missing angles in Square 

To find missing angles in a square, we can use the following angle relationships:

All angles in a square are right angles (90 degrees).

Opposite angles in a square are congruent.

Adjacent angles in a square are supplementary (they add up to 180 degrees).

Here are some steps to follow when finding missing angles in a square:

Identify the angle relationships that apply to the given angles and the missing angles.

Use the given information to set up an equation or system of equations that relates the missing angle(s) to the known angles.

Solve the equation(s) to find the value(s) of the missing angle(s).

Check your answer(s) by verifying that the angles satisfy the angle relationships for a square.

Note that in a square, all angles are 90 degrees, so it is typically straightforward to find missing angles by subtracting the known angles from 90 degrees.

Finding missing angles in Vertical angles 

To find missing angles in vertical angles, we can use the following angle relationship:

Vertical angles are congruent (they have the same measure).

Here are some steps to follow when finding missing angles in vertical angles:

Identify the vertical angles in the given diagram.

Set up an equation that relates the two vertical angles, using the fact that they are congruent.

Solve the equation to find the value of the missing angle.

Check your answer by verifying that the vertical angles have the same measure.

Note that vertical angles are formed by the intersection of two lines or rays, so it is important to identify the correct angles that are vertical to each other. It can be helpful to draw a diagram and label the angles to make it easier to see which angles are vertical.

Finding missing angles in Exterior angles 

To find missing angles in exterior angles of a polygon, we can use the following angle relationships:

The sum of the measures of the exterior angles of a polygon is 360 degrees.

The measure of an exterior angle of a polygon is equal to the sum of the measures of the two remote interior angles.

Here are some steps to follow when finding missing angles in exterior angles:

Identify the exterior angle in the given diagram.

Identify the two remote interior angles that form the exterior angle.

Use the given information to set up an equation that relates the exterior angle to the two remote interior angles.

Solve the equation to find the value of the missing angle.

Check your answer by verifying that the exterior angle is equal to the sum of the measures of the two remote interior angles.

Note that the sum of the measures of the exterior angles of a polygon is always 360 degrees, regardless of the number of sides of the polygon. Also, the remote interior angles are the angles that are not adjacent to the exterior angle, but are inside the polygon and on the same side of the exterior angle

Solved Examples

Finding missing angles in triangles:

Given a triangle with angles measuring 50 degrees and 70 degrees, find the measure of the third angle.

Solution:

Since the sum of the measures of the angles in a triangle is 180 degrees, we can set up an equation:

50 + 70 + x = 180

Simplifying, we get:

x = 180 – 50 – 70

x = 60

Therefore, the measure of the missing angle is 60 degrees.

Finding missing angles in trapezoids:

Given a trapezoid with one angle measuring 100 degrees and the other three angles unknown, find the measures of the other three angles.In an isosceles trapezoid, ABCD is the size of the angle B = 123° Determine the size of the other three unknown angles.

Solution:

B=123 degrees 

A=B=123=123 (In Isosceles trapezium, The angles on either side of the bases are the same size/measure (congruent).

B+C=180  (In Isosceles trapezium, Adjacent angles (next to each other) along the sides are supplementary. This means that their measures add up to 180 degrees.)

C=180−B

C=180−123

C= 57 degrees

D=C (In Isosceles trapezium, The angles on either side of the bases are the same size/measure (congruent).

Angle D will be 57 degrees

The other three sides will be 

B= 123 degrees

C= 57 degrees

D = 57 degrees

Finding missing angles in vertical angles:

Given two vertical angles with measures 80 degrees and x degrees, find the measure of the missing angle.

Solution:

Since vertical angles are congruent, we can set up an equation:

80 = x

Therefore, the measure of the missing angle is 80 degrees.

Finding missing angles in exterior angles:

Given a regular hexagon with one exterior angle measuring 50 degrees, find the measure of one interior angle.

Solution:

Since the sum of the measures of the exterior angles of a polygon is 360 degrees, and a hexagon has six exterior angles, we can set up an equation:

6x = 360

x = 60

Therefore, each exterior angle measures 60 degrees. Since the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles, we can set up another equation:

50 = 2y

y = 25

Therefore, one of the interior angles measures 25 degrees.

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