Calculate Angle FCD: Complex Geometry with 90°, 55°, and 95° Angles

Q: How do I know which triangle to use to find the missing angle?

Look for a triangle that contains the missing angle as one of its vertices. Triangle EDC contains angle ECD, and we know two of its angles (70° and 90°), so we can find the third!

Q: What if I get confused about which angles to add?

Draw or trace the angle you're looking for on the diagram. Angle FCD goes from ray CF to ray CD, passing through point E. This shows you need FCE + ECD.

Q: Why does triangle EDC have a 90° angle?

The problem states that angle D equals 90°. In triangle EDC, angle D is at vertex D, which is the right angle (90°) mentioned in the given information.

Q: Can I solve this problem without using the triangle angle sum?

Not easily! The 180° triangle rule is essential here because angle ECD isn't directly given. You need this rule to find ECD = 20°, which you then add to FCE = 30° to get FCD = 50°.

Angle Addition with Triangle Properties

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

Angle BCE is equal to 55 degrees

Angle DEB is equal to 95 degrees

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find angle FCD

00:04 The angle is angle C between points F and D

00:16 The full angle (FCD) equals the sum of its parts

00:38 Let's substitute the known angle value

00:54 The sum of angles in a triangle equals 180

01:04 Let's substitute the angle values and solve to find angle ECD

01:32 Let's isolate angle ECD

01:46 This is the size of angle ECD

02:00 Let's substitute this value in our equation and solve to find angle FCD

02:04 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

Angle BCE is equal to 55 degrees

Angle DEB is equal to 95 degrees

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

Step-by-step solution

Let's break down angle FCD for an angle addition exercise:

$FCD=FCE+ECD$

Let's write down the known information from the question:

$FCD=30+ECD$

Since angle ECD is not given to us, we will calculate it in the following way:

Let's look at triangle EDC, where we have 2 angles.

Since we know that the sum of angles in a triangle equals 180 degrees, let's write down the data in the formula:

$ECD+CED+EDC=180$

$ECD+70+90=180$

Let's move terms and keep the appropriate sign:

$ECD=180-90-70$

$ECD=20$

Now we can substitute ECD in the formula we wrote earlier:

$FCD=30+ECD$

$FCD=30+20$

$FCD=50$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Triangle Rule: Sum of angles in any triangle equals 180 degrees
Technique: Find missing angle: $ECD = 180 - 70 - 90 = 20°$
Check: Verify angle addition: $FCD = FCE + ECD = 30 + 20 = 50°$ ✓

Common Mistakes

Avoid these frequent errors

Adding angles without finding missing parts first
Don't just add the given angles FCE = 30° and assume that's the answer! This ignores the ECD portion completely. The diagram shows angle FCD spans both FCE and ECD sections. Always identify all component angles before adding.

Practice Quiz

Test your knowledge with interactive questions

Indicates which angle is greater

FAQ

Everything you need to know about this question

Why can't I just use the 30° angle shown in the diagram?

The 30° angle is only part of angle FCD! Looking at the diagram, angle FCD includes both FCE (30°) and ECD. You need to find the missing ECD angle first, then add them together.

How do I know which triangle to use to find the missing angle?

Look for a triangle that contains the missing angle as one of its vertices. Triangle EDC contains angle ECD, and we know two of its angles (70° and 90°), so we can find the third!

What if I get confused about which angles to add?

Draw or trace the angle you're looking for on the diagram. Angle FCD goes from ray CF to ray CD, passing through point E. This shows you need FCE + ECD.

Why does triangle EDC have a 90° angle?

The problem states that angle D equals 90°. In triangle EDC, angle D is at vertex D, which is the right angle (90°) mentioned in the given information.

Can I solve this problem without using the triangle angle sum?

Not easily! The 180° triangle rule is essential here because angle ECD isn't directly given. You need this rule to find ECD = 20°, which you then add to FCE = 30° to get FCD = 50°.

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