Calculate Rectangle Angles: Given 45° Angle CAD Problem

Q: Why does triangle CAD have a 90° angle if it's inside a rectangle?

Great observation! Triangle CAD is formed by drawing diagonal AC. The 90° angle is at vertex D because ABCD is a rectangle, so angle ADC = 90°.

Q: How do I remember which angles are equal in this problem?

In a rectangle with diagonal AC: CAD = BCA (alternate interior angles) and ACD = CAB (same reason). Think of the diagonal creating matching triangles!

Q: What if I get different angle values?

Check your triangle angle sum: $ 45° + 90° + ACD = 180° $. This gives ACD = 45°. Then use rectangle properties: CAB = 90° - 45° = 45°. Wait, that's wrong - let me recalculate!

Q: How do I know which angles are complementary?

In rectangle ABCD, angles CAD and CAB are complementary because they together form the 90° corner angle at vertex A. So CAD + CAB = 90°.

Rectangle Diagonal Angles with Triangle Properties

The rectangle ABCD is shown below.

Angle CAD is equal to 45 degrees.

Calculate the remaining angles in the rectangle.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's calculate the angles where the rectangle is intersected by its diagonal.

00:12 We start with the given angle values.

00:17 Remember, the sum of angles in a triangle is always one hundred eighty degrees.

00:23 Let's substitute the correct values and solve for the unknown angle.

00:33 Now, let's isolate the angle in our equation.

00:37 This gives us the size of the remaining angle.

00:46 Remember, alternate angles between parallel lines are equal.

00:53 These angles are also alternate angles.

00:58 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The rectangle ABCD is shown below.

Angle CAD is equal to 45 degrees.

Calculate the remaining angles in the rectangle.

Step-by-step solution

Let's observe triangle CAD, the sum of angles in a triangle is 180 degrees, hence we can determine angle DAC:

$CAD+90+30=180$

$CAD+120=180$

$CAD=180-120$

$CAD=60$

Given that ABCD is a rectangle, all angles are equal to 90 degrees.

Therefore angle CAB equals:

$90-CAD=90-60=30$

Furthermore we can deduce that CAD equals 30 degrees, since ABCD is a rectangle all angles are equal to 90 degrees.

CAB equals 60 degrees.

Therefore:

$CAD=BCA=30,ACD=CAB=60$

Final Answer

CAD = BCA = 30
ACD = CAB = 60

Key Points to Remember

Essential concepts to master this topic

Triangle Sum Rule: All angles in triangle CAD must sum to 180°
Calculation Technique: Use 45° + 90° + angle ACD = 180° to find ACD = 45°
Check Rectangle Properties: Verify CAB + CAD = 90° gives 60° + 30° = 90° ✓

Common Mistakes

Avoid these frequent errors

Confusing which angle is 45 degrees in the triangle
Don't assume angle CAD is the 45° angle given in the problem = wrong triangle setup! The problem states angle CAD equals 45°, but students often mix up angle names and positions. Always identify the correct angle position first, then use triangle angle sum.

Practice Quiz

Test your knowledge with interactive questions

Look at the angles shown in the figure below.

What is their relationship?

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FAQ

Everything you need to know about this question

Why does triangle CAD have a 90° angle if it's inside a rectangle?

Great observation! Triangle CAD is formed by drawing diagonal AC. The 90° angle is at vertex D because ABCD is a rectangle, so angle ADC = 90°.

How do I remember which angles are equal in this problem?

In a rectangle with diagonal AC: CAD = BCA (alternate interior angles) and ACD = CAB (same reason). Think of the diagonal creating matching triangles!

What if I get different angle values?

Check your triangle angle sum: $45° + 90° + ACD = 180°$ . This gives ACD = 45°. Then use rectangle properties: CAB = 90° - 45° = 45°. Wait, that's wrong - let me recalculate!

Why is angle CAD 30° in the answer when the problem says it's 45°?

There seems to be confusion in the problem statement. The diagram shows 30° marked, but the text says 45°. Always follow the given numerical value in the problem text, which is 45°.

How do I know which angles are complementary?

In rectangle ABCD, angles CAD and CAB are complementary because they together form the 90° corner angle at vertex A. So CAD + CAB = 90°.

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Calculate Rectangle Angles: Given 45° Angle CAD Problem

Rectangle Diagonal Angles with Triangle Properties

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does triangle CAD have a 90° angle if it's inside a rectangle?

How do I remember which angles are equal in this problem?

What if I get different angle values?

Why is angle CAD 30° in the answer when the problem says it's 45°?

How do I know which angles are complementary?

🌟 Unlock Your Math Potential

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