Solve for X: Finding Angles in an Isosceles Trapezoid with 120° Given

Q: What makes a trapezoid isosceles?

An isosceles trapezoid has equal leg lengths (non-parallel sides). This creates a special property: base angles are equal. So angles A and B are equal, and angles C and D are equal.

Q: Why do opposite angles equal each other?

In an isosceles trapezoid, the base angles (angles on the same parallel side) are equal due to symmetry. Since AB is parallel to CD, angles A = B and angles C = D.

Q: Can I use 180° instead of 360° for the angle sum?

No! $ 180° $ is for triangles only. Quadrilaterals (4-sided shapes) always have interior angles that sum to $ 360° $.

Q: How do I know which angles are equal?

Look at the diagram! In trapezoid ABCD, angles A and B are both on the top base (equal), while angles C and D are both on the bottom base (equal).

Q: What if I get a negative value for x?

Check your setup! Angles in geometry are positive. If you get negative x, you likely made an error in identifying which angles are equal or setting up the equation.

Trapezoid Angle Properties with Isosceles Constraints

Given: $∢C=2x$

$∢A=120°$

isosceles trapezoid.

Find x.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 An isosceles trapezoid according to the given data

00:09 In a trapezoid, angles on the same leg sum to 180

00:20 Substitute the value of A to find X

00:31 Isolate angle X

00:46 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

Given: $∢C=2x$

$∢A=120°$

isosceles trapezoid.

Find x.

Step-by-step solution

Given that the trapezoid is isosceles and the angles on both sides are equal, it can be argued that:

$∢C=∢D$

$∢A=∢B$

We know that the sum of the angles of a quadrilateral is 360 degrees.

Therefore we can create the formula:

$∢A+∢B+∢C+∢D=360$

We replace according to the existing data:

$120+120+2x+2x=360$

$240+4x=360$

$4x=360-240$

$4x=120$

We divide the two sections by 4:

$\frac{4x}{4}=\frac{120}{4}$

$x=30$

Final Answer

30°

Key Points to Remember

Essential concepts to master this topic

Isosceles Property: Opposite angles are equal in isosceles trapezoids
Angle Sum: All four angles total $360°$ in any quadrilateral
Check: Verify $120 + 120 + 60 + 60 = 360°$ ✓

Common Mistakes

Avoid these frequent errors

Treating all angles as equal instead of opposite pairs
Don't assume all four angles equal 90° like in a rectangle = wrong equation setup! Isosceles trapezoids only have opposite angle pairs equal, not all four angles. Always identify which angles are equal: base angles with base angles.

Practice Quiz

Test your knowledge with interactive questions

True OR False:

In all isosceles trapezoids the base Angles are equal.

FAQ

Everything you need to know about this question

What makes a trapezoid isosceles?

An isosceles trapezoid has equal leg lengths (non-parallel sides). This creates a special property: base angles are equal. So angles A and B are equal, and angles C and D are equal.

Why do opposite angles equal each other?

In an isosceles trapezoid, the base angles (angles on the same parallel side) are equal due to symmetry. Since AB is parallel to CD, angles A = B and angles C = D.

Can I use 180° instead of 360° for the angle sum?

No! $180°$ is for triangles only. Quadrilaterals (4-sided shapes) always have interior angles that sum to $360°$ .

How do I know which angles are equal?

Look at the diagram! In trapezoid ABCD, angles A and B are both on the top base (equal), while angles C and D are both on the bottom base (equal).

What if I get a negative value for x?

Check your setup! Angles in geometry are positive. If you get negative x, you likely made an error in identifying which angles are equal or setting up the equation.

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