Find Angle DAC in a Deltoid: Using 2x and 60° Relationships

Q: What makes a deltoid different from other quadrilaterals?

A deltoid has two pairs of adjacent equal sides. This creates special properties: one diagonal bisects angles at both ends, while the other diagonal bisects the deltoid into two congruent triangles.

Q: Why does AC bisect angle BAD?

In deltoid ABCD, since AB = AD and CB = CD, diagonal AC connects the vertices where equal sides meet. This symmetry makes AC an angle bisector at both A and C.

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

Not efficiently! The triangle angle sum is the key insight. Other approaches like using deltoid angle properties directly would require more complex relationships that aren't given in this problem.

Deltoid Properties with Angle Bisector Relationships

ABCD is a deltoid.

$∢DAC=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate angle DAC

00:05 The diagonal in a rhombus bisects the angle

00:19 The sum of angles in a triangle equals 180

00:26 Let's isolate X

00:43 This is the unknown X

00:48 Now let's substitute X and calculate angle DAC

00:54 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

ABCD is a deltoid.

$∢DAC=\text{?}$

Step-by-step solution

As we know that ABCD is a deltoid, and AC is the bisector of an angle and therefore:

$BAC=CAD=2X$

Now we focus on the triangle BAD and calculate the sum of the angles since we know that the sum of the angles in a triangle is 180 degrees:

$2X+2X+2X+60=180$

$6X+60=180$

$180-60=6X$

$120=6X$

We divide the two sections by 6: $\frac{120}{6}=\frac{6x}{6}$

$20=x$

Now we can calculate the angle DAC:

$20\times2=40$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Deltoid Property: AC bisects angle BAD making both angles equal
Triangle Sum: $2x + 2x + 2x + 60 = 180$ gives x = 20
Verification: Check that all deltoid angles sum to 360°: 40 + 40 + 60 + 220 = 360° ✓

Common Mistakes

Avoid these frequent errors

Forgetting that AC is an angle bisector in deltoids
Don't treat angles BAC and CAD as different unknowns = overcomplicated equations! Students miss that deltoids have special symmetry properties. Always remember that in a deltoid, the diagonal connecting vertices with equal sides bisects the angle at that vertex.

Practice Quiz

Test your knowledge with interactive questions

Identify the angle shown in the figure below?

FAQ

Everything you need to know about this question

What makes a deltoid different from other quadrilaterals?

A deltoid has two pairs of adjacent equal sides. This creates special properties: one diagonal bisects angles at both ends, while the other diagonal bisects the deltoid into two congruent triangles.

Why does AC bisect angle BAD?

In deltoid ABCD, since AB = AD and CB = CD, diagonal AC connects the vertices where equal sides meet. This symmetry makes AC an angle bisector at both A and C.

How do I set up the equation for triangle BAD?

Triangle BAD has angles: 2x (angle BAC), 2x (angle CAD), and 2x + 60° (angle ABD). Since triangle angles sum to 180°: $2x + 2x + (2x + 60) = 180$

Can I solve this without using the triangle sum?

Not efficiently! The triangle angle sum is the key insight. Other approaches like using deltoid angle properties directly would require more complex relationships that aren't given in this problem.

What if I got x = 30 instead of x = 20?

Check your equation setup! If you got x = 30, you might have forgotten that angle ABD = 2x + 60°, not just 60°. The correct equation is $6x + 60 = 180$ .

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