Math / Geometry

QuestionSAT $/ A C T$ Given $A B=B D=B C=D C$ ano $m \angle A B C=114$ , what is $m \angle B A D$ ?

Studdy Solution

STEP 1

What is this asking? Find the measure of angle $\angle BAD$ in a quadrilateral where three sides are equal and one angle is given. Watch out! Don't forget that equal sides often mean equal angles, and the sum of angles in a triangle is always $180^\circ$ .

STEP 2

1. Understand the shape
2. Use angle properties
3. Calculate the desired angle

STEP 3

Alright, let's break this down!
We have a quadrilateral $ABCD$ with equal sides: $AB = BD = BC = DC$ .
This means it's a rhombus!
In a rhombus, opposite angles are equal, and adjacent angles are supplementary (they add up to $180^\circ$ ).

STEP 4

We know $\angle ABC = 114^\circ$ .
Since $ABCD$ is a rhombus, the opposite angle $\angle ADC$ is also $114^\circ$ .

STEP 5

Now, let's use the property that adjacent angles in a rhombus are supplementary.
So, $\angle BCD + \angle ABC = 180^\circ$ .
Therefore, $\angle BCD = 180^\circ - 114^\circ = 66^\circ$ .

STEP 6

Now, let's focus on triangle $ABD$ .
Since $AB = BD$ , triangle $ABD$ is isosceles.
This means $\angle BAD = \angle BDA$ .

STEP 7

The sum of angles in triangle $ABD$ is $180^\circ$ .
So, we have: $\angle BAD + \angle BDA + \angle ABD = 180^\circ$

STEP 8

Since $\angle ABD = \angle ABC = 114^\circ$ , we can plug this into our equation: $\angle BAD + \angle BDA + 114^\circ = 180^\circ$

STEP 9

Because $\angle BAD = \angle BDA$ , we can write: $2 \cdot \angle BAD + 114^\circ = 180^\circ$

STEP 10

Subtract $114^\circ$ from both sides: $2 \cdot \angle BAD = 66^\circ$

STEP 11

Divide both sides by 2 to find $\angle BAD$ : $\angle BAD = 33^\circ$

STEP 12

The measure of $\angle BAD$ is $\mathbf{33^\circ}$ .

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