Question

In a rhombus ABCD, \[AC = AB\]. State the measure of \[\angle D\] and \[\angle BAD\].

Answer 1

Hint:
Here, we will find the measure of the angles. We will use the properties of the rhombus to find the measure of the sides. Then we will take into consideration the two triangles formed by the diagonal. Using the properties of triangles we will find the measure of the \[\angle D\]. We will then use the property of parallelogram to find the measure of \[\angle BAD\]. Rhombus is a four sided quadrilateral which is also a parallelogram.

Complete Step by step Solution:
We are given a rhombus ABCD.

We are given that \[AC = AB\].
We know that all the sides of a rhombus are equal. Therefore,
\[AB = BC = CD = DA\]
Since all the sides are equal, then the diagonal of a rhombus is also equal.
So, we get
\[AB = BC = CD = DA = AC\] where AC is the diagonal of the rhombus.
We have two triangles on the rhombus with the diagonal as base common to both the triangles.
Since all the sides of a rhombus are equal, then the two triangles so formed also have equal sides.
So, the two triangles are an equilateral triangle.
So, \[\Delta ACD\] and \[\Delta ABC\] are the two triangles.
Now, in \[\Delta ACD\], we get
\[AC = CD = DA\]
As all sides are equal so the triangle is an equilateral triangle.
We know that the sum of angles of a triangle is \[180^\circ \].
So, each angle of an equilateral triangle is \[60^\circ \].
So, we get \[\angle D = 60^\circ \]
We know that rhombus is also a parallelogram.
We also know that the sum of adjacent sides of a parallelogram is \[180^\circ \].
\[\angle D + \angle A = 180^\circ \]
Substituting \[\angle D = 60^\circ \] in the above equation, we get
\[ \Rightarrow 60^\circ + \angle A = 180^\circ \]
Subtracting \[60^\circ \] from both the sides, we get
\[ \Rightarrow \angle A = 180^\circ - 60^\circ \]
\[ \Rightarrow \angle A = 120^\circ \]
\[ \Rightarrow \angle BAD = 120^\circ \]

Therefore, the measure of \[\angle D = 60^\circ \] and \[\angle BAD = 120^\circ \].

Note:
Here, we need to keep in mind the properties of a rhombus. All the sides of a rhombus are equal, so the rhombus is a type of square. The opposite sides of a rhombus are parallel so the rhombus is a type of parallelogram. Also, the opposite angles of a rhombus are equal. Since we have two equilateral triangles and we know that the angles of an equilateral triangle is \[60^\circ \], so we get
\[\angle BAC + \angle CAD = 60^\circ + 60^\circ \]
\[ \Rightarrow \angle BAD = 120^\circ \]