Question

an arrowhead and a parallelogram are joined together. an equilateral triangle is formed with one of the parallelograms diagonals, as shown below.

calculate the size of angle (c).

Explanation:

⚡ Using what you learned: properties of triangles · properties of quadrilaterals

Step 1: Analyze the equilateral triangle

The dashed line represents a diagonal of the parallelogram. We are told that an equilateral triangle is formed with this diagonal.

Looking at the tick marks on the joined figure:

• The two outer sides of the arrowhead have single tick marks.
• The diagonal of the parallelogram also has a single tick mark.
• Since they form an equilateral triangle, all three sides of this triangle are equal in length, and each of its interior angles is \( 60^\circ \).

Step 2: Find the angles of the parallelogram

The bottom-left interior angle of the parallelogram is given as \( 74^\circ \).

• Opposite angles in a parallelogram are equal, so the top-right interior angle is also \( 74^\circ \).
• The diagonal splits this top-right corner into two parts. One part is given as \( 41^\circ \).
• Therefore, the other part of this angle (inside the left triangle of the parallelogram) is:

$$ 74^\circ - 41^\circ = 33^\circ $$

Step 3: Determine the angle of the equilateral triangle

Since the triangle formed by the two outer sides of the arrowhead and the diagonal is equilateral, its interior angle at the top-right vertex is:

$$ 60^\circ $$

Step 4: Calculate angle c

The top-right vertex of the combined shape is made up of three adjacent angles that share this vertex:

1. The angle inside the equilateral triangle: \( 60^\circ \)
2. The angle \( c \) from the arrowhead
3. The angle inside the parallelogram: \( 33^\circ \)

From the first diagram, we can see that the top-right angle of the equilateral triangle is composed of the angle \( c \) plus the upper angle of the split parallelogram corner:

$$ c + 33^\circ = 60^\circ $$

Solving for \( c \):

$$ c = 60^\circ - 33^\circ = 27^\circ $$

Answer:

\( 27^\circ \) (or \( 27 \))