Question

a kite and an isosceles triangle are formed from five straight lines.

work out the size of the angle marked (u).

Explanation:

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

Step 1: Find the missing angles of the kite

A kite is a quadrilateral with two pairs of equal adjacent sides. The tick marks show:

• The two upper sides of the kite are equal (marked with double ticks).
• The two lower sides of the kite are equal (marked with triple ticks).

In any kite, the angles between the non-equal sides are equal. This means the left angle and the right angle of the kite are equal:

$$ \text{Left angle} = \text{Right angle} = 86^\circ $$

The sum of the interior angles of any quadrilateral is \( 360^\circ \). Let the top angle of the kite be \( x \):

$$ x + 86^\circ + 86^\circ + 122^\circ = 360^\circ $$

$$ x + 294^\circ = 360^\circ $$

$$ x = 360^\circ - 294^\circ = 66^\circ $$

Step 2: Relate the kite's top angle to the triangle

The top vertex of the kite and the bottom vertex of the small triangle meet at an intersection of two straight lines, forming vertically opposite angles.

Vertically opposite angles are equal, so the bottom angle of the small triangle is also:

$$ 66^\circ $$

Step 3: Use the properties of the isosceles triangle to find \( u \)

The small triangle at the top is isosceles, as indicated by the single tick marks on its two legs. This means the two base angles at the top are equal.

The angle \( u \) is one of these equal base angles. Let the other base angle also be \( u \).

The sum of angles in a triangle is \( 180^\circ \):

$$ u + u + 66^\circ = 180^\circ $$

$$ 2u + 66^\circ = 180^\circ $$

$$ 2u = 114^\circ $$

$$ u = 57^\circ $$

Answer:

\( 57^\circ \)