Question

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Find m∠1 and m∠3 in the kite. The diagram is not drawn to scale.

Answer 1

m∠2 = 90°

m∠3 = 51°

Explanation:

The given figure represents a kite

Definition of kite

A kite is a quadrilateral that has 2 pairs of equal-length sides and these sides are adjacent to each other.

Properties of a kite:

1. The two angles are equal where the unequal sides meet.
2. It can be viewed as a pair of congruent triangles with a common base.
3. It has 2 diagonals that intersect each other at right angles.
4. The longer or main diagonal bisects the other diagonal.
5. A kite is symmetrical about its main diagonal.
6. The shorter diagonal divides the kite into 2 isosceles triangles.

Let O be the intersection of the two diagonals

Then we have two similar triangles
ΔAOD and ΔAOB

m∠DAO is given as 39°

Using property 3, angle AOB = 90°. This is m∠2.
So m∠2 = 90°

Using property 2, ΔAOD is congruent to ΔAOB

Therefore m ∠AOD is also 90°

In triangle AOD, the three angles are 39°, 90° and ∠3

They must add up to 180°

==> 39 + 90 + m∠3 = 180°

==> 129 + m∠3 = 180°

Subtract 129 both sides:
==> 129 - 129 + m∠3 = 180 - 129
==> m∠3 = 51°