Question

1. what is the measure of angle y?

• \\(30^\circ\\)
• \\(90^\circ\\)
• \\(45^\circ\\)
• \\(60^\circ\\)

Explanation:

⚡ Using what you learned: special right triangles (45-45-90 and 30-60-90)

Step 1: Analyze triangle VYZ

In the right-angled triangle \(\triangle VYZ\) (where \(\angle V = 90^\circ\)):

$$ \angle Z = 60^\circ $$

$$ \angle YVZ = 90^\circ $$

$$ \angle VZY = 60^\circ \implies \angle ZVY = 90^\circ - 60^\circ = 30^\circ $$

Using the properties of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with the side adjacent to \(60^\circ\) being \(VZ = 15\text{ cm}\):

$$ VZ = 15\text{ cm} $$

$$ YZ = 2 \times VZ = 30\text{ cm} $$

$$ VY = VZ \sqrt{3} = 15\sqrt{3}\text{ cm} $$

Step 2: Analyze triangle ABV

Line segment \(AB\) is perpendicular to \(XY\), making \(\angle VAB = 90^\circ\). Since \(AB \parallel VZ\), \(ABVY\) forms a rectangle, meaning:

$$ AB = VZ = 15\text{ cm} $$

In the right-angled triangle \(\triangle VAB\) (where \(\angle A = 90^\circ\)):

$$ \angle ABV = 45^\circ $$

$$ \angle AVB = 90^\circ - 45^\circ = 45^\circ $$

Since \(\triangle VAB\) is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle:

$$ AV = AB = 15\text{ cm} $$

$$ VB = AB\sqrt{2} = 15\sqrt{2}\text{ cm} $$

Step 3: Find the length of XY

Since \(ABVY\) is a rectangle:

$$ VY = AB + \text{height} \implies AV = 15\text{ cm} $$

$$ XY = XA + AV $$

In the right-angled triangle \(\triangle XAB\) (where \(\angle XAB = 90^\circ\)):

$$ \angle AXB = y $$

$$ \tan(y) = \frac{AB}{XA} = \frac{15}{XA} $$

Since \(X, B, Z\) lie on a straight line, we can look at the large right triangle \(\triangle XYZ\) (where \(\angle XZY = 60^\circ\) is not directly a right angle, but \(\triangle XYZ\) has \(\angle XYZ = 90^\circ\)):

$$ \tan(60^\circ) = \frac{XY}{YZ} $$

$$ \sqrt{3} = \frac{XY}{15} \implies XY = 15\sqrt{3}\text{ cm} $$

Since \(XY = XA + AV\):

$$ 15\sqrt{3} = XA + 15 \implies XA = 15\sqrt{3} - 15 = 15(\sqrt{3} - 15 \text{ is not needed directly if we use the whole triangle } \triangle XYZ) $$

Let's use the large right triangle \(\triangle XYZ\) directly:

$$ \angle XYZ = 90^\circ $$

$$ \angle XZY = 60^\circ $$

$$ \angle YXZ = 90^\circ - 60^\circ = 30^\circ $$

Since angle \(y\) is \(\angle YXZ\):

$$ y = 30^\circ $$

Answer:

30°