**Step 1: Use the property of cyclic quadrilaterals.**
Opposite angles in a cyclic quadrilateral add up to 180°.
Therefore, f + 63° = 180° and g + h = 180°.
**Step 2: Find the value of f.**
f = 180° - 63°
f = 117°
**Step 3: Use the ratio to find the value of g.**
We are given f : g = 2 : 3.
We know f = 117°. So, 2 parts of the ratio correspond to 117°.
Value of 1 part = 117° / 2 = 58.5°.
The ratio for g is 3 parts.
g = 3 × 58.5° = 175.5°.
**Step 4: Find the value of h.**
Using the cyclic quadrilateral property again: g + h = 180°.
175.5° + h = 180°
h = 180° - 175.5°
h = 4.5°
**Step 5: Write the ratio f : h and simplify.**
f : h = 117 : 4.5
To simplify a ratio with a decimal, we can multiply both sides by a number to make them integers. Multiply by 2:
(117 × 2) : (4.5 × 2)
234 : 9
Now, we simplify this ratio by dividing by the greatest common divisor. Both are divisible by 9.
234 / 9 = 26
9 / 9 = 1
So, the simplified ratio is 26 : 1.
**Answer: 26 : 1**
**1. Find angle f:**
A cyclic quadrilateral is a four-sided shape whose vertices all lie on the circumference of a circle. A property of cyclic quadrilaterals is that opposite angles sum to 180°.
Angle f is opposite the 63° angle.
So, f + 63° = 180°
f = 180° - 63° = 117°.
**2. Find angle g:**
We are given the ratio f : g = 2 : 3.
We know f = 117°, which corresponds to the 2 parts of the ratio.
2 parts = 117°
1 part = 117° / 2 = 58.5°
Angle g corresponds to 3 parts.
g = 3 × 58.5° = 175.5°.
**3. Find angle h:**
Angle g and angle h are opposite angles in the cyclic quadrilateral, so they also sum to 180°.
g + h = 180°
175.5° + h = 180°
h = 180° - 175.5° = 4.5°.
**4. Find the ratio f : h:**
We need to find the ratio f : h in its simplest form.
f : h = 117 : 4.5
To remove the decimal, we can multiply both sides of the ratio by 2.
(117 × 2) : (4.5 × 2)
234 : 9
Now we simplify by dividing both sides by their greatest common divisor. Let's try dividing by 9.
234 ÷ 9 = 26
9 ÷ 9 = 1
The simplified ratio is 26 : 1.
Expert